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Abstract: We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equations with a sign-changing potential V that converges at infinity and a nonlinear term being asymptotically linear at infinity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is that, due to the convergency of V at infinity, the classical methods such as periodic translation technique and compact inclusion method cannot be employed directly to deal with the lack of compactness of the Cerami sequence. New techniques are developed in this work to overcome these two major difficulties. This enables us to establish the existence of a ground state solution and derive a necessary and sufficient condition for a special case. To the best of our knowledge, this is the first attempt in the literature on the existence of a ground state solution for the strongly indefinite problem under no periodicity condition on the bounded potential and the nonlinear term being asymptotically linear at infinity. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation. PubDate: 2019-03-12 DOI: 10.1007/s10884-019-09743-4
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Abstract: In this paper, we investigate a delayed reaction–diffusion–advection equation, which models the population dynamics in the advective heterogeneous environment. The existence of the nonconstant positive steady state and associated Hopf bifurcation are obtained. A weighted inner product associated with the advection rate is introduced to compute the normal forms, which is the main difference between Hopf bifurcation for delayed reaction–diffusion–advection model and that for delayed reaction–diffusion model. Moreover, we find that the spatial scale and advection can affect Hopf bifurcation in the heterogenous environment. PubDate: 2019-03-08 DOI: 10.1007/s10884-019-09739-0
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Abstract: We consider a model of multi-species competition in the chemostat that includes demographic stochasticity and discrete delays. We prove that for any given initial data, there exists a unique global positive solution for the stochastic delayed system. By employing the method of stochastic Lyapunov functionals, we determine the asymptotic behaviors of the stochastic solution and show that although the sample path fluctuate, it remains positive and the expected time average of the distance between the stochastic solution and the equilibrium of the associated deterministic delayed chemostat model is eventually small, i.e. we obtain an analogue of the competition exclusion principle when the noise intensities are relatively small. Numerical simulations are carried out to illustrate our theoretical results. PubDate: 2019-03-07 DOI: 10.1007/s10884-019-09741-6
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Abstract: In this paper, we investigate the dynamics of population inhabiting a favorable environment surrounded by a region where survival is impossible. We use a parabolic partial differential equation with nonlinearity changing signs to describe the dynamics. By using a recent result on the existence of nonzero fixed point for r-nowhere normal-outward compact maps, we establish new results on existence of nonzero nonnegative or positive steady-state solutions of such boundary value problems. Additionally, we obtain the convergence of positive solutions of such parabolic boundary value problems in the Banach space \(C^1\) , which is different from the previous results that considered weak convergence or uniform convergence. PubDate: 2019-03-07 DOI: 10.1007/s10884-019-09742-5
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Abstract: This paper concerns the Cauchy problem of the compressible non-resistive magnetohydrodynamic equations on the whole two-dimensional space with vacuum as far field density. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, we prove that there exists a unique local strong solution provided the initial density and the initial magnetic field decay not too slow at infinity. In particular, there is no need to require any Cho–Choe–Kim type compatibility conditions. PubDate: 2019-03-06 DOI: 10.1007/s10884-019-09740-7
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Abstract: We show that solutions of the periodic KdV equations $$\begin{aligned} u_t+\gamma u +u_{xxx}+uu_x=f, \end{aligned}$$ are asymptotically determined by their values at three points. That is if there exists \(x_1,x_2,x_3\) such that \(0< x_3-x_2<<x_3-x_1<<1\) and $$\begin{aligned} \lim _{t\rightarrow +\infty } u_1(t,x_j)-u_2(t,x_j) =0, \; \mathrm{for} \; j=1,2,3, \end{aligned}$$ for two solutions \(u_1,u_2\) of the KdV equation above, then $$\begin{aligned} \lim _{t\rightarrow +\infty } u_1(t)-u_2(t) _{H^1}=0. \end{aligned}$$ PubDate: 2019-02-26 DOI: 10.1007/s10884-019-09737-2
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Abstract: The Hopf bifurcations for the classical Gray–Scott system and the Schnakenberg system in an one-dimensional interval are considered. For each system, the existence of time-periodic solutions near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of limit cycles are determined and it is shown that the Hopf bifurcation is supercritical for the Schnakenberg system, and hence the bifurcating periodic solutions are linearly stable. For the Gray–Scott system, there is a critical feeding rate, when the feeding rate of the system is greater than this critical feeding rate, the Hopf bifurcation is supercritical which implies the bifurcating periodic solutions are linearly stable, while when the feeding rate is smaller than this critical feeding rate, the Hopf bifurcation is subcritical, implying the bifurcating periodic solutions are linearly unstable. PubDate: 2019-02-22 DOI: 10.1007/s10884-019-09736-3
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Abstract: In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equation under nonlocal perturbation $$\begin{aligned} u_{tt}-\triangle u +M_\xi u +\left( \int _{\mathbb {T}^d} u^2 dx\right) u=0,\quad t\in \mathbb {R},\ x\in \mathbb {T}^d\ \end{aligned}$$ where \(M_\xi \) is a real Fourier multiplier. PubDate: 2019-02-22 DOI: 10.1007/s10884-019-09738-1
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Abstract: In this paper, we have solved the center-focus problem for a family of high-order polynomial differential systems. We give the sufficient and necessary conditions for the corresponding periodic differential equation to have a center and also show that this center is a CC-center. PubDate: 2019-02-21 DOI: 10.1007/s10884-019-09735-4
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Abstract: In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations \(\partial _{t}^{2} u - \varDelta u + \nu (-\varDelta )^{\sigma } \partial _{t} u=0\) , where \(\nu >0\) and \(0< \sigma \le 1\) . Our result is the approximation formula of the solution by a constant multiple of a special function as \(t \rightarrow \infty \) , which states that the asymptotic profiles of the solutions are classified into 5 patterns depending on the values \(\nu \) and \(\sigma \) . Here we emphasize that our main interest of the paper is in the case \(\sigma \in (\frac{1}{2},1)\) . PubDate: 2019-02-11 DOI: 10.1007/s10884-019-09731-8
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Abstract: By applying the random topological degree theory we develop the methods of random multivalent guiding functions and use them for the study of periodic solutions for random differential equations in finite dimensional spaces. PubDate: 2019-02-11 DOI: 10.1007/s10884-019-09734-5
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Abstract: In this paper we consider a system of parabolic reaction–diffusion equations with strong competition and two related scalar reaction–diffusion equations. We show that in certain space periodic media with large periods, there exist periodic, non-constant, non-trivial, stable stationary states. We compare our results with already known results about the existence and nonexistence of such solutions. Finally, we provide ecological interpretations for these results. PubDate: 2019-02-11 DOI: 10.1007/s10884-019-09732-7
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Abstract: The possibility of rate-induced tipping (R-tipping) away from an attracting fixed point has been thoroughly explored in 1-dimensional systems. In these systems, it is impossible to have R-tipping away from a path of quasi-stable equilibria that is forward basin stable (FBS), but R-tipping is guaranteed for paths that are non-FBS of a certain type. We will investigate whether these results carry over to multi-dimensional systems. In particular, we will show that the same conditions guaranteeing R-tipping in 1-dimension also guarantee R-tipping in higher dimensions; however, it is possible to have R-tipping away from a path that is FBS even in 2-dimensional systems. We will propose a different condition, forward inflowing stability (FIS), which we show is sufficient to prevent R-tipping in all dimensions. The condition, while natural, is difficult to verify in concrete examples. Monotone systems are a class for which FIS is implied by an easily verifiable condition. As a result, we see how the additional structure of these systems makes predicting the possibility of R-tipping straightforward in a fashion similar to 1-dimension. In particular, we will prove that the FBS and FIS conditions in monotone systems reduce to comparing the relative positions of equilibria over time. An example of a monotone system is given that demonstrates how these ideas are applied to determine exactly when R-tipping is possible. PubDate: 2019-02-11 DOI: 10.1007/s10884-019-09730-9
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Abstract: The original version of this article unfortunately contained a mistake in Eq. 1.4. The correct equation is given below. PubDate: 2019-02-06 DOI: 10.1007/s10884-019-09728-3
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Abstract: We establish the robustness of the notion of an exponential dichotomy for a nonautonomous linear delay equation. We consider the general cases of equations satisfying the Carathéodory conditions and of noninvertible evolution families defined by the linear equations. The stable and unstable spaces of the perturbed equations are obtained as images of linear operators that in their turn are fixed points of appropriate contraction maps. PubDate: 2019-02-06 DOI: 10.1007/s10884-019-09733-6
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Abstract: This article determines all possible (proper as well as pseudo) equilibrium arrangements under a repulsive power law force of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by \(V_s(r)= s^{-1}\left( r^{-s}-1 \right) \) when the real parameter \(s \ne 0\) , and by \(V_0(r) := \lim _{s\rightarrow 0}V_s(r) = -\ln r\) ; here, r is the chordal distance between the particles in the pair. The bifurcation diagram which exhibits all these equilibrium arrangements together as functions of s features three obvious “universal” equilibria, which do not depend on s, and two not-so-obvious continuous families of s-dependent non-universal isosceles triangular equilibria. The two continuous families of non-universal equilibria are disconnected, yet they bifurcate off of a common universal limiting equilibrium (the equilateral triangular configuration), at \(s=-4\) , where the graph of the total Riesz energy of the 3-particle configurations has the shape of a “monkey saddle.” In addition, one of the families of non-universal equilibria also bifurcates off of another universal equilibrium (the antipodal arrangement), at \(s=-2\) . While the bifurcation at \(s=-4\) is analytical, the one at \(s=-2\) is not. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N-point equilibrium problems on \({\mathbb {S}}^d\) for small N. PubDate: 2019-01-30 DOI: 10.1007/s10884-019-09729-2
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Abstract: We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: $$\begin{aligned} \partial _tu=\partial _{xx}(u^m) + x ^{\sigma }u, \end{aligned}$$ with \(\sigma >0\) . Through this study, we show that the non-homogeneous coefficient \( x ^{\sigma }\) has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known autonomous case \(\sigma =0\) . Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent \(\sigma \) is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when \(\sigma >0\) is sufficiently small, while for \(\sigma >0\) sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates. PubDate: 2019-01-14 DOI: 10.1007/s10884-018-09727-w
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Abstract: In this article, we consider a small rigid body moving in a viscous fluid filling the whole \(\mathbb R^2\) . We assume that the diameter of the rigid body goes to 0, that the initial velocity has bounded energy and that the density of the rigid body goes to infinity. We prove that the rigid body has no influence on the limit equation by showing convergence of the solutions towards a solution of the Navier–Stokes equations in the full plane \(\mathbb {R}^{2}\) . PubDate: 2019-01-07 DOI: 10.1007/s10884-018-9718-3
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Abstract: The vanishing shear viscosity limit to an initial-boundary value problem of planar MHD equations for compressible viscous heat-conducting fluids is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at small shear viscosity, both the boundary-layer thickness and the boundary-layer corrector are discussed. PubDate: 2019-01-04 DOI: 10.1007/s10884-018-9726-3
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Abstract: In this paper, we are interested in studying the existence of the nonstationary solutions in the global attractor of the 2D Navier–Stokes equations with constant energy and enstrophy. We particularly focus on a subclass of these solutions that the geometric structures have a supplementary stability property which were called “chained ghost solutions” in Tian and Zhang (Indiana Univ Math J 64:1925–1958, 2015). By solving a Galerkin system of a particular form, we show the nonexistence of the chained ghost solutions for certain cases. PubDate: 2019-01-02 DOI: 10.1007/s10884-018-9721-8
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