Dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions

• Published in: Journal of Differential Equations 2017-12, Vol.263 (11), p.7709-7752
• Main Authors: Pao, C.V., Ruan, W.H.
• Format: Article
• Language: English
• Subjects: Asymptotic behavior; Lotka–Volterra models; Quasilinear reaction diffusion systems; Upper and lower solutions; Weak and classical solutions
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2017.08.024

This paper is concerned with a system of quasilinear reaction–diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions. The equations are allowed to be degenerate and the boundary conditions are of the nonlinear type. The main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions; show that the weak solution evolves into the classical solution, and analyze the asymptotic behavior of the solution using quasi-solutions of the steady-state system. The general results are applied to a degenerate Lotka–Volterra competition model. Conditions are given for the solution to exist globally, to evolve into the classical solution, and to be attracted into a sector formed by quasi-solutions of the elliptic system. Especially for the Neumann problem we give a simple condition for the solution to converge to a unique constant steady-state solution which is a global attractor.