Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Cristina Lois-Prados, Radu Precup The paper deals with a non-autonomous Lotka–Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel’skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expressions for the functional response of predators are discussed.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Mario Fuest The Neumann initial–boundary problem for the chemotaxis system (⋆)ut=Δu−∇⋅(u∇v)+κ( x )u−μ( x )up,0=Δv−m(t) Ω +u,m(t)≔∫Ωu(⋅,t)is studied in a ball Ω=BR(0)⊂R2, R>0 for p≥1 and sufficiently smooth functions κ,μ:[0,R]→[0,∞).We prove that whenever μ′,−κ′≥0 as well as μ(s)≤μ1s2p−2 for all s∈[0,R] and some μ1>0 then for all m0>8π there exists u0∈C0(Ω¯) with ∫Ωu0=m0...

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Dugyu Kim Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body R3∖Ω which is also moving in the direction of the axis of rotation with nonzero constant velocity −ke1. We assume that the angular velocity ω= ω e1 is also constant and the external force is given by f=divF. Then the motion is described by a variant of the stationary Navier–Stokes equations with the velocity ke1 at infinity. Our main result is the existence of at least one solution u satisfying u−ke1∈L3(Ω) for arbitrarily large F∈L3∕2(Ω). The uniqueness is also proved by assuming that ω + k +‖F‖L3∕2(Ω) is sufficiently small in comparison with the viscosity ν. Moreover, we establish several regularity results to obtain an existence theorem for weak solutions u satisfying ∇u∈L3∕2(Ω) and u−ke1∈L3(Ω).

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Ling Wan, Lan Zhang We prove the global existence and uniqueness of strong solutions for an initial boundary value problem modeling the motion of the compressible micropolar fluids in one dimensional space. Compared with former studies, we are concerned with the nonisentropic case with constant transport coefficients and the initial density is allowed to have vacuum. Our analysis is based on the nonlinear energy method and the crucial step is to derive the uniform upper and lower bounds on the ratio of the density to its initial value.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Swati Mishra, Ranjit Kumar Upadhyay Fear of predators is an important drive for predator–prey interactions, which increases survival probability but cost the overall population size of the prey. In this paper, we have extended our previous work spatiotemporal dynamics of predator–prey interactions with fear effect by introducing the cross-diffusion. The conditions for cross-diffusion-driven instability are obtained using the linear stability analysis. The standard multiple scale analysis is used to derive the amplitude equations for the excited modes near Turing bifurcation threshold by taking the cross-diffusion coefficient as a bifurcation parameter. From the stability analysis of amplitude equations, the conditions for the emergence of various ecologically realistic Turing patterns such as spot, stripe, and mixture of spots and stripes are identified. Analytical results are verified with the help of numerical simulations. Turing bifurcation diagrams are plotted taking diffusion coefficients as control parameters. The effect of the cross-diffusion coefficients on the homogeneous steady state and pattern structures of the self-diffusive model is illustrated using the simulation techniques. It is also observed that the level of fear has stabilizing effect on the cross-diffusion induced instability and spot patterns change to stripe, then a mixture of spots and stripes and finally to the labyrinthine type of patterns with an increase in the level of fear.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Daomin Cao, Jie Wan, Guodong Wang In this paper, we construct two types of vortex patch equilibria for the two-dimensional Euler equations in a disc. The first type is called the “N+1 type” equilibrium, in which a central vortex patch is surrounded by N identical patches with opposite signs, and the other type is called the “2N type” equilibrium, in which the centers of N identical positive patches and N negative patches lie evenly on a circle. The construction is performed by solving a variational problem for the vorticity in which the kinetic energy is maximized subject to some symmetry constraints, and then analyzing the asymptotic behavior as the vorticity strength goes to infinity.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Qiao Liu We study the initial boundary value problem of a simplified Ericksen–Leslie system modeling the incompressible nematic liquid crystal flows in two dimensions of space, where the equations of the velocity field are characterized by a time-dependent external force g(t) and a no-slip boundary condition, and the equations for the molecular orientation are subjected to a time-dependent Dirichlet boundary condition h(t). Based on the recently addressed well-posedness and regularity results of the system, we present a rigorous proof to show the existence of optimal distributed controls, the control-to-state operator is Fréchet differentiable and first-order necessary optimality conditions for an associated optimal control problem.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Shihe Xu, Dan Su In this paper we study a free boundary problem for tumor growth with angiogenesis and a necrotic core. The problem has two free boundaries: the outer boundary R(t) and the necrotic core boundary ρ(t). In order to receive nutrients σ, the tumor attracts blood vessels at a rate proportional to α, so that the Robin condition ∂σ∂r+α(σ−σ̄)=0 holds on the outer boundary, where σ̄ is the nutrients concentration outside the tumor. The sufficient conditions for the existence, uniqueness and stability of the stationary solution to the model are given. The results show that there exists a positive constant σ0 such that if the external nutrient supply σ̄ is greater than or equal to σ0, there exists a unique stationary solution with a necrotic core and this stationary solution is stable. Meanwhile, we also discuss the asymptotic behavior of the transient solutions when the external nutrient supply is below the threshold value σ0. The effects of the external nutrients supply and the connection between the non-necrotic stage and the necrotic stage are also discussed.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Jaume Giné, Claudia Valls In this paper we study the existence of nonlinear oscillations of a modified Leslie–Gower model around the positive equilibrium point. It is proved that at least one limit cycle can exist bifurcating from it but that this point is never a center, that is, there does not exist an infinite number of nonlinear oscillations around it.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Julieta Bollati, María F. Natale, José A. Semitiel, Domingo A. Tarzia One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet, a Neumann or a Robin type condition at fixed face x=0. Moreover, it is proved that the solution of the problem with the Robin type condition converges to the solution of the problem with the Dirichlet condition at the fixed face. Computational examples are provided.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Da Xu We study the finite element approximations to a general class of nonlinear and nonlocal hyperbolic integro-differential equations with L1 convolution kernels. The continuous time Galerkin procedures are defined and global existence of a unique discrete solution is derived. Moreover, optimal error estimates are shown in the L∞(H01(Ω))-norms. For the completely discrete scheme, linearized backward Euler method is defined and error estimates in l∞(H01(Ω))-norm are proved. Several numerical experiments are reported to confirm our theoretical findings.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Denisa Stancu-Dumitru The asymptotic behaviour of the sequence of positive solutions for a family of torsional creep-type problems involving anisotropic rapidly growing differential operators is studied in a bounded domain from the Euclidian space RN. We prove that the sequence of solutions converges uniformly on the domain to a certain distance function defined in accordance with the anisotropy of the problem.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Shihe Xu In this paper we consider a free boundary problem for tumor growth with angiogenesis and time delays in the process of proliferation. The model is established by using reaction–diffusion dynamics and taking a time delay into account. In order to get more nutrients the tumor will attract blood vessels. Assume α(t) is the rate at which the tumor attracts blood vessels, so that ∂σ∂r+α(t)(σ−σ̄)=0 holds on the boundary, where σ̄ is the concentration of nutrients externally supplied to the tumor. When α is a constant, the stability of the unique stationary solution is proved. When α depends on time, we show that (i) R(t) will remain bounded if α(t) is bounded; (ii) limt→∞R(t)=0 if limt→∞α(t)=0; (iii) if α(t) is almost periodic and the nutrients supply outside the tumor is sufficient, there exists an almost periodic R(t).

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Qidi Zhang In this paper, we explain how weighted Strichartz estimates could be exploited to deal with the long time existence problem for the weighted semilinear wave equation with small data. When the solution blows up in finite time, we obtain the estimates for the lifespan of the solution.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Jie Wang, Jia-Feng Cao In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Debadatta Adak, Nandadulal Bairagi, Robert Hakl In this paper we analyze a delay-induced predator–prey–parasite model with prey harvesting, where the predator–prey interaction is represented by Leslie–Gower type model with type II functional response. Infection is assumed to spread horizontally from one infected prey to another susceptible prey following mass action law. Spreading of disease is not instantaneous but mediated by a time lag to take into account the time required for incubation process. Both the susceptible and infected preys are subjected to linear harvesting. The analysis is accomplished in two phases. First we analyze the delay-induced predator–prey–parasite system in absence of harvesting and proved the local & global dynamics of different (six) equilibrium points. It is proved that the delay has no influence on the stability of different equilibrium points except the interior one. Delay may cause instability in an otherwise stable interior equilibrium point of the system and larger delay may even produce chaos if the infection rate is also high. In the second phase, we explored the dynamics of the delay-induced harvested system. It is shown that harvesting of prey population can suppress the abrupt fluctuations in the population densities and can stabilize the system when it exceeds some threshold value.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Satoru Iwasaki, Atsushi Yagi We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their ω-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Liang Zhang, Shuang-Ming Wang Dengue fever is a typical mosquito-borne disease transmitted by Aedes aegypti and Aedes albopictus mosquitoes to human, which has become a major public health concern worldwide. In this paper, we incorporate the extrinsic incubation period (EIP) of dengue virus, spatial and temporal (seasonal) heterogeneity, crowing effect for host population into Dengue fever transmission, and propose a time-periodic and nonlocal delayed reaction–diffusion model of Dengue fever. The basic reproduction number R0 is introduced for the model system, and it serves a threshold type parameter that determines the transmission dynamics of Dengue fever. Numerical simulations indicate the influence of the aforementioned virus factors on the spread of the disease.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Xiaoyan Gao, Sadia Ishag, Shengmao Fu, Wanjun Li, Weiming Wang The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Dariusz Pączka We examine a mathematical model that describes a quasistatic adhesive contact between a viscoplastic body and deformable foundation. The material’s behaviour is described by the rate-type constitutive law which involves functions with a non-polynomial growth. The contact is modelled by the normal compliance condition with limited penetration and adhesion, a subdifferential friction condition also depending on adhesion, and the evolution of bonding field is governed by an ordinary differential equation. We present the variational formulation of this problem which is a system of an almost history-dependent variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. The results on existence and uniqueness of solution to an abstract almost history-dependent inclusion and variational–hemivariational inequality in the reflexive Orlicz–Sobolev space are proved and applied to the adhesive contact problem.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Wei Wang, Wanbiao Ma, Zhaosheng Feng In this paper, we are concerned with a diffusive viral infection dynamical model with general infection mechanism and distinct dispersal rates. In a general setting in which the model parameters are spatially heterogeneous, it is shown that if ℛ0≤1, the infection-free steady state is globally asymptotically stable; while if ℛ0>1, the model is uniformly persistent. The asymptotic profiles of the infection steady state are discussed as the dispersal rate of uninfected CD4+ T cells approaches zero by means of the persistence theory of semidynamical systems and the eigenvalue theory of elliptic equations.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): A. Chakib, A. Hadri, A. Laghrib This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Manuel Falconi, Yrina Vera-Damián, Claudio Vidal The aim of this paper is to analyze general dynamics features of a new Intraguild Predation (IGP) model where the top predator feeds only on the mesopredator and also affects its consumption rate. Important dynamical aspects of the model are described. Specifically, we prove that the trajectories of the associated system are bounded and defined for all positive time; there is a trapping domain; there are open subsets of parameters, such that the system in the first octant has at most five equilibrium solutions and at most three of them are of co-existence. Here we characterize the existence of Hopf bifurcations and we prove that this model exhibits either one, two or three small amplitude periodic solutions which arise from a zero-Hopf bifurcation. We prove the existence of alternative stable states. Finally, some numerical computations have been given in order to support our analytical results. The importance of some parameters of the model is discussed, in particular the role of the interference rate. The numerical exploration suggests that the equilibrium biomass of each of the three species grows as the level of interference grows.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Shouming Zhou, Zhijun Qiao, Chunlai Mu In this paper, we consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and Hölder continuity of the data-to-solution map for a generalized cross-coupled Camassa–Holm system with waltzing peakons on both the periodic and the non-periodic case. In light of a Galerkin-type approximation scheme, the system is shown well-posed in the Sobolev spaces Hs×Hs,s>5∕2 in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous. Furthermore, we prove the Hölder continuity in the Hr×Hr topology when 0≤r

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Xiongxiong Bao, Wan-Tong Li This paper is concerned with propagation phenomena of a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats. We first show that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions. Next, we prove that if the wave speed is greater than the spreading speed, there exists a time and space periodic traveling wave solution connecting the stable positive time and space periodic steady state and 0. Finally, we apply these results to two species partially degenerate competition systems and a partially degenerate epidemic model with nonlocal dispersal in time and space periodic habitats.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Mahendra Panthee, Francisco J. Vielma Leal In this work we study the controllability and stabilization of the linearized Benjamin equation which models the unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We show that the linearized Benjamin equation with periodic boundary conditions is exactly controllable and exponentially stabilizable with any given decay rate in Hps(T) with s≥0.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Yan Li In this paper, an attraction–repulsion chemotaxis system with p-Laplacian diffusion ut=∇⋅( ∇u p−2∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),x∈Ωis considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn, n≥2. χ,ξ,α,β,γ,δ are positive parameters. Global bounded weak solution is constructed for any p>1 under the following cases:⋅ Case I: ξγ−χα>0 with p>1;⋅ Case II: ξγ−χα≤0 with p>3nn+1;⋅ Case III:

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Łukasz Płociniczak In this paper we analyse a dynamical system based on the so-called KCG (Källén, Crafoord, Ghil) conceptual climate model. This model describes an evolution of the globally averaged temperature and the average extent of the ice sheets. In the nondimensional form the equations can be simplified facilitating the subsequent analysis. We consider the limiting case of a stationary snow line for which the phase plane can be completely analysed and the type of each critical point can be determined. One of them can exhibit the Hopf bifurcation and we find sufficient conditions for its existence. Those, in turn, have a straightforward physical meaning and indicate that the model predicts internal oscillations of the climate. Using the typical real-world values of appearing parameters we conclude that the obtained results are in the same ballpark as the conditions on our planet during the quaternary ice ages. Our analysis is a rigorous justification of a generalization of some previous results by KCG and other authors.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Huafei Di, Yadong Shang, Zefang Song This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity utt−Δu−Δut=φp(u)log u in a bounded domain Ω⊂Rn. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Lin Zhao, Zhi-Cheng Wang, Shigui Ruan Coinfection of hosts with multiple strains or serotypes of the same agent, such as different influenza virus strains, different human papilloma virus strains, and different dengue virus serotypes, is not only a very serious public health issue but also a very challenging mathematical modeling problem. In this paper, we study a time-periodic two-strain SIS epidemic model with diffusion and latent period. We first define the basic reproduction number R0i and introduce the invasion number Rˆ0i for each strain i(i=1,2), which can determine the ability of each strain to invade the other single-strain. The main question that we investigate is the threshold dynamics of the model. It is shown that if R0i⩽1(i=1,2), then the disease-free periodic solution is globally attractive; if R0i>1⩾R0j(i≠j,i,j=1,2), then competitive exclusion, where the jth strain dies out and the ith strain persists, is a possible outcome; and if Rˆ0i>1(i=1,2), then the disease persists uniformly. Finally we present the basic framework of threshold dynamics of the system by using numerical simulations, some of which are different from that of the corresponding multi-strain SIS ODE models.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Haiyan Yin, Limei Zhu In this paper, we study an asymptotic behavior of a solution to the outflow problem for a two-phase model with magnetic field. Our idea mainly comes from [1] and [2] which investigate the asymptotic stability and convergence rates of stationary solutions to the outflow problem for an isentropic Navier–Stokes equation. For the two-phase model with magnetic field, we also obtain the asymptotic stability and convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Ping Liu, Bowen Yang In this paper, the dynamics of a diffusive predator–prey model with modified Leslie–Gower term and strong Allee effect on prey under homogeneous Neumann boundary condition is considered. Firstly, we obtain the qualitative properties of the system including the existence of the global positive solution and the local and global asymptotical stability of the constant equilibria. In addition, we investigate a priori estimate and the nonexistence of nonconstant positive steady state solutions. Finally, we establish the existence and local structure of steady state patterns and time-periodic patterns for the system.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): P. Álvarez-Caudevilla, E. Colorado, Alejandro Ortega This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator, (−Δ)αu=λu+(−Δ)β u p−1uinΩ,(−Δ)ju=0on∂Ω,forj∈Z,0≤j

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Andrei V. Faminskii Initial–boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global well-posedness in classes of regular solutions in the cases of periodic and Neumann boundary conditions, as well as on internal regularity of solutions for all types of boundary conditions are established. Also in the case of Dirichlet boundary conditions one result on long-time decay of regular solutions is obtained.

Abstract: Publication date: February 2020Source: Nonlinear Analysis: Real World Applications, Volume 51Author(s): Ningkui Sun We consider a reaction–diffusion–advection equation of the form: ut=uxx−β(t)ux+f(t,u) for x∈[0,h(t)), where β(t) is a T-periodic function, f(t,u) is a T-periodic Fisher–KPP type of nonlinearity with a(t)≔fu(t,0) changing sign, h(t) is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers c̄ and B(β̃) with B(β̃)>c̄>0, β̄≔1T∫0Tβ(t)dt and β̃(t)≔β(t)−β̄, such that a vanishing–spreading dichotomy result holds when β̄