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Journal of Differential Equations
Journal Prestige (SJR): 2.525 Citation Impact (citeScore): 2 Number of Followers: 1 Subscription journal ISSN (Print) 00220396  ISSN (Online) 10902732 Published by Elsevier [3302 journals] 
 Markov shifts and topological entropy of families of homoclinic tangles
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Bráulio Garcia, Valentín Mendoza The existence of a homoclinic orbit in dynamical systems implies chaotic behaviour with positive entropy. In this work, we determine explicitly the Markov shifts associated to certain Smale horseshoe homoclinic orbits which allow us to compute a lower bound for the topological entropy that such a system can have. It is done associating a heteroclinic orbit which belongs to the same isotopy class and then determining the Markov partition of the dynamical core of an end periodic mapping.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Bráulio Garcia, Valentín Mendoza The existence of a homoclinic orbit in dynamical systems implies chaotic behaviour with positive entropy. In this work, we determine explicitly the Markov shifts associated to certain Smale horseshoe homoclinic orbits which allow us to compute a lower bound for the topological entropy that such a system can have. It is done associating a heteroclinic orbit which belongs to the same isotopy class and then determining the Markov partition of the dynamical core of an end periodic mapping.
 Steadystate solutions of a reaction–diffusion system arising from
intraguild predation and internal storage Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Hua Nie, SzeBi Hsu, FengBin Wang Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Hua Nie, SzeBi Hsu, FengBin Wang Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.
 A generalization of the Aubin–Lions–Simon compactness lemma for
problems on moving domains Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Boris Muha, Sunčica Čanić This work addresses an extension of the Aubin–Lions–Simon compactness result to generalized Bochner spaces L2(0,T;H(t)), where H(t) is a family of Hilbert spaces, parameterized by t. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin–Lions–Simon compactness result holds. Concrete examples of the application of the compactness theorem are presented, including a classical problem for the incompressible, Navier–Stokes equations defined on a given noncylindrical domain, and a class of fluid–structure interaction problems for the incompressible, Navier–Stokes equations, coupled to the elastodynamics of a Koiter shell. The compactness result presented in this manuscript is crucial in obtaining constructive existence proofs to nonlinear, moving boundary problems, using Rothe's method.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Boris Muha, Sunčica Čanić This work addresses an extension of the Aubin–Lions–Simon compactness result to generalized Bochner spaces L2(0,T;H(t)), where H(t) is a family of Hilbert spaces, parameterized by t. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin–Lions–Simon compactness result holds. Concrete examples of the application of the compactness theorem are presented, including a classical problem for the incompressible, Navier–Stokes equations defined on a given noncylindrical domain, and a class of fluid–structure interaction problems for the incompressible, Navier–Stokes equations, coupled to the elastodynamics of a Koiter shell. The compactness result presented in this manuscript is crucial in obtaining constructive existence proofs to nonlinear, moving boundary problems, using Rothe's method.
 On the critical points of the flight return time function of perturbed
closed orbits Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Adriana Buică, Jaume Giné, Maite Grau We deal here with planar analytic systems x˙=X(x,ε) which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by TΣ(q,ε) the time needed by a perturbed orbit that starts from q∈Σ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of x˙=X(x,0) is a continuable critical orbit in a family of the form x˙=X(x,ε) if, for any q∈Γ and any Σ that passes through q, there exists qε∈Σ a critical point of TΣ(⋅,ε) such that qε→q as ε→0. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of x˙=X(x,0) is a continuable critical orbit in any family of the form x˙=X(x,ε). We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x˙=X(x,0).
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Adriana Buică, Jaume Giné, Maite Grau We deal here with planar analytic systems x˙=X(x,ε) which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by TΣ(q,ε) the time needed by a perturbed orbit that starts from q∈Σ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of x˙=X(x,0) is a continuable critical orbit in a family of the form x˙=X(x,ε) if, for any q∈Γ and any Σ that passes through q, there exists qε∈Σ a critical point of TΣ(⋅,ε) such that qε→q as ε→0. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of x˙=X(x,0) is a continuable critical orbit in any family of the form x˙=X(x,ε). We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x˙=X(x,0).
 Optimal lower eigenvalue estimates for HodgeLaplacian and applications
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Qing Cui, Linlin Sun We consider the eigenvalue problem for HodgeLaplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold M¯. We first assume the pull back Weitzenböck operator of M¯ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of HodgeLaplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of M¯ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when M¯ is the space form.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Qing Cui, Linlin Sun We consider the eigenvalue problem for HodgeLaplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold M¯. We first assume the pull back Weitzenböck operator of M¯ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of HodgeLaplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of M¯ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when M¯ is the space form.
 A refined convergence result in homogenization of second order parabolic
systems Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Weisheng Niu, Yao Xu We derive the sharp O(ε) convergence rate in L2(0,T;Lq0(Ω)),q0=2d/(d−1) in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in L2 settings derived in [7], [28] for second order parabolic systems with timedependent coefficients.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Weisheng Niu, Yao Xu We derive the sharp O(ε) convergence rate in L2(0,T;Lq0(Ω)),q0=2d/(d−1) in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in L2 settings derived in [7], [28] for second order parabolic systems with timedependent coefficients.
 Upper estimates for the number of periodic solutions to multidimensional
systems Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Maoan Han, Hao Sun, Zalman Balanov The maximal number of zeros of multidimensional real analytic maps with small parameter is studied by means of the multidimensional generalization of Rouché's theorem. The obtained result is applied to study the maximal number of periodic solutions to multidimensional differential systems. An application to a class of threedimensional autonomous systems is given.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Maoan Han, Hao Sun, Zalman Balanov The maximal number of zeros of multidimensional real analytic maps with small parameter is studied by means of the multidimensional generalization of Rouché's theorem. The obtained result is applied to study the maximal number of periodic solutions to multidimensional differential systems. An application to a class of threedimensional autonomous systems is given.
 Blowup analysis for integral equations on bounded domains
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Qianqiao Guo Consider the integral equationfq−1(x)=∫Ωf(y) x−y n−αdy,f(x)>0,x∈Ω‾, where Ω⊂Rn is a smooth bounded domain. For 1
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Qianqiao Guo Consider the integral equationfq−1(x)=∫Ωf(y) x−y n−αdy,f(x)>0,x∈Ω‾, where Ω⊂Rn is a smooth bounded domain. For 1
 On the characterization of the controllability property for linear control
systems on nonnilpotent, solvable threedimensional Lie groups Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Víctor Ayala, Adriano Da Silva In this paper we show that a complete characterization of the controllability property for linear control system on threedimensional solvable nonnilpotent Lie groups is possible by the LARC and the knowledge of the eigenvalues of the derivation associated with the drift of the system.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Víctor Ayala, Adriano Da Silva In this paper we show that a complete characterization of the controllability property for linear control system on threedimensional solvable nonnilpotent Lie groups is possible by the LARC and the knowledge of the eigenvalues of the derivation associated with the drift of the system.
 Fine regularity for elliptic and parabolic anisotropic Robin problems with
variable exponents Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): MariaMagdalena Boureanu, Alejandro VélezSantiago We investigate a class of quasilinear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded nonsmooth domains, which may include nonLipschitz domains, such as domains with fractal boundary and rough domains. We obtain solvability and global regularity results for both the elliptic and parabolic Robin problem. Some a priori estimates, as well as fine properties for the corresponding nonlinear semigroups, are established. As a consequence, we generalize the global regularity theory for the Robin problem over nonsmooth domains by extending it for the first time to the variable exponent case, and furthermore, to the anisotropic variable exponent case.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): MariaMagdalena Boureanu, Alejandro VélezSantiago We investigate a class of quasilinear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded nonsmooth domains, which may include nonLipschitz domains, such as domains with fractal boundary and rough domains. We obtain solvability and global regularity results for both the elliptic and parabolic Robin problem. Some a priori estimates, as well as fine properties for the corresponding nonlinear semigroups, are established. As a consequence, we generalize the global regularity theory for the Robin problem over nonsmooth domains by extending it for the first time to the variable exponent case, and furthermore, to the anisotropic variable exponent case.
 Vanishing viscosity limit of short wave–long wave interactions in
planar magnetohydrodynamics Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Daniel R. Marroquin We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schrödinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an auroratype phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters, to a solution of the limit decoupled system involving the compressible Euler equations and a nonlinear Schrödinger equation. The vanishing viscosity limit serves to justify the SW–LW interactions in the limit equations as, in this setting, the SW–LW interactions cannot be defined in a straightforward way, due to the possible occurrence of vacuum.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Daniel R. Marroquin We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schrödinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an auroratype phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters, to a solution of the limit decoupled system involving the compressible Euler equations and a nonlinear Schrödinger equation. The vanishing viscosity limit serves to justify the SW–LW interactions in the limit equations as, in this setting, the SW–LW interactions cannot be defined in a straightforward way, due to the possible occurrence of vacuum.
 Phase portraits of piecewise linear continuous differential systems with
two zones separated by a straight line Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Shimin Li, Jaume Llibre This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Shimin Li, Jaume Llibre This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.
 On the existence of oscillating solutions in nonmonotone MeanField Games
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Marco Cirant For nonmonotone single and twopopulations timedependent MeanField Game systems we obtain the existence of an infinite number of branches of nontrivial solutions. These nontrivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutions predicted by bifurcation theory, and to study further properties of branches far away from bifurcation points.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Marco Cirant For nonmonotone single and twopopulations timedependent MeanField Game systems we obtain the existence of an infinite number of branches of nontrivial solutions. These nontrivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutions predicted by bifurcation theory, and to study further properties of branches far away from bifurcation points.
 Global classical solvability and generic infinitetime blowup in
quasilinear Keller–Segel systems with bounded sensitivities Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Michael Winkler The chemotaxis system(⋆){ut=∇⋅(D(u,v)∇u)−∇⋅(S(u,v)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, along with initial conditions involving suitably regular and nonnegative data.It is firstly asserted that if the positive smooth function D decays at most algebraically with respect to u, then for any smooth nonnegative and bounded S fulfilling a further mild assumption especially satisfied when S≡S(u) with S(0)=0, (⋆) possesses a globally defined classical solution.If Ω is a ball, then under appropriate assumptions on D and S generalizing the prototypical choices in(⋆⋆)D(u,v)=(u+1)m−1andS(u,v)=u(u+1)σ−1,u≥0,v≥0, with m∈R and σ∈R such that(⋆⋆⋆)m−n−2n
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Michael Winkler The chemotaxis system(⋆){ut=∇⋅(D(u,v)∇u)−∇⋅(S(u,v)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, along with initial conditions involving suitably regular and nonnegative data.It is firstly asserted that if the positive smooth function D decays at most algebraically with respect to u, then for any smooth nonnegative and bounded S fulfilling a further mild assumption especially satisfied when S≡S(u) with S(0)=0, (⋆) possesses a globally defined classical solution.If Ω is a ball, then under appropriate assumptions on D and S generalizing the prototypical choices in(⋆⋆)D(u,v)=(u+1)m−1andS(u,v)=u(u+1)σ−1,u≥0,v≥0, with m∈R and σ∈R such that(⋆⋆⋆)m−n−2n
 The L
p
dual Minkowski problem for p > 1 and q > 0
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Károly J. Böröczky, Ferenc Fodor General Lp dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. Lp dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrovtype variational formulas. Lutwak, Yang and Zhang [24] formulated the Lp dual Minkowski problem, which concerns the characterization of Lp dual curvature measures. In this paper, we solve the existence part of the Lp dual Minkowski problem for p>1 and q>0, and we also discuss the regularity of the solution.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Károly J. Böröczky, Ferenc Fodor General Lp dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. Lp dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrovtype variational formulas. Lutwak, Yang and Zhang [24] formulated the Lp dual Minkowski problem, which concerns the characterization of Lp dual curvature measures. In this paper, we solve the existence part of the Lp dual Minkowski problem for p>1 and q>0, and we also discuss the regularity of the solution.
 Local boundedness of solutions to nonlocal parabolic equations modeled on
the fractional pLaplacian Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Martin Strömqvist We state and prove estimates for the local boundedness of subsolutions of nonlocal, possibly degenerate, parabolic integrodifferential equations of the form∂tu(x,t)+P.V.∫RnK(x,y,t) u(x,t)−u(y,t) p−2(u(x,t)−u(y,t))dy=0, (x,t)∈Rn×R, where P.V. means in the principle value sense, p∈(1,∞) and the kernel obeys K(x,y,t)≈ x−y n+ps for some s∈(0,1), uniformly in (x,y,t)∈Rn×Rn×R.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Martin Strömqvist We state and prove estimates for the local boundedness of subsolutions of nonlocal, possibly degenerate, parabolic integrodifferential equations of the form∂tu(x,t)+P.V.∫RnK(x,y,t) u(x,t)−u(y,t) p−2(u(x,t)−u(y,t))dy=0, (x,t)∈Rn×R, where P.V. means in the principle value sense, p∈(1,∞) and the kernel obeys K(x,y,t)≈ x−y n+ps for some s∈(0,1), uniformly in (x,y,t)∈Rn×Rn×R.
 Twolocus clines maintained by diffusion and recombination in a
heterogeneous environment Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Linlin Su, KingYeung Lam, Reinhard Bürger We study existence and stability of stationary solutions of a system of semilinear parabolic partial differential equations that occurs in population genetics. It describes the evolution of gamete frequencies in a geographically structured population of migrating individuals in a bounded habitat. Fitness of individuals is determined additively by two recombining, diallelic genetic loci that are subject to spatially varying selection. Migration is modeled by diffusion. Of most interest are spatially nonconstant stationary solutions, socalled clines. In a twolocus cline all four gametes are present in the population, i.e., it is an internal stationary solution. We provide conditions for existence and linear stability of a twolocus cline if recombination is either sufficiently weak or sufficiently strong relative to selection and diffusion. For strong recombination, we also prove uniqueness and global asymptotic stability. For arbitrary recombination, we determine the stability properties of the monomorphic equilibria, which represent fixation of a single gamete.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Linlin Su, KingYeung Lam, Reinhard Bürger We study existence and stability of stationary solutions of a system of semilinear parabolic partial differential equations that occurs in population genetics. It describes the evolution of gamete frequencies in a geographically structured population of migrating individuals in a bounded habitat. Fitness of individuals is determined additively by two recombining, diallelic genetic loci that are subject to spatially varying selection. Migration is modeled by diffusion. Of most interest are spatially nonconstant stationary solutions, socalled clines. In a twolocus cline all four gametes are present in the population, i.e., it is an internal stationary solution. We provide conditions for existence and linear stability of a twolocus cline if recombination is either sufficiently weak or sufficiently strong relative to selection and diffusion. For strong recombination, we also prove uniqueness and global asymptotic stability. For arbitrary recombination, we determine the stability properties of the monomorphic equilibria, which represent fixation of a single gamete.
 L
1convergence rates to the Barenblatt solution for the damped
compressible Euler equations Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Shifeng Geng, Feimin Huang In our previous work [17], it is shown that any L∞ weak entropy solution of damped compressible Euler equation converges to the Barrenblatt's solution with finite mass in L1 norm with a convergence rate for 1
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Shifeng Geng, Feimin Huang In our previous work [17], it is shown that any L∞ weak entropy solution of damped compressible Euler equation converges to the Barrenblatt's solution with finite mass in L1 norm with a convergence rate for 1
 Wellposedness of the free boundary problem for incompressible
elastodynamics Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Xianpeng Hu, Yongting Huang The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress FF⊤−I and the transpose of the deformation gradient F⊤−I are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the localintime existence of a unique smooth solution is proved using a geometric argument in the spirit of [19].
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Xianpeng Hu, Yongting Huang The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress FF⊤−I and the transpose of the deformation gradient F⊤−I are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the localintime existence of a unique smooth solution is proved using a geometric argument in the spirit of [19].
 Controllability results for the Moore–Gibson–Thompson equation arising
in nonlinear acoustics Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Carlos Lizama, Sebastián Zamorano We show that the Moore–Gibson–Thomson equationτ∂ttty+α∂tty−c2Δy−bΔ∂ty=k∂tt(y2)+χω(t)u, is controlled by a force that is supported on an moving subset ω(t) of the domain, satisfying a geometrical condition. Using the concept of approximately outer invertible map, a generalized implicit function theorem and assuming that γ:=α−τc2b>0, the local null controllability in the nonlinear case is established. Moreover, the analysis of the critical value γ=0 for the linear equation is included.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Carlos Lizama, Sebastián Zamorano We show that the Moore–Gibson–Thomson equationτ∂ttty+α∂tty−c2Δy−bΔ∂ty=k∂tt(y2)+χω(t)u, is controlled by a force that is supported on an moving subset ω(t) of the domain, satisfying a geometrical condition. Using the concept of approximately outer invertible map, a generalized implicit function theorem and assuming that γ:=α−τc2b>0, the local null controllability in the nonlinear case is established. Moreover, the analysis of the critical value γ=0 for the linear equation is included.
 Translating solitons in Riemannian products
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Jorge H.S. de Lira, Francisco Martín In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product (R×P,dt2+g0) and the parallel field is X=∂t. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in R×P can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in R×P. When g0 is rotationally invariant and its sectional curvature is nonpositive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several nonexistence results.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Jorge H.S. de Lira, Francisco Martín In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product (R×P,dt2+g0) and the parallel field is X=∂t. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in R×P can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in R×P. When g0 is rotationally invariant and its sectional curvature is nonpositive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several nonexistence results.
 Stability of peakons for the generalized modified Camassa–Holm
equation Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Zihua Guo, Xiaochuan Liu, Xingxing Liu, Changzheng Qu In this paper, we study orbital stability of peakons for the generalized modified Camassa–Holm (gmCH) equation, which is a natural higherorder generalization of the modified Camassa–Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function h(t,x) and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Qu et al. 2013) [36] successfully to the higherorder case, and is helpful to understand how higherorder nonlinearities affect the dispersion dynamics.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): Zihua Guo, Xiaochuan Liu, Xingxing Liu, Changzheng Qu In this paper, we study orbital stability of peakons for the generalized modified Camassa–Holm (gmCH) equation, which is a natural higherorder generalization of the modified Camassa–Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function h(t,x) and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Qu et al. 2013) [36] successfully to the higherorder case, and is helpful to understand how higherorder nonlinearities affect the dispersion dynamics.
 Pulsating fronts and frontlike entire solutions for a
reaction–advection–diffusion competition model in a periodic habitat Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): LiJun Du, WanTong Li, ShiLiang Wu This paper is devoted to the study of pulsating fronts and pulsating frontlike entire solutions for a reaction–advection–diffusion model of two competing species in a periodic habitat. Under certain assumptions, the competition system admits a leftward and a rightward pulsating fronts in the bistable case. In this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts. Some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the xaxis, which turn out to be unique and Liapunov stable 2dimensional manifolds of solutions, furthermore, the leftward and rightward pulsating fronts are on the boundary of these 2dimensional manifolds. The others behave as the two pulsating fronts propagating from one side of the xaxis, the faster one then invades the slower one as t→+∞. These kinds of pulsating frontlike entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat.
 Abstract: Publication date: 5 June 2019Source: Journal of Differential Equations, Volume 266, Issue 12Author(s): LiJun Du, WanTong Li, ShiLiang Wu This paper is devoted to the study of pulsating fronts and pulsating frontlike entire solutions for a reaction–advection–diffusion model of two competing species in a periodic habitat. Under certain assumptions, the competition system admits a leftward and a rightward pulsating fronts in the bistable case. In this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts. Some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the xaxis, which turn out to be unique and Liapunov stable 2dimensional manifolds of solutions, furthermore, the leftward and rightward pulsating fronts are on the boundary of these 2dimensional manifolds. The others behave as the two pulsating fronts propagating from one side of the xaxis, the faster one then invades the slower one as t→+∞. These kinds of pulsating frontlike entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat.
 Robustly topological mixing of Kan's map
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Shaobo Gan, Yi Shi In 1994, I. Kan constructed a smooth map on the annulus admitting two physical measures, whose basins are intermingled. In this paper, we prove that Kan's map is C2 robustly topologically mixing.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Shaobo Gan, Yi Shi In 1994, I. Kan constructed a smooth map on the annulus admitting two physical measures, whose basins are intermingled. In this paper, we prove that Kan's map is C2 robustly topologically mixing.
 Cauchy problem for the ellipsoidal BGK model for polyatomic particles
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Sa Jun Park, SeokBae Yun We establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Sa Jun Park, SeokBae Yun We establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level.
 Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov
and the Selkov systems Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Hebai Chen, Yilei Tang The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Hebai Chen, Yilei Tang The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.
 Random attractor for the 3D viscous primitive equations driven by
fractional noises Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Guoli Zhou We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of largescale ocean and atmosphere dynamics under nonperiodic boundary conditions and driven by infinitedimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the timeuniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the nonperiodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15], [16]). The main idea of our method is that we first derive that Palmost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain timeuniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish timeuniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C([0,T];(H1(℧))3), in view of our method, we only need to obtain the timeuniform a priori estimates in the solution space (H1(℧))3 to prove the existence of random attractor for this stochastic system, while the common method need to establish timeuniform a priori estimates for the solution in the functional space (H2(℧))3. However, timeuniform a priori estimates in (H2(℧))3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the longtime behavior of stochastic nematic liquid crystals equations.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Guoli Zhou We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of largescale ocean and atmosphere dynamics under nonperiodic boundary conditions and driven by infinitedimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the timeuniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the nonperiodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15], [16]). The main idea of our method is that we first derive that Palmost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain timeuniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish timeuniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C([0,T];(H1(℧))3), in view of our method, we only need to obtain the timeuniform a priori estimates in the solution space (H1(℧))3 to prove the existence of random attractor for this stochastic system, while the common method need to establish timeuniform a priori estimates for the solution in the functional space (H2(℧))3. However, timeuniform a priori estimates in (H2(℧))3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the longtime behavior of stochastic nematic liquid crystals equations.
 Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and
smooth bounded and unbounded domains Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): N.A. Larkin, M.V. Padilha Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): N.A. Larkin, M.V. Padilha Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.
 Existence for evolutionary problems with linear growth by stability
methods Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Verena Bögelein, Frank Duzaar, Leah Schätzler, Christoph Scheven We establish that solutions to the Cauchy–Dirichlet problem∂tu−div(Dξf(x,Du))=0 for functionals f:Ω×RN×n→[0,∞) of linear growth can be obtained as limits of solutions to flows with pgrowth in the limit p↓1. The result can be interpreted on the one hand as a stability result. On the other hand it provides an existence result for general flows with linear growth.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Verena Bögelein, Frank Duzaar, Leah Schätzler, Christoph Scheven We establish that solutions to the Cauchy–Dirichlet problem∂tu−div(Dξf(x,Du))=0 for functionals f:Ω×RN×n→[0,∞) of linear growth can be obtained as limits of solutions to flows with pgrowth in the limit p↓1. The result can be interpreted on the one hand as a stability result. On the other hand it provides an existence result for general flows with linear growth.
 Singular Sturmian separation theorems on unbounded intervals for linear
Hamiltonian systems Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Peter Šepitka, Roman Šimon Hilscher In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Peter Šepitka, Roman Šimon Hilscher In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.
 Completely degenerate lowerdimensional invariant tori for Hamiltonian
system Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Shengqing Hu, Bin Liu We study the persistence of lowerdimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Shengqing Hu, Bin Liu We study the persistence of lowerdimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.
 Compactness of signchanging solutions to scalar curvaturetype equations
with bounded negative part Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Bruno Premoselli, Jérôme Vétois We consider the equation Δgu+hu= u 2⁎−2u in a closed Riemannian manifold (M,g), where h∈C0,θ(M), θ∈(0,1) and 2⁎=2nn−2, n:=dim(M)≥3. We obtain a sharp compactness result on the sets of signchanging solutions whose negative part is a priori bounded. We obtain this result under the conditions that n≥7 and h
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Bruno Premoselli, Jérôme Vétois We consider the equation Δgu+hu= u 2⁎−2u in a closed Riemannian manifold (M,g), where h∈C0,θ(M), θ∈(0,1) and 2⁎=2nn−2, n:=dim(M)≥3. We obtain a sharp compactness result on the sets of signchanging solutions whose negative part is a priori bounded. We obtain this result under the conditions that n≥7 and h
 Blowup phenomena for the Liouville equation with a singular source of
integer multiplicity Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Teresa D'Aprile We are concerned with the existence of blowingup solutions to the following boundary value problem−Δu=λa(x)eu−4πNδ0 in Ω,u=0 on ∂Ω, where Ω is a smooth and bounded domain in R2 such that 0∈Ω, a(x) is a positive smooth function, N is a positive integer and λ>0 is a small parameter. Here δ0 defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution uλ blowing up at 0 and satisfying λ∫Ωa(x)euλ→8π(N+1) as λ→0+.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Teresa D'Aprile We are concerned with the existence of blowingup solutions to the following boundary value problem−Δu=λa(x)eu−4πNδ0 in Ω,u=0 on ∂Ω, where Ω is a smooth and bounded domain in R2 such that 0∈Ω, a(x) is a positive smooth function, N is a positive integer and λ>0 is a small parameter. Here δ0 defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution uλ blowing up at 0 and satisfying λ∫Ωa(x)euλ→8π(N+1) as λ→0+.
 Minimalspeed selection of traveling waves to the Lotka–Volterra
competition model Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Ahmad Alhasanat, Chunhua Ou In this paper the minimalspeed determinacy of traveling wave fronts of a twospecies competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upperlower solution technique on the cooperative system to study the traveling waves as well as its minimalspeed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Ahmad Alhasanat, Chunhua Ou In this paper the minimalspeed determinacy of traveling wave fronts of a twospecies competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upperlower solution technique on the cooperative system to study the traveling waves as well as its minimalspeed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.
 The role of protection zone on species spreading governed by a
reaction–diffusion model with strong Allee effect Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Kai Du, Rui Peng, Ningkui Sun It is known that a species dies out in the long run for small initial data if its evolution obeys a reaction of bistable nonlinearity. Such a phenomenon, which is termed as the strong Allee effect, is well supported by numerous evidence from ecosystems, mainly due to the environmental pollution as well as unregulated harvesting and hunting. To save an endangered species, in this paper we introduce a protection zone that is governed by a Fisher–KPP nonlinearity, and examine the dynamics of a reaction–diffusion model with strong Allee effect and protection zone. We show the existence of two critical values 0
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Kai Du, Rui Peng, Ningkui Sun It is known that a species dies out in the long run for small initial data if its evolution obeys a reaction of bistable nonlinearity. Such a phenomenon, which is termed as the strong Allee effect, is well supported by numerous evidence from ecosystems, mainly due to the environmental pollution as well as unregulated harvesting and hunting. To save an endangered species, in this paper we introduce a protection zone that is governed by a Fisher–KPP nonlinearity, and examine the dynamics of a reaction–diffusion model with strong Allee effect and protection zone. We show the existence of two critical values 0
 Asymptotic stability of superposition of stationary solutions and
rarefaction waves for 1D Navier–Stokes/Allen–Cahn system Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Haiyan Yin, Changjiang Zhu In this paper, we investigate the large time behavior of the solutions to the inflow problem for the onedimensional Navier–Stokes/Allen–Cahn system in the half space. First, we assume that the spaceasymptotic states (ρ+,u+,χ+) and the boundary data (ρb,ub,χb) satisfy some conditions so that the timeasymptotic state of solutions for the inflow problem is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Then, we show the existence of the stationary solution by the center manifold theorem. Finally, we prove that the nonlinear wave is asymptotically stable when the initial data is a small perturbation of the nonlinear wave. The proof is mainly based on the energy method by taking into account the effect of the concentration χ and the complexity of nonlinear wave.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Haiyan Yin, Changjiang Zhu In this paper, we investigate the large time behavior of the solutions to the inflow problem for the onedimensional Navier–Stokes/Allen–Cahn system in the half space. First, we assume that the spaceasymptotic states (ρ+,u+,χ+) and the boundary data (ρb,ub,χb) satisfy some conditions so that the timeasymptotic state of solutions for the inflow problem is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Then, we show the existence of the stationary solution by the center manifold theorem. Finally, we prove that the nonlinear wave is asymptotically stable when the initial data is a small perturbation of the nonlinear wave. The proof is mainly based on the energy method by taking into account the effect of the concentration χ and the complexity of nonlinear wave.
 Positive solutions for a class of singular quasilinear Schrödinger
equations with critical Sobolev exponent Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Zhouxin Li We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth−Δu−λc(x)u−κα(Δ( u 2α)) u 2α−2u= u q−2u+ u 2⁎−2u,u∈D1,2(RN), via variational methods, where λ≥0, c:RN→R+, κ>0, 0
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Zhouxin Li We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth−Δu−λc(x)u−κα(Δ( u 2α)) u 2α−2u= u q−2u+ u 2⁎−2u,u∈D1,2(RN), via variational methods, where λ≥0, c:RN→R+, κ>0, 0
 Uniform stability of semilinear wave equations with arbitrary local memory
effects versus frictional dampings Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): KunPeng Jin, Jin Liang, TiJun Xiao This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memoryeffect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of g′ controlled by g.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): KunPeng Jin, Jin Liang, TiJun Xiao This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memoryeffect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of g′ controlled by g.
 Asymptotic regularity of trajectory attractor and trajectory statistical
solution for the 3D globally modified Navier–Stokes equations Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Caidi Zhao, Tomás Caraballo We first prove the existence and regularity of the trajectory attractor for a threedimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Caidi Zhao, Tomás Caraballo We first prove the existence and regularity of the trajectory attractor for a threedimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.
 Selfsimilar solutions for dyadic models of the Euler equations
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): InJee Jeong We show existence of selfsimilar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin [1]. The proof is based on the analysis of certain dynamical systems on the plane.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): InJee Jeong We show existence of selfsimilar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin [1]. The proof is based on the analysis of certain dynamical systems on the plane.
 Long time behavior of solutions to the 3D Hallmagnetohydrodynamics
system with one diffusion Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Mimi Dai, Han Liu This paper studies the asymptotic behavior of smooth solutions to the generalized Hallmagnetohydrodynamics system (1.1) with one single diffusion on the whole space R3. We establish that, in the inviscid resistive case, the energy ‖b(t)‖22 vanishes and ‖u(t)‖22 converges to a constant as time tends to infinity provided the velocity is bounded in W1−α,3α(R3); in the viscous nonresistive case, the energy ‖u(t)‖22 vanishes and ‖b(t)‖22 converges to a constant provided the magnetic field is bounded in W1−β,∞(R3). In summary, one single diffusion, being as weak as (−Δ)αb or (−Δ)βu with small enough α,β, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Mimi Dai, Han Liu This paper studies the asymptotic behavior of smooth solutions to the generalized Hallmagnetohydrodynamics system (1.1) with one single diffusion on the whole space R3. We establish that, in the inviscid resistive case, the energy ‖b(t)‖22 vanishes and ‖u(t)‖22 converges to a constant as time tends to infinity provided the velocity is bounded in W1−α,3α(R3); in the viscous nonresistive case, the energy ‖u(t)‖22 vanishes and ‖b(t)‖22 converges to a constant provided the magnetic field is bounded in W1−β,∞(R3). In summary, one single diffusion, being as weak as (−Δ)αb or (−Δ)βu with small enough α,β, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.
 Asymptotics for periodic systems
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Lassi Paunonen, David Seifert This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a socalled Ritt operator under a natural ‘resonance’ condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the onedimensional wave equation with periodic damping.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Lassi Paunonen, David Seifert This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a socalled Ritt operator under a natural ‘resonance’ condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the onedimensional wave equation with periodic damping.
 The transport equation in the scaling invariant Besov or
Essén–Janson–Peng–Xiao space Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Jie Xiao This paper addresses a wellposedness of the weak solution to the transport equation (describing how a scalar quantity is transported in a space) with an initial data in the scaling invariant Besov or Essén–Janson–Peng–Xiao space via the boundedness of the left and right compositions acting on each space.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Jie Xiao This paper addresses a wellposedness of the weak solution to the transport equation (describing how a scalar quantity is transported in a space) with an initial data in the scaling invariant Besov or Essén–Janson–Peng–Xiao space via the boundedness of the left and right compositions acting on each space.
 On the concentration phenomenon of L
2subcritical constrained minimizers
for a class of Kirchhoff equations with potentials Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Gongbao Li, Hongyu Ye In this paper, we study the existence and concentration behavior of minimizers for iV(c)=infu∈ScIV(u), here Sc={u∈H1(RN) ∫RNV(x) u 20} andIV(u)=12∫RN(a ∇u 2+V(x) u 2)+b4(∫RN ∇u 2)2−1p∫RN u p, where N=1,2,3 and a,b>0 are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of iV(c) for 2
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Gongbao Li, Hongyu Ye In this paper, we study the existence and concentration behavior of minimizers for iV(c)=infu∈ScIV(u), here Sc={u∈H1(RN) ∫RNV(x) u 20} andIV(u)=12∫RN(a ∇u 2+V(x) u 2)+b4(∫RN ∇u 2)2−1p∫RN u p, where N=1,2,3 and a,b>0 are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of iV(c) for 2
 Geometric stability switch criteria in delay differential equations with
two delays and delay dependent parameters Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Qi An, Edoardo Beretta, Yang Kuang, Chuncheng Wang, Hao Wang Most modeling efforts involve multiple physical or biological processes. All physical or biological processes take time to complete. Therefore, multiple time delays occur naturally and shall be considered in more advanced modeling efforts. Carefully formulated models of such natural processes often involve multiple delays and delay dependent parameters. However, a general and practical theory for the stability analysis of models with more than one discrete delay and delay dependent parameters is nonexistent. The main purpose of this paper is to present a practical geometric method to study the stability switching properties of a general transcendental equation which may result from a stability analysis of a model with two discrete time delays and delay dependent parameters that dependent only on one of the time delay. In addition to simple and illustrative examples, we present a detailed application of our method to the study of a two discrete delay SIR model.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Qi An, Edoardo Beretta, Yang Kuang, Chuncheng Wang, Hao Wang Most modeling efforts involve multiple physical or biological processes. All physical or biological processes take time to complete. Therefore, multiple time delays occur naturally and shall be considered in more advanced modeling efforts. Carefully formulated models of such natural processes often involve multiple delays and delay dependent parameters. However, a general and practical theory for the stability analysis of models with more than one discrete delay and delay dependent parameters is nonexistent. The main purpose of this paper is to present a practical geometric method to study the stability switching properties of a general transcendental equation which may result from a stability analysis of a model with two discrete time delays and delay dependent parameters that dependent only on one of the time delay. In addition to simple and illustrative examples, we present a detailed application of our method to the study of a two discrete delay SIR model.
 Noncoercive Lyapunov functions for infinitedimensional systems
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Andrii Mironchenko, Fabian Wirth We show that the existence of a noncoercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinitedimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new noncoercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Andrii Mironchenko, Fabian Wirth We show that the existence of a noncoercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinitedimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new noncoercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.
 Nonlocal scalar field equations: Qualitative properties, asymptotic
profiles and local uniqueness of solutions Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Mousomi Bhakta, Debangana Mukherjee We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu= u p−2u− u q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p2⁎, the solution asymptotically coincides with a groundstate solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Mousomi Bhakta, Debangana Mukherjee We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu= u p−2u− u q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p2⁎, the solution asymptotically coincides with a groundstate solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.
 The 1:1 resonance in Hamiltonian systems
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Heinz Hanßmann, Igor Hoveijn Twodegreeoffreedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:−1. A twodegreeoffreedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to S1symmetric systems. The question we wish to address is about the codimension of such a system in 1:1 resonance with respect to leftrightequivalence, where the right action is S1equivariant. The result is a codimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining nonmodal parameters are the ones found in the linear unfolding of this system.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Heinz Hanßmann, Igor Hoveijn Twodegreeoffreedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:−1. A twodegreeoffreedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to S1symmetric systems. The question we wish to address is about the codimension of such a system in 1:1 resonance with respect to leftrightequivalence, where the right action is S1equivariant. The result is a codimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining nonmodal parameters are the ones found in the linear unfolding of this system.
 Lagrange stability for impulsive Duffing equations
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Jianhua Shen, Lu Chen, Xiaoping Yuan This work discusses the boundedness of solutions for impulsive Duffing equation with timedependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasiperiodic solutions clustering at infinity. This result extends some wellknown results on Duffing equations to impulsive Duffing equations.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Jianhua Shen, Lu Chen, Xiaoping Yuan This work discusses the boundedness of solutions for impulsive Duffing equation with timedependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasiperiodic solutions clustering at infinity. This result extends some wellknown results on Duffing equations to impulsive Duffing equations.
 Spectral theory approach for a class of radial indefinite variational
problems Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Liliane A. Maia, Mayra Soares Considering the radial nonlinear Schrödinger equation(Pr)−Δu+V(x)u=g(x,u) inRN,N≥3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring problem (Pr) is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating problem (Pr) to an indefinite functional in order to apply our Linking Theorem for Cerami sequences in [8] to get a nontrivial critical point for this functional. Our goal is to make use of spectral properties of operator A:=Δ+V(x) restricted to Hrad1(RN), the space of radially symmetric functions in H1(RN), for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Liliane A. Maia, Mayra Soares Considering the radial nonlinear Schrödinger equation(Pr)−Δu+V(x)u=g(x,u) inRN,N≥3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring problem (Pr) is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating problem (Pr) to an indefinite functional in order to apply our Linking Theorem for Cerami sequences in [8] to get a nontrivial critical point for this functional. Our goal is to make use of spectral properties of operator A:=Δ+V(x) restricted to Hrad1(RN), the space of radially symmetric functions in H1(RN), for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.
 Uniqueness for an inverse problem for a nonlinear parabolic system with an
integral term by onepoint Dirichlet data Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Dietmar Hömberg, Shuai Lu, Masahiro Yamamoto We consider an inverse problem arising in laserinduced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a onepoint measurement.
 Abstract: Publication date: 15 May 2019Source: Journal of Differential Equations, Volume 266, Issue 11Author(s): Dietmar Hömberg, Shuai Lu, Masahiro Yamamoto We consider an inverse problem arising in laserinduced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a onepoint measurement.
 Solvability for a driftdiffusion system with Robin boundary conditions
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): A. Heibig, A. Petrov, C. Reichert This paper focuses on a driftdiffusion system subjected to boundedly non dissipative Robin boundary conditions. A general existence result with large initial conditions is established by using suitable L1, L2 and trace estimates. Finally, two examples coming from the corrosion and the selfgravitation model are analyzed.
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): A. Heibig, A. Petrov, C. Reichert This paper focuses on a driftdiffusion system subjected to boundedly non dissipative Robin boundary conditions. A general existence result with large initial conditions is established by using suitable L1, L2 and trace estimates. Finally, two examples coming from the corrosion and the selfgravitation model are analyzed.
 On the fourth order Schrödinger equation in four dimensions: Dispersive
estimates and zero energy resonances Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): William R. Green, Ebru Toprak We study the fourth order Schrödinger operator H=(−Δ)2+V for a decaying potential V in four dimensions. In particular, we show that the t−1 decay rate holds in the L1→L∞ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of t−1(logt)−2 is attained for large t, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds.
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): William R. Green, Ebru Toprak We study the fourth order Schrödinger operator H=(−Δ)2+V for a decaying potential V in four dimensions. In particular, we show that the t−1 decay rate holds in the L1→L∞ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of t−1(logt)−2 is attained for large t, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds.
 Hilltype formula for Hamiltonian system with Lagrangian boundary
conditions Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): Xijun Hu, Yuwei Ou, Penghui Wang In this paper, we build up Hilltype formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the Nreversible symmetry periodic orbits in nbody problem naturally, where N is an antisymplectic orthogonal matrix with N2=I. The Hilltype formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Kreintype trace formula and give nontrivial estimation for the eigenvalue problem. Combined with the Maslovtype index theory, we give some new stability criteria for the Nreversible symmetry periodic solutions of Hamiltonian systems. As an application, we study the linear stability of elliptic relative equilibria in planar 3body problem.
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): Xijun Hu, Yuwei Ou, Penghui Wang In this paper, we build up Hilltype formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the Nreversible symmetry periodic orbits in nbody problem naturally, where N is an antisymplectic orthogonal matrix with N2=I. The Hilltype formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Kreintype trace formula and give nontrivial estimation for the eigenvalue problem. Combined with the Maslovtype index theory, we give some new stability criteria for the Nreversible symmetry periodic solutions of Hamiltonian systems. As an application, we study the linear stability of elliptic relative equilibria in planar 3body problem.
 Dynamic and asymptotic behavior of singularities of certain weak KAM
solutions on the torus Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): Piermarco Cannarsa, Qinbo Chen, Wei Cheng For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristic semiflows associated with the HamiltonJacobi equations, and build the relation between the ωlimit sets of the semiflows and the projected Aubry sets.
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): Piermarco Cannarsa, Qinbo Chen, Wei Cheng For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristic semiflows associated with the HamiltonJacobi equations, and build the relation between the ωlimit sets of the semiflows and the projected Aubry sets.
 Mean LiYorke chaos for random dynamical systems
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): Yunping Wang, Ercai Chen, Xiaoyao Zhou In this paper, we studied the complexity of some random dynamical systems with positive topological entropy. The random dynamical systems are usually generated by stochastic partial differential equations (SPDEs) and contain randomness in many ways. We proved that there exists mean LiYorke chaotic phenomenon in some random dynamical systems with positive entropy.
 Abstract: Publication date: Available online 21 March 2019Source: Journal of Differential EquationsAuthor(s): Yunping Wang, Ercai Chen, Xiaoyao Zhou In this paper, we studied the complexity of some random dynamical systems with positive topological entropy. The random dynamical systems are usually generated by stochastic partial differential equations (SPDEs) and contain randomness in many ways. We proved that there exists mean LiYorke chaotic phenomenon in some random dynamical systems with positive entropy.
 Nonlinear multiparameter eigenvalue problems: Linearised and nonlinearised
solutions Abstract: Publication date: Available online 20 March 2019Source: Journal of Differential EquationsAuthor(s): V.Yu. Kurseeva, S.V. Tikhov, D.V. Valovik The paper focuses on a nonlinear multiparameter eigenvalue problem called P. This problem involves n spectral parameters and also depends on n2 numerical parameters, where n⩾2 is an integer. If these numerical parameters vanish, P degenerates into n linear (oneparameter) eigenvalue problems called Pi0 (i=1,n‾). In connection with P one can consider another n nonlinear oneparameter eigenvalue problems called Pi. The problems Pi have eigenvalues with as well as without linear counterparts. The paper suggests to consider Pi as ‘nonunperturbed’ instead of Pi0. Using properties of eigenvalues of Pi, one manages to prove existence of eigentuples of P. Among the eigentuples found in this way, there are eigetuples with as well as without linear counterparts. Results of the paper are found with a nonclassical approach. Applications of the found results to nonlinear optics are shown.
 Abstract: Publication date: Available online 20 March 2019Source: Journal of Differential EquationsAuthor(s): V.Yu. Kurseeva, S.V. Tikhov, D.V. Valovik The paper focuses on a nonlinear multiparameter eigenvalue problem called P. This problem involves n spectral parameters and also depends on n2 numerical parameters, where n⩾2 is an integer. If these numerical parameters vanish, P degenerates into n linear (oneparameter) eigenvalue problems called Pi0 (i=1,n‾). In connection with P one can consider another n nonlinear oneparameter eigenvalue problems called Pi. The problems Pi have eigenvalues with as well as without linear counterparts. The paper suggests to consider Pi as ‘nonunperturbed’ instead of Pi0. Using properties of eigenvalues of Pi, one manages to prove existence of eigentuples of P. Among the eigentuples found in this way, there are eigetuples with as well as without linear counterparts. Results of the paper are found with a nonclassical approach. Applications of the found results to nonlinear optics are shown.
 Homogenization of an advection equation with locally stationary random
coefficients Abstract: Publication date: Available online 18 March 2019Source: Journal of Differential EquationsAuthor(s): Tymoteusz Chojecki, Tomasz Komorowski In the paper we consider the solution of an advection equation with rapidly changing coefficients ∂tuε+(1/ε)V(t/ε2,x/ε)⋅∇xuε=0 for t0 is some small parameter and the drift term (V(t,x))(t,x)∈R1+d is assumed to be a ddimensional, vector valued random field with incompressible spatial realizations. We prove that when the field is Gaussian, locally stationary, quasiperiodic in the x variable and strongly mixing in time the solutions uε(t,x) converge in law, as ε→0, to u0(x(T;t,x)), where (x(s;t,x))s≥t is a diffusion satisfying x(t;t,x)=x. The averages of uε(T,x) converge then to the solution of the corresponding Kolmogorov backward equation.
 Abstract: Publication date: Available online 18 March 2019Source: Journal of Differential EquationsAuthor(s): Tymoteusz Chojecki, Tomasz Komorowski In the paper we consider the solution of an advection equation with rapidly changing coefficients ∂tuε+(1/ε)V(t/ε2,x/ε)⋅∇xuε=0 for t0 is some small parameter and the drift term (V(t,x))(t,x)∈R1+d is assumed to be a ddimensional, vector valued random field with incompressible spatial realizations. We prove that when the field is Gaussian, locally stationary, quasiperiodic in the x variable and strongly mixing in time the solutions uε(t,x) converge in law, as ε→0, to u0(x(T;t,x)), where (x(s;t,x))s≥t is a diffusion satisfying x(t;t,x)=x. The averages of uε(T,x) converge then to the solution of the corresponding Kolmogorov backward equation.
 On the simplicity of eigenvalues of two nonhomogeneous EulerBernoulli
beams connected by a point mass Abstract: Publication date: Available online 18 March 2019Source: Journal of Differential EquationsAuthor(s): Jamel Ben Amara, Hedi Bouzidi In this paper we consider a linear system modeling the vibrations of two nonhomogeneous EulerBernoulli beams connected by a point mass. This system is generated by the following equationsρ(x)ytt(t,x)+(σ(x)yxx(t,x))xx−(q(x)yx(t,x))x=0,t>0,x∈(−1,0)∪(0,1),Mytt(t,0)=(Ty(t,x)) x=0−−(Ty(t,x)) x=0+,t>0, with hinged boundary conditions at both ends, where M>0, ρ(x)>0,σ(x)>0, q(x)≥0 and Ty=(σ(x)yxx)x−q(x)yx for x∈(−1,0)∪(0,1). We prove that all the associated eigenvalues (λn)n≥1 are algebraically simple, furthermore the corresponding eigenfunctions (ϕn)n≥1 satisfy ϕn′(−1
 Abstract: Publication date: Available online 18 March 2019Source: Journal of Differential EquationsAuthor(s): Jamel Ben Amara, Hedi Bouzidi In this paper we consider a linear system modeling the vibrations of two nonhomogeneous EulerBernoulli beams connected by a point mass. This system is generated by the following equationsρ(x)ytt(t,x)+(σ(x)yxx(t,x))xx−(q(x)yx(t,x))x=0,t>0,x∈(−1,0)∪(0,1),Mytt(t,0)=(Ty(t,x)) x=0−−(Ty(t,x)) x=0+,t>0, with hinged boundary conditions at both ends, where M>0, ρ(x)>0,σ(x)>0, q(x)≥0 and Ty=(σ(x)yxx)x−q(x)yx for x∈(−1,0)∪(0,1). We prove that all the associated eigenvalues (λn)n≥1 are algebraically simple, furthermore the corresponding eigenfunctions (ϕn)n≥1 satisfy ϕn′(−1
 Measure Nexpansive systems
 Abstract: Publication date: Available online 18 March 2019Source: Journal of Differential EquationsAuthor(s): K. Lee, C.A. Morales, B. San Martin The Nexpansive systems have been recently studied in the literature [6], [7], [9], [14]. Here we characterize them as those homeomorphisms for which every Borel probability measure is Nexpansive. In particular, the strongly measure expansive homeomorphisms in the sense of [8] are precisely the homeomorphisms for which every invariant measure is 1expansive. We also characterize the 1expansive measures for equicontinuous homeomorphisms as the convex sum of finitely many Dirac measures supported on isolated points. In particular, such measures do not exist on metric spaces without isolated points. Furthermore, we consider Nexpansive measure for flows and prove that a flow is Nexpansive in the sense of [9] if and only if every Borel probability measure is Nexpansive. Finally, we obtain a lower bound of the topological entropy of the Nexpansive flows as the exponential growth rate of the number of periodic orbits.
 Abstract: Publication date: Available online 18 March 2019Source: Journal of Differential EquationsAuthor(s): K. Lee, C.A. Morales, B. San Martin The Nexpansive systems have been recently studied in the literature [6], [7], [9], [14]. Here we characterize them as those homeomorphisms for which every Borel probability measure is Nexpansive. In particular, the strongly measure expansive homeomorphisms in the sense of [8] are precisely the homeomorphisms for which every invariant measure is 1expansive. We also characterize the 1expansive measures for equicontinuous homeomorphisms as the convex sum of finitely many Dirac measures supported on isolated points. In particular, such measures do not exist on metric spaces without isolated points. Furthermore, we consider Nexpansive measure for flows and prove that a flow is Nexpansive in the sense of [9] if and only if every Borel probability measure is Nexpansive. Finally, we obtain a lower bound of the topological entropy of the Nexpansive flows as the exponential growth rate of the number of periodic orbits.
 Unique recovery of lower order coefficients for hyperbolic equations from
data on disjoint sets Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Yavar Kian, Yaroslav Kurylev, Matti Lassas, Lauri Oksanen We consider a restricted DirichlettoNeumann map ΛS,RT associated with the operator ∂t2−Δg+A+q where Δg is the LaplaceBeltrami operator of a Riemannian manifold (M,g), and A and q are a vector field and a function on M. The restriction ΛS,RT corresponds to the case where the Dirichlet traces are supported on (0,T)×S and the Neumann traces are restricted on (0,T)×R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that ΛS,RT determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T/2. We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be nonselfadjoint, and in particular, the corresponding physical system may have dissipation of energy.
 Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Yavar Kian, Yaroslav Kurylev, Matti Lassas, Lauri Oksanen We consider a restricted DirichlettoNeumann map ΛS,RT associated with the operator ∂t2−Δg+A+q where Δg is the LaplaceBeltrami operator of a Riemannian manifold (M,g), and A and q are a vector field and a function on M. The restriction ΛS,RT corresponds to the case where the Dirichlet traces are supported on (0,T)×S and the Neumann traces are restricted on (0,T)×R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that ΛS,RT determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T/2. We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be nonselfadjoint, and in particular, the corresponding physical system may have dissipation of energy.
 Stability of strong solutions for a model of incompressible two–phase
flow under thermal fluctuations Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Eduard Feireisl, Madalina Petcu We consider a model of a two–phase flow based on the phase field approach, where the fluid bulk velocity obeys the standard Navier–Stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the Allen–Cahn equations. Possible thermal fluctuations are incorporated through a random forcing term in the Allen–Cahn equation. We show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality. This fact is used for showing the weak–strong uniqueness principle both pathwise and in law.
 Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Eduard Feireisl, Madalina Petcu We consider a model of a two–phase flow based on the phase field approach, where the fluid bulk velocity obeys the standard Navier–Stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the Allen–Cahn equations. Possible thermal fluctuations are incorporated through a random forcing term in the Allen–Cahn equation. We show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality. This fact is used for showing the weak–strong uniqueness principle both pathwise and in law.
 Recovery of an embedded obstacle and the surrounding medium for Maxwell's
system Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Youjun Deng, Hongyu Liu, Xiaodong Liu In this paper, we are concerned with the inverse electromagnetic scattering problem of recovering a complex scatterer by the corresponding electric farfield data. The complex scatterer consists of an inhomogeneous medium and a possibly embedded perfectly electric conducting (PEC) obstacle. The farfield data are collected corresponding to incident plane waves with a fixed incident direction and a fixed polarisation, but frequencies from an open interval. It is shown that the embedded obstacle can be uniquely recovered by the aforementioned farfield data, independent of the surrounding medium. Furthermore, if the surrounding medium is piecewise homogeneous, then the medium can be recovered as well. Those unique recovery results are new to the literature. Our argument is based on lowfrequency expansions of the electromagnetic fields and certain harmonic analysis techniques.
 Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Youjun Deng, Hongyu Liu, Xiaodong Liu In this paper, we are concerned with the inverse electromagnetic scattering problem of recovering a complex scatterer by the corresponding electric farfield data. The complex scatterer consists of an inhomogeneous medium and a possibly embedded perfectly electric conducting (PEC) obstacle. The farfield data are collected corresponding to incident plane waves with a fixed incident direction and a fixed polarisation, but frequencies from an open interval. It is shown that the embedded obstacle can be uniquely recovered by the aforementioned farfield data, independent of the surrounding medium. Furthermore, if the surrounding medium is piecewise homogeneous, then the medium can be recovered as well. Those unique recovery results are new to the literature. Our argument is based on lowfrequency expansions of the electromagnetic fields and certain harmonic analysis techniques.
 The Γlimit of traveling waves in the FitzHughNagumo system
 Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): ChaoNien Chen, Yung Sze Choi, Nicola Fusco Patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems. A common theme in investigating such systems is to identify the intrinsic factors responsible for such selforganization. The Γconvergence is a wellknown technique applicable to variational formulations of concentration phenomena of stable patterns. Recently a geometric variational functional associated with the Γlimit of standing waves of the FitzHughNagumo system has been built. This article studies the Γlimit of traveling waves. To the best of our knowledge, this is the first attempt to expand the scope of applicability of Γconvergence to cover nonstationary problems.
 Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): ChaoNien Chen, Yung Sze Choi, Nicola Fusco Patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems. A common theme in investigating such systems is to identify the intrinsic factors responsible for such selforganization. The Γconvergence is a wellknown technique applicable to variational formulations of concentration phenomena of stable patterns. Recently a geometric variational functional associated with the Γlimit of standing waves of the FitzHughNagumo system has been built. This article studies the Γlimit of traveling waves. To the best of our knowledge, this is the first attempt to expand the scope of applicability of Γconvergence to cover nonstationary problems.
 The pointinteraction approximation for the fields generated by contrasted
bubbles at arbitrary fixed frequencies Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Habib Ammari, Durga Prasad Challa, Anupam Pal Choudhury, Mourad Sini We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of M small bubbles, distributed in a bounded domain Ω, with relative densities having contrasts of the order aβ,β>0, where a models their relative maximum diameter, a≪1. We provide useful and natural conditions on the number M, the minimum distance and the contrasts parameter β of the small bubbles under which the point interaction approximation (called also the FoldyLax approximation) is valid.With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order aβ, β∈(32,2) and in this case we can handle any fixed frequency. In the particular and important case β=2, we can handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.
 Abstract: Publication date: Available online 15 March 2019Source: Journal of Differential EquationsAuthor(s): Habib Ammari, Durga Prasad Challa, Anupam Pal Choudhury, Mourad Sini We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of M small bubbles, distributed in a bounded domain Ω, with relative densities having contrasts of the order aβ,β>0, where a models their relative maximum diameter, a≪1. We provide useful and natural conditions on the number M, the minimum distance and the contrasts parameter β of the small bubbles under which the point interaction approximation (called also the FoldyLax approximation) is valid.With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order aβ, β∈(32,2) and in this case we can handle any fixed frequency. In the particular and important case β=2, we can handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.
 Bifurcation analysis of an SIRS epidemic model with a generalized
nonmonotone and saturated incidence rate Abstract: Publication date: Available online 14 March 2019Source: Journal of Differential EquationsAuthor(s): Min Lu, Jicai Huang, Shigui Ruan, Pei Yu In this paper, we study a susceptibleinfectiousrecovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate kI2S1+βI+αI2, in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddlenode bifurcation, BogdanovTakens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value α=α0 for the psychological effect, and two critical values k=k0,k1(k0α0, or α≤α0 and k≤k0, the disease will die out for all positive initial populations; (ii) when α=α0 and k0k1, the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when αk0, the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and datafitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.
 Abstract: Publication date: Available online 14 March 2019Source: Journal of Differential EquationsAuthor(s): Min Lu, Jicai Huang, Shigui Ruan, Pei Yu In this paper, we study a susceptibleinfectiousrecovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate kI2S1+βI+αI2, in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddlenode bifurcation, BogdanovTakens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value α=α0 for the psychological effect, and two critical values k=k0,k1(k0α0, or α≤α0 and k≤k0, the disease will die out for all positive initial populations; (ii) when α=α0 and k0k1, the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when αk0, the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and datafitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.
 Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary
conditions Abstract: Publication date: Available online 13 March 2019Source: Journal of Differential EquationsAuthor(s): FeiYing Yang, WanTong Li, Shigui Ruan In this paper we study a nonlocal dispersal susceptibleinfectedsusceptible (SIS) epidemic model with Neumann boundary condition, where the spatial movement of individuals is described by a nonlocal (convolution) diffusion operator, the transmission rate and recovery rate are spatially heterogeneous, and the total population number is constant. We first define the basic reproduction number R0 and discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of R0. Then we consider the impacts of the large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease. The obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease. In particular, our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease.
 Abstract: Publication date: Available online 13 March 2019Source: Journal of Differential EquationsAuthor(s): FeiYing Yang, WanTong Li, Shigui Ruan In this paper we study a nonlocal dispersal susceptibleinfectedsusceptible (SIS) epidemic model with Neumann boundary condition, where the spatial movement of individuals is described by a nonlocal (convolution) diffusion operator, the transmission rate and recovery rate are spatially heterogeneous, and the total population number is constant. We first define the basic reproduction number R0 and discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of R0. Then we consider the impacts of the large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease. The obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease. In particular, our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease.
 Optimal regularity of stochastic evolution equations in Mtype 2 Banach
spaces Abstract: Publication date: Available online 8 March 2019Source: Journal of Differential EquationsAuthor(s): Jialin Hong, Chuying Huang, Zhihui Liu In this paper, we prove the wellposedness and optimal trajectory regularity for the solution of stochastic evolution equations driven by general multiplicative noises in martingale type 2 Banach spaces. The main idea of our method is to combine the approach in [9] dealing with Hilbert setting and a version of Burkholder inequality in Mtype 2 Banach space. Applying our main results to the stochastic heat equation gives a positive answer to an open problem proposed in [10].
 Abstract: Publication date: Available online 8 March 2019Source: Journal of Differential EquationsAuthor(s): Jialin Hong, Chuying Huang, Zhihui Liu In this paper, we prove the wellposedness and optimal trajectory regularity for the solution of stochastic evolution equations driven by general multiplicative noises in martingale type 2 Banach spaces. The main idea of our method is to combine the approach in [9] dealing with Hilbert setting and a version of Burkholder inequality in Mtype 2 Banach space. Applying our main results to the stochastic heat equation gives a positive answer to an open problem proposed in [10].
 A criterion for the triviality of the centralizer for vector fields and
applications Abstract: Publication date: Available online 8 March 2019Source: Journal of Differential EquationsAuthor(s): Wescley Bonomo, Paulo Varandas In this paper we establish a criterion for the triviality of the C1centralizer for vector fields and flows. In particular we deduce the triviality of the centralizer at homoclinic classes of Cr vector fields (r≥1). Furthermore, we show that the set of flows whose C1centralizer is trivial include: (i) C1generic volume preserving flows, (ii) C2generic Hamiltonian flows on a generic and full Lebesgue measure set of energy levels, and (iii) C1open set of nonhyperbolic vector fields (that admit a Lorenz attractor). We also provide a criterion for the triviality of the C0centralizer of continuous flows.
 Abstract: Publication date: Available online 8 March 2019Source: Journal of Differential EquationsAuthor(s): Wescley Bonomo, Paulo Varandas In this paper we establish a criterion for the triviality of the C1centralizer for vector fields and flows. In particular we deduce the triviality of the centralizer at homoclinic classes of Cr vector fields (r≥1). Furthermore, we show that the set of flows whose C1centralizer is trivial include: (i) C1generic volume preserving flows, (ii) C2generic Hamiltonian flows on a generic and full Lebesgue measure set of energy levels, and (iii) C1open set of nonhyperbolic vector fields (that admit a Lorenz attractor). We also provide a criterion for the triviality of the C0centralizer of continuous flows.

R + 3
in R + 3 Abstract: Publication date: Available online 6 March 2019Source: Journal of Differential EquationsAuthor(s): Jinrui Huang, Changyou Wang, Huanyao Wen In this paper, we obtain optimal timedecay rates in Lr(R+3) for r≥1 of global strong solutions to the nematic liquid crystal flows in R+3, provided the initial data has small L3(R+3)norm.
 Abstract: Publication date: Available online 6 March 2019Source: Journal of Differential EquationsAuthor(s): Jinrui Huang, Changyou Wang, Huanyao Wen In this paper, we obtain optimal timedecay rates in Lr(R+3) for r≥1 of global strong solutions to the nematic liquid crystal flows in R+3, provided the initial data has small L3(R+3)norm.
 Regularity for multiphase variational problems
 Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Cristiana De Filippis, Jehan Oh We prove C1,νregularity for local minimizers of the multiphase energy:w↦∫Ω Dw p+a(x) Dw q+b(x) Dw sdx, under sharp assumptions relating the couples (p,q) and (p,s) to the Hölder exponents of the modulating coefficients a(⋅) and b(⋅), respectively.
 Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Cristiana De Filippis, Jehan Oh We prove C1,νregularity for local minimizers of the multiphase energy:w↦∫Ω Dw p+a(x) Dw q+b(x) Dw sdx, under sharp assumptions relating the couples (p,q) and (p,s) to the Hölder exponents of the modulating coefficients a(⋅) and b(⋅), respectively.
 The 2D Boussinesq equations with vertical dissipation and linear stability
of shear flows Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Lizheng Tao, Jiahong Wu This paper studies the linear stability of a steadystate solution with the velocity being a shear flow to the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations model many fluid phenomena when the Boussinesq approximation applies such as the RayleighBenard convection, atmospheric fronts and oceanic circulation. The vertically dissipative 2D Boussinesq equations model geophysical fluids in certain physical regimes. Whether or not the vertical dissipation can damp perturbations near the equilibrium with the velocity being a shear and the temperature being zero is an important but difficult problem. Assuming the spatial domain is periodic in the horizontal direction and halfline in the vertical direction with no flux boundary condition, we show that any perturbation satisfying the linearized equation around this equilibrium is infinitely smooth in the x−variable and decays exponentially in time and in the horizontal Fourier mode, even though the linearized system involves only vertical dissipation.
 Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Lizheng Tao, Jiahong Wu This paper studies the linear stability of a steadystate solution with the velocity being a shear flow to the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations model many fluid phenomena when the Boussinesq approximation applies such as the RayleighBenard convection, atmospheric fronts and oceanic circulation. The vertically dissipative 2D Boussinesq equations model geophysical fluids in certain physical regimes. Whether or not the vertical dissipation can damp perturbations near the equilibrium with the velocity being a shear and the temperature being zero is an important but difficult problem. Assuming the spatial domain is periodic in the horizontal direction and halfline in the vertical direction with no flux boundary condition, we show that any perturbation satisfying the linearized equation around this equilibrium is infinitely smooth in the x−variable and decays exponentially in time and in the horizontal Fourier mode, even though the linearized system involves only vertical dissipation.
 Bifurcations of small limit cycles in Liénard systems with cubic
restoring terms Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Yun Tian, Maoan Han, Fangfang Xu In this paper, we study bifurcations of smallamplitude limit cycles of Liénard systems of the form x˙=y−F(x), y˙=−g(x), where g(x) is a cubic polynomial, and F(x) is a smooth or piecewise smooth polynomial of degree n. By using involutions, we obtain sharp upper bounds of the number of smallamplitude limit cycles produced around a singular point for some systems of this type.
 Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Yun Tian, Maoan Han, Fangfang Xu In this paper, we study bifurcations of smallamplitude limit cycles of Liénard systems of the form x˙=y−F(x), y˙=−g(x), where g(x) is a cubic polynomial, and F(x) is a smooth or piecewise smooth polynomial of degree n. By using involutions, we obtain sharp upper bounds of the number of smallamplitude limit cycles produced around a singular point for some systems of this type.
 Robin eigenvalues on domains with peaks
 Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Hynek Kovařík, Konstantin Pankrashkin Let Ω⊂RN, N≥2, be a bounded domain with an outward powerlike peak which is assumed not too sharp in a suitable sense. We consider the Laplacian u↦−Δu in Ω with the Robin boundary condition ∂nu=αu on ∂Ω with ∂n being the outward normal derivative and α>0 being a parameter. We show that for large α the associated eigenvalues Ej(α) behave as Ej(α)∼−ϵjαν, where ν>2 and ϵj>0 depend on the dimension and the peak geometry. This is in contrast with the wellknown estimate Ej(α)=O(α2) for the Lipschitz domains.
 Abstract: Publication date: Available online 5 March 2019Source: Journal of Differential EquationsAuthor(s): Hynek Kovařík, Konstantin Pankrashkin Let Ω⊂RN, N≥2, be a bounded domain with an outward powerlike peak which is assumed not too sharp in a suitable sense. We consider the Laplacian u↦−Δu in Ω with the Robin boundary condition ∂nu=αu on ∂Ω with ∂n being the outward normal derivative and α>0 being a parameter. We show that for large α the associated eigenvalues Ej(α) behave as Ej(α)∼−ϵjαν, where ν>2 and ϵj>0 depend on the dimension and the peak geometry. This is in contrast with the wellknown estimate Ej(α)=O(α2) for the Lipschitz domains.
 The Cauchy problem for shallow water waves of large amplitude in Besov
space Abstract: Publication date: Available online 2 March 2019Source: Journal of Differential EquationsAuthor(s): Lili Fan, Wei Yan In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local wellposedness in Besov space B2,13/2. Then, we give a blowup criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.
 Abstract: Publication date: Available online 2 March 2019Source: Journal of Differential EquationsAuthor(s): Lili Fan, Wei Yan In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local wellposedness in Besov space B2,13/2. Then, we give a blowup criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.
 Principle of linearized stability and instability for parabolic partial
differential equations with statedependent delay Abstract: Publication date: Available online 2 March 2019Source: Journal of Differential EquationsAuthor(s): Yunfei Lv, Yongzhen Pei, Rong Yuan In this paper, the stability properties of a parabolic partial differential equation with statedependent delay are investigated by the heuristic approach. The previous works [1], [2] obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions, and resolved the problem of linearization for this equation. Here, we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation, and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution. By a biological application, we finally verify all hypotheses for an age structured diffusive model with statedependent delay and consider its stability behavior.
 Abstract: Publication date: Available online 2 March 2019Source: Journal of Differential EquationsAuthor(s): Yunfei Lv, Yongzhen Pei, Rong Yuan In this paper, the stability properties of a parabolic partial differential equation with statedependent delay are investigated by the heuristic approach. The previous works [1], [2] obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions, and resolved the problem of linearization for this equation. Here, we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation, and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution. By a biological application, we finally verify all hypotheses for an age structured diffusive model with statedependent delay and consider its stability behavior.

R N
and signchanging potentials Abstract: Publication date: Available online 1 March 2019Source: Journal of Differential EquationsAuthor(s): Shibo Liu, Zhihan Zhao We obtain existence and multiplicity results for fourth order elliptic equations on RN involving uΔ(u2) and signchanging potentials. Our results generalize some recent results on this kind of problems. To study this kind of problems, we first consider the case that the potential V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential V is bounded so that the working space is exactly H2(RN), which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term uΔ(u2) in the equation.
 Abstract: Publication date: Available online 1 March 2019Source: Journal of Differential EquationsAuthor(s): Shibo Liu, Zhihan Zhao We obtain existence and multiplicity results for fourth order elliptic equations on RN involving uΔ(u2) and signchanging potentials. Our results generalize some recent results on this kind of problems. To study this kind of problems, we first consider the case that the potential V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential V is bounded so that the working space is exactly H2(RN), which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term uΔ(u2) in the equation.
 Nonlocal dispersal equations in timeperiodic media: Principal spectral
theory, limiting properties and longtime dynamics Abstract: Publication date: Available online 26 February 2019Source: Journal of Differential EquationsAuthor(s): Zhongwei Shen, HoangHung Vo The present paper is devoted to the investigation of the following nonlocal dispersal equationut(t,x)=Dσm[∫ΩJσ(x−y)u(t,y)dy−u(t,x)]+f(t,x,u(t,x)),t>0,x∈Ω‾, where Ω⊂RN is a bounded and connected domain with smooth boundary, m∈[0,2) is the cost parameter, D>0 is the dispersal rate, σ>0 characterizes the dispersal range, Jσ=1σNJ(⋅σ) is the scaled dispersal kernel, and f is a timeperiodic nonlinear function of generalized KPP type. This paper is a continuation of the works of Berestycki et al. [3], [4], where f was assumed to be timeindependent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at u≡0. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important supinf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate D and the dispersal range characterized by σ on the principal spectrum point and the positive timeperiodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive timeperiodic solution as D,σ→0+ or ∞. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual L2 variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the timedependence of the nonlinearity f. Finally, we establish the maximum principle for timeperiodic nonlocal dispersal operators.
 Abstract: Publication date: Available online 26 February 2019Source: Journal of Differential EquationsAuthor(s): Zhongwei Shen, HoangHung Vo The present paper is devoted to the investigation of the following nonlocal dispersal equationut(t,x)=Dσm[∫ΩJσ(x−y)u(t,y)dy−u(t,x)]+f(t,x,u(t,x)),t>0,x∈Ω‾, where Ω⊂RN is a bounded and connected domain with smooth boundary, m∈[0,2) is the cost parameter, D>0 is the dispersal rate, σ>0 characterizes the dispersal range, Jσ=1σNJ(⋅σ) is the scaled dispersal kernel, and f is a timeperiodic nonlinear function of generalized KPP type. This paper is a continuation of the works of Berestycki et al. [3], [4], where f was assumed to be timeindependent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at u≡0. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important supinf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate D and the dispersal range characterized by σ on the principal spectrum point and the positive timeperiodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive timeperiodic solution as D,σ→0+ or ∞. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual L2 variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the timedependence of the nonlinearity f. Finally, we establish the maximum principle for timeperiodic nonlocal dispersal operators.
 On the influence of gravity on densitydependent incompressible periodic
fluids Abstract: Publication date: Available online 26 February 2019Source: Journal of Differential EquationsAuthor(s): VanSang Ngo, Stefano Scrobogna The present work is devoted to the analysis of densitydependent, incompressible fluids in a 3D torus, when the Froude number ε goes to zero. We consider the very general case where the initial data do not have a zero horizontal average, where we only have smoothing effect on the velocity but not on the density and where we can have resonant phenomena on the domain. We explicitly determine the limit system when ε→0 and prove its global wellposedness. Finally, we prove that for large initial data, the densitydependent, incompressible fluid system is globally wellposed, provided that ε is small enough.
 Abstract: Publication date: Available online 26 February 2019Source: Journal of Differential EquationsAuthor(s): VanSang Ngo, Stefano Scrobogna The present work is devoted to the analysis of densitydependent, incompressible fluids in a 3D torus, when the Froude number ε goes to zero. We consider the very general case where the initial data do not have a zero horizontal average, where we only have smoothing effect on the velocity but not on the density and where we can have resonant phenomena on the domain. We explicitly determine the limit system when ε→0 and prove its global wellposedness. Finally, we prove that for large initial data, the densitydependent, incompressible fluid system is globally wellposed, provided that ε is small enough.
 Travelling wave solutions for a nonlocal evolutionaryepidemic system
 Abstract: Publication date: Available online 26 February 2019Source: Journal of Differential EquationsAuthor(s): L. Abi Rizk, J.B. Burie, A. Ducrot In this work we study the travelling wave solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plantpathogen interaction. Here the mutation process is described using a nonlocal convolution operator in the phenotype space.Using dynamical system ideas coupled with refined estimates on the asymptotic behaviour of the profiles, we prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ode system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c⋆, provided some parameters condition expressed using the principle eigenvalue of some integral operator. It is also used to prove that any travelling wave solution connects two determined stationary states.
 Abstract: Publication date: Available online 26 February 2019Source: Journal of Differential EquationsAuthor(s): L. Abi Rizk, J.B. Burie, A. Ducrot In this work we study the travelling wave solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plantpathogen interaction. Here the mutation process is described using a nonlocal convolution operator in the phenotype space.Using dynamical system ideas coupled with refined estimates on the asymptotic behaviour of the profiles, we prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ode system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c⋆, provided some parameters condition expressed using the principle eigenvalue of some integral operator. It is also used to prove that any travelling wave solution connects two determined stationary states.
 Invariant Cantor manifolds of quasiperiodic solutions for the derivative
nonlinear Schrödinger equation Abstract: Publication date: Available online 25 February 2019Source: Journal of Differential EquationsAuthor(s): Meina Gao, Jianjun Liu This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditionsiut+uxx+i(f(x,u,u¯))x=0,x∈T:=R/2πZ, where f is an analytic function of the formf(x,u,u¯)=μ u 2u+f≥4(x,u,u¯),0≠μ∈R, and f≥4(x,u,u¯) denotes terms of order at least four in u,u¯. We show the above equation possesses Cantor families of smooth quasiperiodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields.
 Abstract: Publication date: Available online 25 February 2019Source: Journal of Differential EquationsAuthor(s): Meina Gao, Jianjun Liu This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditionsiut+uxx+i(f(x,u,u¯))x=0,x∈T:=R/2πZ, where f is an analytic function of the formf(x,u,u¯)=μ u 2u+f≥4(x,u,u¯),0≠μ∈R, and f≥4(x,u,u¯) denotes terms of order at least four in u,u¯. We show the above equation possesses Cantor families of smooth quasiperiodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields.
 Entire solutions to reactiondiffusion equations in multiple halflines
with a junction Abstract: Publication date: Available online 25 February 2019Source: Journal of Differential EquationsAuthor(s): Shuichi Jimbo, Yoshihisa Morita There exists a traveling front wave to a bistable reactiondiffusion equation in a whole line under a certain condition of reaction term f(u). We deal with the bistable reactiondiffusion equation with the same f(u) in a domain Ω which is a graph of special type, that is, a union of halflines starting at a common point, so the domain has a unique junction of the halflines. The aim of our study is to show the existence of nontrivial entire solutions, which are classical solutions defined for all (x,t)∈Ω×R. We prove that there are entire solutions which converge to the front waves in some of halflines and converge to zero in the remaining halflines as t→−∞. We also give a condition under that the entire solutions exhibit the blocking of the front propagation. This blocking is caused by the emergence of stationary solutions. The stability/instability of the stationary solutions are proved.
 Abstract: Publication date: Available online 25 February 2019Source: Journal of Differential EquationsAuthor(s): Shuichi Jimbo, Yoshihisa Morita There exists a traveling front wave to a bistable reactiondiffusion equation in a whole line under a certain condition of reaction term f(u). We deal with the bistable reactiondiffusion equation with the same f(u) in a domain Ω which is a graph of special type, that is, a union of halflines starting at a common point, so the domain has a unique junction of the halflines. The aim of our study is to show the existence of nontrivial entire solutions, which are classical solutions defined for all (x,t)∈Ω×R. We prove that there are entire solutions which converge to the front waves in some of halflines and converge to zero in the remaining halflines as t→−∞. We also give a condition under that the entire solutions exhibit the blocking of the front propagation. This blocking is caused by the emergence of stationary solutions. The stability/instability of the stationary solutions are proved.
 Dichotomous solutions for semilinear illposed equations with sectorially
dichotomous operator Abstract: Publication date: Available online 25 February 2019Source: Journal of Differential EquationsAuthor(s): Lianwang Deng, Dongmei Xiao In this paper, we study a class of semilinear illposed equations with sectorially dichotomous operator S on Banach space Z. Firstly we give a direct sum decomposition of Z, Z+⊕Z−=Z, corresponding to spectrum of S such that hyperbolic bisectorial operator S can be split into two sectorial operators S Z+ and −S Z− on Z+ and Z−, respectively. Then we construct the intermediate spaces between whole space Z and domain D(S) of sectorially dichotomous operator S. Following ElBialy's works, we propose the dichotomous initial condition for this semilinear illposed equation, and obtain the existence, uniqueness, continuous dependence on the dichotomous initial value, regularity and Zαestimate of dichotomous solutions. As applications of the results, we give the existence and uniqueness of local solutions for an elliptic PDE in infinite cylindrical domain and an abstract semilinear illposed equation with nondense domain.
 Abstract: Publication date: Available online 25 February 2019Source: Journal of Differential EquationsAuthor(s): Lianwang Deng, Dongmei Xiao In this paper, we study a class of semilinear illposed equations with sectorially dichotomous operator S on Banach space Z. Firstly we give a direct sum decomposition of Z, Z+⊕Z−=Z, corresponding to spectrum of S such that hyperbolic bisectorial operator S can be split into two sectorial operators S Z+ and −S Z− on Z+ and Z−, respectively. Then we construct the intermediate spaces between whole space Z and domain D(S) of sectorially dichotomous operator S. Following ElBialy's works, we propose the dichotomous initial condition for this semilinear illposed equation, and obtain the existence, uniqueness, continuous dependence on the dichotomous initial value, regularity and Zαestimate of dichotomous solutions. As applications of the results, we give the existence and uniqueness of local solutions for an elliptic PDE in infinite cylindrical domain and an abstract semilinear illposed equation with nondense domain.
 Nullcontrollability properties of the wave equation with a second order
memory term Abstract: Publication date: Available online 22 February 2019Source: Journal of Differential EquationsAuthor(s): Umberto Biccari, Sorin Micu We study the internal controllability of a wave equation with memory in the principal part, defined on the onedimensional torus T=R/2πZ. We assume that the control is acting on an open subset ω(t)⊂T, which is moving with a constant velocity c∈R∖{−1,0,1}. The main result of the paper shows that the equation is null controllable in a sufficiently large time T and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated with our problem and from the application of the classical moment method.
 Abstract: Publication date: Available online 22 February 2019Source: Journal of Differential EquationsAuthor(s): Umberto Biccari, Sorin Micu We study the internal controllability of a wave equation with memory in the principal part, defined on the onedimensional torus T=R/2πZ. We assume that the control is acting on an open subset ω(t)⊂T, which is moving with a constant velocity c∈R∖{−1,0,1}. The main result of the paper shows that the equation is null controllable in a sufficiently large time T and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated with our problem and from the application of the classical moment method.
 Long time dynamics of Schrödinger and wave equations on flat tori
 Abstract: Publication date: Available online 21 February 2019Source: Journal of Differential EquationsAuthor(s): M. Berti, A. Maspero We consider a class of linear time dependent Schrödinger equations and quasiperiodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a tϵ (∀ϵ>0) upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasiperiodic solutions.Both results are based on “clusterization properties” of the eigenvalues of the Laplacian on a flat torus and on suitable “separation properties” of the singular sites of Schrödinger and wave operators, which are integers, in space–time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [20] to control the speed of growth of the Sobolev norms, and Berti–Corsi–Procesi abstract Nash–Moser theorem [8] to construct quasiperiodic solutions.
 Abstract: Publication date: Available online 21 February 2019Source: Journal of Differential EquationsAuthor(s): M. Berti, A. Maspero We consider a class of linear time dependent Schrödinger equations and quasiperiodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a tϵ (∀ϵ>0) upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasiperiodic solutions.Both results are based on “clusterization properties” of the eigenvalues of the Laplacian on a flat torus and on suitable “separation properties” of the singular sites of Schrödinger and wave operators, which are integers, in space–time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [20] to control the speed of growth of the Sobolev norms, and Berti–Corsi–Procesi abstract Nash–Moser theorem [8] to construct quasiperiodic solutions.
 Global existence and convergence rates to a chemotaxisfluids system with
mixed boundary conditions Abstract: Publication date: Available online 19 February 2019Source: Journal of Differential EquationsAuthor(s): Yingping Peng, Zhaoyin Xiang In this paper, we investigate the large time behavior of strong solutions to a chemotaxisfluids system in an unbounded domain with mixed boundary conditions. Based on the anisotropic Lp technique, the elliptic estimates and Stokes estimates, we first establish the global existence of strong solution around the equilibrium state (0,csatn,0) with the help of the continuity arguments, where csatn is the saturation value of oxygen inside the fluid. Then we use De Giorgi's technique and energy method to show that such a solution will converge to (0,csatn,0) with an explicit convergence rate in the chemotaxisfree case. Our assumptions and results are consistent with the experimental descriptions and the numerical analysis. The novelty here consists of deriving some new elliptic estimates and Stokes estimates, and choosing a suitable weight in De Giorgi's technique to deal with the mixed boundary conditions.
 Abstract: Publication date: Available online 19 February 2019Source: Journal of Differential EquationsAuthor(s): Yingping Peng, Zhaoyin Xiang In this paper, we investigate the large time behavior of strong solutions to a chemotaxisfluids system in an unbounded domain with mixed boundary conditions. Based on the anisotropic Lp technique, the elliptic estimates and Stokes estimates, we first establish the global existence of strong solution around the equilibrium state (0,csatn,0) with the help of the continuity arguments, where csatn is the saturation value of oxygen inside the fluid. Then we use De Giorgi's technique and energy method to show that such a solution will converge to (0,csatn,0) with an explicit convergence rate in the chemotaxisfree case. Our assumptions and results are consistent with the experimental descriptions and the numerical analysis. The novelty here consists of deriving some new elliptic estimates and Stokes estimates, and choosing a suitable weight in De Giorgi's technique to deal with the mixed boundary conditions.
 Renormalized solutions to parabolic equations in time and space dependent
anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon
 Abstract: Publication date: Available online 19 February 2019Source: Journal of Differential EquationsAuthor(s): Iwona Chlebicka, Piotr Gwiazda, Anna ZatorskaGoldstein The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in RN. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic Nfunction M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large ξ and small changes of time and space variables. Its instances are logHölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (possibly changing in time). New delicate approximationintime result is proven and applied in the construction of renormalized solutions.
 Abstract: Publication date: Available online 19 February 2019Source: Journal of Differential EquationsAuthor(s): Iwona Chlebicka, Piotr Gwiazda, Anna ZatorskaGoldstein The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in RN. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic Nfunction M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large ξ and small changes of time and space variables. Its instances are logHölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (possibly changing in time). New delicate approximationintime result is proven and applied in the construction of renormalized solutions.
 New singular standing wave solutions of the nonlinear Schrodinger equation
 Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): W.C. Troy We prove existence, and asymptotic behavior as r→∞, of a family of singular solutions of(1)y″+2ry′+y y p−1−y=0,0
 Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): W.C. Troy We prove existence, and asymptotic behavior as r→∞, of a family of singular solutions of(1)y″+2ry′+y y p−1−y=0,0
 Nonlinear Schrödinger equation in the Bopp–Podolsky electrodynamics:
Solutions in the electrostatic case Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): Pietro d'Avenia, Gaetano Siciliano We study the following nonlinear Schrödinger–Bopp–Podolsky system{−Δu+ωu+q2ϕu= u p−2u−Δϕ+a2Δ2ϕ=4πu2 inR3 with a,ω>0. We prove existence and nonexistence results depending on the parameters q,p. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schrödinger–Poisson system as a→0.
 Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): Pietro d'Avenia, Gaetano Siciliano We study the following nonlinear Schrödinger–Bopp–Podolsky system{−Δu+ωu+q2ϕu= u p−2u−Δϕ+a2Δ2ϕ=4πu2 inR3 with a,ω>0. We prove existence and nonexistence results depending on the parameters q,p. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schrödinger–Poisson system as a→0.
 Global wellposedness of the freesurface incompressible Euler equations
with damping Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): Jiali Lian We consider a layer of an incompressible inviscid fluid, bounded below by a fixed solid boundary and above by a free moving boundary, in a horizontally periodic setting. The fluid dynamics is governed by the gravitydriven incompressible Euler equations with damping, and the effect of surface tension is neglected on the free surface. We prove that the problem is globally wellposed for the small initial data and that solutions decay to the equilibrium at an almost exponential rate.
 Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): Jiali Lian We consider a layer of an incompressible inviscid fluid, bounded below by a fixed solid boundary and above by a free moving boundary, in a horizontally periodic setting. The fluid dynamics is governed by the gravitydriven incompressible Euler equations with damping, and the effect of surface tension is neglected on the free surface. We prove that the problem is globally wellposed for the small initial data and that solutions decay to the equilibrium at an almost exponential rate.
 High order Melnikov method: Theory and application
 Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): Fengjuan Chen, Qiudong Wang Let D(t0,ε) be the splitting distance of the stable and unstable manifold of a timeperiodic second order equation. We expand D(t0,ε) as a formal power series in ε asD(t0,ε)=E0(t0)+εE1(t0)+⋯+εnEn(t0)+⋯. In this paper we derive an explicit integral formula for E1(t0). We also evaluate E1(t0) to prove the existence of homoclinic tangles for an equation to which the Poincaré/Melnikov method fails to apply.
 Abstract: Publication date: Available online 14 February 2019Source: Journal of Differential EquationsAuthor(s): Fengjuan Chen, Qiudong Wang Let D(t0,ε) be the splitting distance of the stable and unstable manifold of a timeperiodic second order equation. We expand D(t0,ε) as a formal power series in ε asD(t0,ε)=E0(t0)+εE1(t0)+⋯+εnEn(t0)+⋯. In this paper we derive an explicit integral formula for E1(t0). We also evaluate E1(t0) to prove the existence of homoclinic tangles for an equation to which the Poincaré/Melnikov method fails to apply.
 Blowup phenomena for linearly perturbed Yamabe problem on manifolds with
umbilic boundary Abstract: Publication date: Available online 12 February 2019Source: Journal of Differential EquationsAuthor(s): Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia We build blowingup solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n≥11.
 Abstract: Publication date: Available online 12 February 2019Source: Journal of Differential EquationsAuthor(s): Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia We build blowingup solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n≥11.
 Existence for a kHessian equation involving supercritical growth
 Abstract: Publication date: Available online 12 February 2019Source: Journal of Differential EquationsAuthor(s): José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla In this paper we use variational techniques to give existence results for the problem{Sk[u]=f(x,−u)inΩu
 Abstract: Publication date: Available online 12 February 2019Source: Journal of Differential EquationsAuthor(s): José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla In this paper we use variational techniques to give existence results for the problem{Sk[u]=f(x,−u)inΩu
 Boundary layers for the subcritical modes of the 3D primitive equations in
a cube Abstract: Publication date: Available online 6 February 2019Source: Journal of Differential EquationsAuthor(s): Makram Hamouda, Daozhi Han, ChangYeol Jung, Krutika Tawri, Roger Temam In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interactioncorner layers. The boundary layer correctors are determined by a phenomenological study reminiscent of the Prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study.
 Abstract: Publication date: Available online 6 February 2019Source: Journal of Differential EquationsAuthor(s): Makram Hamouda, Daozhi Han, ChangYeol Jung, Krutika Tawri, Roger Temam In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interactioncorner layers. The boundary layer correctors are determined by a phenomenological study reminiscent of the Prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study.
 Wong–Zakai approximation for the stochastic
Landau–Lifshitz–Gilbert equations Abstract: Publication date: Available online 5 February 2019Source: Journal of Differential EquationsAuthor(s): Zdzisław Brzeźniak, Utpal Manna, Debopriya Mukherjee In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with nonzero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a twodimensional unit sphere S2 in R3. The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.
 Abstract: Publication date: Available online 5 February 2019Source: Journal of Differential EquationsAuthor(s): Zdzisław Brzeźniak, Utpal Manna, Debopriya Mukherjee In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with nonzero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a twodimensional unit sphere S2 in R3. The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.
 Existence and regularity of the solutions of some singular
Monge–Ampère equations Abstract: Publication date: Available online 1 February 2019Source: Journal of Differential EquationsAuthor(s): Haodi Chen, Genggeng Huang In this paper, we investigate the following singular Monge–Ampère equation(0.1){detD2u=1(Hu)n+k+2u⁎kinΩ⊂⊂Rn,u=0,on∂Ω where k≥0, H
 Abstract: Publication date: Available online 1 February 2019Source: Journal of Differential EquationsAuthor(s): Haodi Chen, Genggeng Huang In this paper, we investigate the following singular Monge–Ampère equation(0.1){detD2u=1(Hu)n+k+2u⁎kinΩ⊂⊂Rn,u=0,on∂Ω where k≥0, H
 Global existence and boundedness of solutions to a chemotaxis system with
singular sensitivity and logistictype source Abstract: Publication date: Available online 1 February 2019Source: Journal of Differential EquationsAuthor(s): Xiangdong Zhao, Sining Zheng We consider the fully parabolic Keller–Segel system with singular sensitivity and logistictype source: ut=Δu−χ∇⋅(uv∇v)+ru−μuk, vt=Δv−v+u under the nonflux boundary conditions in a smooth bounded convex domain Ω⊂Rn, χ,r,μ>0, k>1. A global very weak solution for the system with n≥2 is obtained under one of the following conditions: (i) r>χ24 for 0max{χ24(1−p02),χ−1} for χ>2 with p0=4(k−1)4+(2−k)kχ2 if k∈(2−1n,2]; (ii) χ22. Furthermore, this global very weak solution should be globally bounded in fact provided rμ and the initial data ‖u0‖L2(Ω),‖
 Abstract: Publication date: Available online 1 February 2019Source: Journal of Differential EquationsAuthor(s): Xiangdong Zhao, Sining Zheng We consider the fully parabolic Keller–Segel system with singular sensitivity and logistictype source: ut=Δu−χ∇⋅(uv∇v)+ru−μuk, vt=Δv−v+u under the nonflux boundary conditions in a smooth bounded convex domain Ω⊂Rn, χ,r,μ>0, k>1. A global very weak solution for the system with n≥2 is obtained under one of the following conditions: (i) r>χ24 for 0max{χ24(1−p02),χ−1} for χ>2 with p0=4(k−1)4+(2−k)kχ2 if k∈(2−1n,2]; (ii) χ22. Furthermore, this global very weak solution should be globally bounded in fact provided rμ and the initial data ‖u0‖L2(Ω),‖