THE SINGLE-POINT QUENCHING OF NONLINEAR DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This article is concerned with the quenching phenomena of the nonlinear degenerate functional reaction-diffusion equation. Some results are obtained on the single-point quenching and the uniqueness of quenching.     

Quenching Time Estimates for Semilinear Parabolic Equations Controlled by Two Absorption Sources in Control System

This paper deals with the quenching solution of the initial boundary value problem for aclass of semilinear reaction-diffusion equation controlled by two absorption sources in control system and estimate upper bound and lower bound of the quenching time. We point that the number of absorption sources influences the time of quenching phenomenon.The solution can solve some boundary value problem in control system.     

Lipschitzian semigroups and abstract functional differential equations

The paper is devoted to studying an abstract functional differential equation by a nonlinear semigroup approach. We first prove in details the equivalence of the well posedness of an abstract functional differential equation and an associated abstract Cauchy problem in the sense of strong solutions. Secondly, a sufficient condition is derived for well posedness of the abstract functional differential equation. Thirdly, we present principles of linearized stability for the abstract functional differential equation. Finally, the results obtained are applied to a reaction-diffusion equation with delays.     

反应-扩散方程的熄灭问题

In this article, we consider a reaction-diffusion differential equation with initial value conditions u(x, 0) = 0 on [0, a] and boundary condition ux + αiu = 0 on Γ= {0, a} × (0, T), and the quenching happens for the reaction-diffusion equation.     

The Hopf Bifurcation for an Abstract Functional Differential Equation and Its Application

ln this peper we obtained the Hopf bifurcation theorem for an abstract functional differential equation by the results of [1]. The asymptotic expression of bifurcation formulae and stability condition were given in detail. Applying the result, we considered the Hopf bifurcation problem for a reaction-diffusion equation with time delay.     

Spectrum comparison for a conserved reaction-diffusion system with a variational property

We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.     

THE GLOBAL DUFORT-FRANKEL DIFFERENCE APPROXIMATION FOR NONLINEAR REACTION-DIFFUSION EQUATIONS

1.IntroductionInthispaperweconsiderthefollowinginitial-valueproblemofnonlinearreactiondiffusionequation:HerefiisaboundeddomaininRd(d<3)withaLipschitzboundaryOffand7isapositiveconstallt.Lettheset{in*l:j(u*)=0}benotemptyandu=ma-c{lu*I:f(u*)=0}.Assumptionont…     

Relaxation in Reactive Flows

We consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. We show that the resulting flows possess relaxation-enhancing properties in the sense of [CoKRZ]. In particular, we show that solutions of the nonlinear problems become small when gravity is sufficiently strong due to the improved interaction with the cold boundary. As an application, we deduce that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit. We also discuss the behavior of the principal eigenvalues of the corresponding advection-diffusion problem and the quenching phenomenon for reaction-diffusion equations. Received: March 2007, Revision: May 2007, Accepted: May 2007     

Dynamics in dumbbell domains I. Continuity of the set of equilibria

We analyze the dynamics of a reaction-diffusion equation with homogeneous Neumann boundary conditions in a dumbbell domain. We provide an appropriate functional setting to treat this problem and, as a first step, we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds.     

On the Unique Solvability of a Nonlinear Functional Evolution Equation

Existence and uniqueness of weak solutions to an abstract functional evolution equation are proved. An application of this theory to a nonlinear nonlocal reaction-diffusion equation is presented.     

退化的半线性反应扩散方程组解的熄灭

In this paper,the initial boundary value problem of semilinear degenerate reaction-diffusion systems is studied.The regularization method and upper and lower solutions technique are employed to show the existence and continuation of a positive classical solution.The location of quenching points is found.The critical length is estimated by the eigenvalue method.     

Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

具有吸引项的反应扩散型和拟抛物方程解的全局死核问题

在文[1、2]文[3、6]基础上,应用文[7、8]的方法和相应结论,在古典解存在且唯一的条件下,结合文[10、11]研究具吸引项的反应扩散方程(1)的死核问题,且进一步讨论了跟一般的非线性抛物型方程(2)解的全局死核问题,得到新的结果和时间估计.     

Finite Element Simulations with Adaptively Moving Mesh for the Reaction Diffusion System

A moving mesh method is proposed for solving reaction-diffusion equations. The finite element method is used to solving the partial different equation system, and an efficient scheme is applied to implement mesh moving. In the practical calculations, the moving mesh step and the problem equation solver are performed alternatively. Serveral numerical examples are presented, including the Gray-Scott, the Activator-Inhibitor and a case with a growing domain. It is illustrated numerically that the moving mesh method costs much lower, compared with the numerical schemes on a fixed mesh. Even in the case of complex pattern dynamics described by the reaction-diffusion systems, the adapted meshes can capture the details successfully.     

长时间计算的二阶LEGENDRE谱格式(I)

本文提出了一类二阶Legendre谱格式,并考虑了反应扩散方程。证明了数值解的存在性和唯一性。模拟了原问题的守恒型和长时间性态。     

Local and non‐local boundary quenching

The quenching problem is examined for a one‐dimensional heat equation with a non‐linear boundary condition that is of either local or non‐local type. Sufficient conditions are derived that establish both quenching and non‐quenching behaviour. The growth rate of the solution near quenching is also given for a power‐law non‐linearity. The analysis is conducted in the context of a nonlinear Volterra integral equation that is equivalent to the initial–boundary value problem. Copyright © 1999 John Wiley & Sons, Ltd.     

关于猝灭问题的一些结果及其应用

本文研究一类含奇异项的半线性抛物方程的初边值问题,给出了该问题的古典解整体存在或发生猝灭现象的判据以及猝灭解的生命跨度估计。同时,还用上述结果研究了一类含超Sobolev临界指标的半线性椭圆方程的Dirichlet边值问题,获得了正解的存在性结果。     

一类广义扩散模型解的存在性

通过考虑具二阶导数项的Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ自由能量泛函,本文导出了一类广义扩散模型,进而采用经典的能量估计方法和对所引入的能量泛函进行精细的分析,获得了所论模型解的存在性和唯一性。     

半线性奇摄动反应扩散方程初边值问题解的渐近性(英文)

本文运用边界层以及角点层函数法构造了一类半线性奇摄动反应扩散方程初边值问题解的渐近展开式 ,并用微分不等式方法证明了该展式达到任一精度的一致有效性     

On Time-space Fractional Reaction-diffusion Equations with Nonlocal Initial Conditions

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.