Existence and asymptotic behavior of solutions for quasilinear parabolic systems

Abstract

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.

Discrete monotone method for space-fractional nonlinear reaction–diffusion equations

“…For example, from the analytical side, such iterative techniques have been applied to investigate the existence and uniqueness of solutions of a wide range of parabolic partial differential equations [1], as well as other analytical features of the solutions. In particular, this approach has been used to establish the existence of positive solutions of quasilinear parabolic systems with Dirichlet boundary conditions [2], to study quasilinear parabolic and elliptic systems with mixed quasimonotone functions [3], to analyze periodic boundary-value problems for differential equations with delay [4], to solve first-order functional-difference equations with nonlinear boundary value conditions [5], to prove the existence and asymptotic behavior of solutions for quasilinear parabolic systems [6], and, recently, to establish the existence, uniqueness, and stability of the solutions of a parabolic model in the formation of porous silicon [7], among other interesting applications.…”