Existence and asymptotic behavior of solutions for quasilinear parabolic systems

This paper is concerned with the existence, uniqueness, and asymptotic behavior of solutions for the quasilinear parabolic systems with mixed quasimonotone reaction functions, the elliptic operators in which are allowed to be degenerate. By the method of the coupled upper and lower solutions, and its monotone iterations, it shows that a pair of coupled upper and lower solutions ensures that the unique positive solution exists and globally stable if the quasisolutions are equal. Moreover, we study the asymptotic behavior of solutions to the Lotka–Volterra model with the density‐dependent diffusion. Copyright © 2013 John Wiley & Sons, Ltd.     

带时滞的具有阶段结构的捕食抛物型方程组

本文对两种群的捕食与被捕食模型,运用了上下解方法及相应的单调迭代序列研究具有时滞的耦合半线性抛物方程组的动力学行为,给出了解的渐近性质及阶段结构对解的性质的影响。     

三种群捕食-被捕食模型中具时滞的抛物系统

本文研究三种群食物链,其中第三种群是第二种群的捕食者,第二种群是第一种群的捕食者。我们用上下解方法研究具时滞的耦合半线性抛物方程组的动力学行为,给出了解的渐近性质。     

Coexistence and asymptotic periodicity in a competitor-competitor-mutualist model

In this paper, the competitor-competitor-mutualist three-species Lotka-Volterra model is discussed. Firstly, by Schauder fixed point theory, the coexistence state of the strongly coupled system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak. Secondly, the existence and asymptotic behavior of T-periodic solutions for the periodic reaction-diffusion system under homogeneous Dirichlet boundary conditions are investigated. Sufficient conditions which guarantee the existence of T-periodic solution are also obtained.     

Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory

Some coupled reaction-diffusion systems arising from chemical diffusion processes and combustion theory are analyzed. This analysis includes the existence and uniqueness of positive time-dependent solutions, upper and lower bounds of the solution, asymptotic behavior and invariant sets, and the stability of steady-state solutions, including an estimate of the stability region. Explicit conditions for the asymptotic behavior and the stability of a steady-state solution are given. These conditions establish some interrelationship among the physical parameters of the diffusion medium, the reaction mechanism, the initial function and the type of boundary condition. Under the same set of physical parameters and reaction function, a comparison between the Neumann type and Dirichlet or third type boundary condition exhibits quite different asymptotic behavior of the solution. For the general nonhomogeneous system, multiple steady-state solutions may exist and only local stability results are obtained. However, for certain models it is possible to obtain global stability of a steady-state solution by either increasing the diffusion coefficients or decreasing the size of the diffusion medium. This fact is demonstrated by a one-dimensional tubular reactor model commonly discussed in the literature.     

Traveling wave solutions in a diffusive system with two preys and one predator

This paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator. By constructing upper and lower solutions, the existence of nontrivial traveling wave solutions is established. The asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles. Applying the theory of asymptotic spreading, the nonexistence of traveling wave solutions is proved.     

On nonlinear reaction-diffusion systems

This paper presents a qualitative analysis for a coupled system of two reaction-diffusion equations under various boundary conditions which arises from a number of physical problems. The nonlinear reaction functions are classified into three basic types according to their relative quasi-monotone property. For each type of reaction functions, an existence-comparison theorem, in terms of upper and lower solutions, is established for the time-dependent system as well as some boundary value problems. Three concrete physical systems arising from epidemics, biochemistry and engineering are taken as representatives of the basic types of reacting problems. Through suitable construction of upper and lower solutions, various qualitative properties of the solution for each system are obtained. These include the existence and bounds of time-dependent solutions, asymptotic behavior of the solution, stability and instability of nontrivial steady-state solutions, estimates of stability regions, and finally the blowing-up property of the solution. Special attention is given to the homogeneous Neumann boundary condition.     

Global asymptotic behavior in a Lotka–Volterra competition system with spatio-temporal delays

This paper is concerned with a Lotka–Volterra competition system with spatio-temporal delays. By using the linearization method, we show the local asymptotic behavior of the nonnegative steady-state solutions. Especially, the global asymptotic stability of the positive steady-state solution is investigated by the method of upper and lower solutions. The result of global asymptotic stability implies that the system has no nonconstant positive steady-state solution.     

Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay

This paper is concerned with the existence, asymptotic stability and uniqueness of traveling wavefronts in a nonlocal diffusion equation with delay. By constructing proper upper and lower solutions, the existence and asymptotic behavior of traveling wavefronts are established. Then the asymptotic stability with phase shift as well as the uniqueness up to translation of traveling wavefronts are proved by applying the idea of squeezing technique.     

Large solutions for quasilinear elliptic equation with nonlinear gradient term

We study the existence, asymptotic behavior near the boundary and uniqueness of large solutions for a class of quasilinear elliptic equation with a nonlinear gradient term. By constructing the suitable blow-up upper and lower solutions, we obtain the existence and the asymptotic behavior of radial large solutions of the problem in balls and then derive the existence of solutions in a general domain by a comparison argument. By using a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of any nonnegative solution of it near the boundary. The uniqueness is shown by a standard argument.     

The asymptotic periodicity in a Schoener’s competitive model

This paper deals with a two species model with Schoener’s competitive interaction. The existence and the asymptotic behavior of T-periodic solutions for the periodic system of quasilinear parabolic equations under nonlinear boundary conditions are given by using upper and lower solutions and corresponding iteration. The numerical simulations are also presented to illustrate our result. It is shown that periodic solutions may exist if the inter-specific competition rates are weak.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behav-ior of the time-dependent solutions to a linear parabolic equation with nonlinear boundarycondition in relation to their corresponding steady state solutions. Then, the above resultsare extended to a semilinear parabolic equation with nonlinear boundary condition by an-alyzing the corresponding eigenvalue problem and using the method of upper and lowersolutions.     

催化反应中的一类奇摄动边值问题

§1. Introduction We are concered with the singularly perturbed boundary value problemε2y″=y3,(1)y(0)=1, y(1)=2,(2)where ε＞0 is a positive small parameter. This problem arises as models for certain catalytic reactions in chemical engineering. The study of that problem has been paid much attention for the boundary layers of the problem exihibit the behavior of nonexponential decay. There have been some works on this subject ［1］-［4］. In particular, Howes and Chang［1］ gave an accurate…     

一类非局部反应扩散问题

本文研究了一类非局部反应扩散问题．利用上、下解，讨论了相应问题解的存在唯一性及其渐近性态．     

On a class of nonlinear Neumann problems

Summary Existence theorems for nonlinear Neumann problems with inhomogeneous boundary conditions are established. It is then investigated under which conditions the solutions are uniformly bounded. Uniqueness results for positive solutions are given and the asymptotic behavior of the solutions of the corresponding parabolic equation is discussed. The main tools are fixed point theorems and the method of upper and lower solutions.     

一类具耦合的化学反应扩散系统的奇摄动

本文研究一类带耦合项的化学反应扩散方程组解的性态,利用上下解理论证得解的存在性,然后在一定的参数环境下考虑了相应的奇摄动问题,并给出一致有效的渐近解.     

Asymptotic analysis of a semilinear elliptic equation in highly oscillating thin domains

In this work we are interested in the asymptotic behavior of a family of solutions of a semilinear elliptic problem with homogeneous Neumann boundary condition defined in a two-dimensional bounded set which degenerates to the unit interval as a positive parameter \({\epsilon}\) goes to zero. Here we also allow that upper and lower boundaries from this singular region present highly oscillatory behavior with different orders and variable profile. Combining results from linear homogenization theory and nonlinear analyzes we get the limit problem showing upper and lower semicontinuity of the solutions at \({\epsilon=0}\).     

Global asymptotic stability of positive equilibrium of three-species Lotka–Volterra mutualism models with diffusion and delay effects

In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction–diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction–diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction–diffusion system. Finally, the numerical simulation is given to illustrate our results.     

Asymptotic behavior of reaction–advection–diffusion population models with Allee effect

In this work, a qualitative analysis is carried out for reaction–advection–diffusion (RAD) systems modeling the interactions between two species with Allee effect. In particular, we study different scenarios: mutualism, competition, and a predator–prey relationship in order to investigate the survival or extinction of both populations. Global existence and uniqueness of positive solutions of the proposed RAD problems are demonstrated. Equilibrium states and asymptotic behavior of solutions are obtained using the monotone method and the upper and lower solutions technique. Numerical simulations by a Crank–Nicolson monotone iterative method of the different asymptotic solution dynamics are shown to illustrate our theoretical results.