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 endowed with Dirichlet boundary condition, in which the elliptic operators are allowed to be degenerate. By the method of the coupled upper and lower solutions and its monotone iterations, it is shown that a pair of coupled upper and lower solutions ensures that the unique positive solution exists and is globally stable if the quasisolutions are equal. Moreover, we study the asymptotic behavior of solutions to the Lotka-Volterra predator-prey model with the density-dependent diffusion.