Existence and dynamics of quasilinear parabolic systems with time delays

Abstract

This paper is concerned with a coupled system of quasilinear parabolic equations where the effect of time delays is taken into consideration in the reaction functions of the system. The partial differential operators in the system may be degenerate and the reaction functions possess some mixed quasimonotone property, including quasimonotone nondecreasing functions. The aim of the paper is to show the existence and uniqueness of a global solution to the parabolic system, the existence of positive quasisolutions or maximal–minimal solutions of the corresponding elliptic system, and the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal–minimal solutions of the elliptic system. Applications are given to three reaction–diffusion models arising from mathematical biology and ecology where the diffusion coefficients are density dependent and are degenerate. This degenerate density-dependent diffusion leads to some interesting distinct asymptotic behavior of the time-dependent solution when compared with density-independent diffusion.