Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:248

Article

Pao, C. V.²  Ruan, W. H.¹ 

[1] Purdue Univ Calumet, Dept Math Comp Sci & Stat, Hammond, IN 46323 USA

[2] N Carolina State Univ, Dept Math, Raleigh, NC 27695 USA

关键词: Quasilinear parabolic and elliptic equations;    Degenerate reaction-diffusion system;    Maximal and minimal solutions;    Asymptotic behavior of solution;    Method of upper and lower solutions;   

DOI  :  10.1016/j.jde.2009.12.011

来源: Elsevier

【 摘 要 】

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D-i(u(i)) may have the property D-i(0) = 0 for some or all i = 1, ... , N, and the boundary condition is u(i) = 0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent Solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a Coupled system of polynomial growth problem. and a two component competition model in ecology. (c) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free