A Remark on Partly Dissipative Reaction Diffusion Systems on IR~n

The long time behavior of the solutions of some partly dissipative reaction diffusion systems is studied.We prove the existence of a compact(L~2×L~2-H~1×L~2)attractor for a partly dissipative reaction diffusion system in R~n.This improves a previous result obtained by A.Rodrigues-Bernal and B.Wang concerning the existence of a compact(L~2×L~2-L~2×L~2)attractor for the same system.     

A remark on partly dissipative reaction diffusion systems on IRn

The long time behavior of the solutions of some partly dissipative reaction diffusion systems is studied. We prove the existence of a compact （L^2 × L^2 - H^1 × L^2） attractor for a partly dissipative reaction diffusion system in Rn. This improves a previous result obtained by A. Rodrigues-Bernal and B. Wang concerning the existence of a compact （L^2 × L^2 - L^2 × L^2） attractor for the same system.     

A FREE-BOUNDARY PROBLEM TO DYNAMIC SYSTEM FOR PURE FOREST

A free-boundary model of nonlinear dynamic system for pure forest is presented, in which the felling rate is unbounded nearby the free boundary. The effiect of unbounded function on a priori estimate and analysis of regularity is overcome, and the existence and uniqueness of the global classical solution to this system are proved.     

Simultaneous Identification of Two Parameters on the Reaction Diffusion System from Discrete Measurement Data

This article deals with an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final time space discretized measurement using the optimization method with the help of the smooth interpolation technique. The main objective of the article is to analyse the asymptotic behavior of the solution of the inverse problem for the linearly coupled reaction diffusion system with respect to the homogeneous Dirichlet boundary condition.     

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.     

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.     

SINGULAR PERTURBATION FOR A REACTION DIFFUSION SYSTEM IN BACTERIA GROWTH

In this paper, a mathematical model for a class of reaction diffusion system in bacteria growth is studied Under suitable conditions the formal asymptotic expansions of solution for the proposed problem is constructed and a uniformly valik solution to any degree of precision is proved by using the method of differential inequalities.     

THE GLOBAL SOLUTION AND BLOWUP OF A SPATIOTEMPORAL EIT PROBLEM WITH A DYNAMICAL BOUNDARY CONDITION

We study a spatiotemporal EIT problem with a dynamical boundary condition for the fractional Dirichlet-to-Neumann operator with a critical exponent.There are three major ingredients in this paper.The first is the finite time blowup and the decay estimate of the global solution with a lower-energy initial value.The second ingredient is the L^q(2 ≤q <∞) estimate of the global solution applying the Moser iteration,which allows us to show that any global solution is a classical solution...     

Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms

This paper deals with the blow-up properties of the solution to a semilin-ear parabolic system with localized nonlinear reaction terms, subject to the null Dirichlet boundary condition. We first give sufficient conditions for that the classical solution blows up in the finite time, secondly give necessary conditions and a sufficient condition for that two components blow up simultaneously, and then obtain the uniform blow-up profiles in the interior. Finally we describe the asymptotic behavior of the blow-up solution in the boundary layer.     

ON A REACTION DIFFUSION SYSTEM OF COMPETING PREDATORS

This paper is concerned with the existence-uniqueness and the asymptotic behavior of the solution of a coupled system of reaction diffusion equations which arose from biochemistry. Under Diricblet boundary condition, the existence and the bifurcation of nonegative steady-state solutions of the system will be studied. The principal methods used in this paper are the monotone method ond the topological degree method.     

Global Dynamics of the Oregonator System

In this work, the existence and properties of a global attractor for the solution semiflow of the Oregonator system are proved. The Oregonator system is the mathematical model of the celebrated Belousov–Zhabotinskii reaction. A rescaling and grouping estimation method is developed to show the absorbing property and the asymptotic compactness of the solution trajectories of this three‐component reaction–diffusion system with quadratic nonlinearity. It is also proved that the fractal dimension of the global attractor is finite and an exponential attractor exists for the Oregonator semiflow. Copyright © 2012 John Wiley & Sons, Ltd.     

反应扩散方程古典解的最大吸引子

利用能量积分和解析半群的有关估计，一类反应扩散方程非负古典解在连续函数空间的最大吸引子的存在性被证明，且非线性项取为任意阶多项式．     

非线性抛物型方程组的最大吸引子

利用能量积分、Sobolev空间的嵌入定理和不变区域,本文证明了一类具有"自然结构条件"的非线性抛物型方程组的最大吸引子的存在性,并给出了一定条件下解的衰减性估计。     

Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.     

Random Attractor for Stochastic Partly Dissipative Systems on Unbounded Domains

In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback random attractor in L²(R^n) × L²(R^n). We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.     

Extended Fisher-Kolmogorov系统的渐近吸引子

考虑了ExtendedFisher-Kolmogorov系统的解的长时间行为,构造了一个有限维解序列即该系统的渐近吸引子,证明了它在长时间后无限趋于方程的整体吸引子,并给出了渐近吸引子的维数估计.     

Cross-diffusion induced Turing instability for a three species food chain model

In this paper, we study a strongly coupled reaction–diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. We first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ODE system. The positive equilibrium solution remains linearly stable for the reaction–diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. We further proved that the positive equilibrium solution is globally asymptotically stable for the reaction–diffusion system without cross-diffusion by constructing a Lyapunov function. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction–diffusion system, hence the instability is driven solely from the effect of cross-diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions.     

The long time behavior for partly dissipative stochastic systems

In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations in \(D\in\mathbb{R}^n\). The main purpose of thie paper is to establish the existence of a compact global random attractor. The existence of a random absorbing set is first discussed for the systems and then an estimate on the solutions is derived when the time is large enough, which ensures the asymptotic compactness of solutions. Finally, establish the existence of the global attractor in \(L^2(D)\times L^2(D)\).     

一类反应扩散方程组最大吸引子的正则性和维数估计

反应扩散方程组的最大吸引子通常是在不变区域内研究,如果不具有不变区域,或者去掉不变区域的限制而在全空间考虑这类问题,其结果如何？本文将证明一类反应扩散方程组在全空间最大吸引子的存在性,并对该吸引子的正则性进行了详细讨论,还给出了该吸引子的维数估计．     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.