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, the elliptic operators in which are allowed to be degenerate. By the method of the coupled upper and lower solutions, and its monotone iterations, it shows that a pair of coupled upper and lower solutions ensures that the unique positive solution exists and globally stable if the quasisolutions are equal. Moreover, we study the asymptotic behavior of solutions to the Lotka–Volterra model with the density‐dependent diffusion. Copyright © 2013 John Wiley & Sons, Ltd.