Quasilinear parabolic and elliptic systems with mixed quasimonotone functions

This paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions. The boundary condition in the system may be Dirichlet, nonlinear, or a combination of these two types. The elliptic operators in the system are allowed to be degenerate. The aim is to show the existence and uniqueness of a classical solution to the parabolic system, the existence of maximal and minimal solutions or quasisolutions of the elliptic system, and the asymptotic behavior of the solution of the parabolic system. This consideration leads to a global attractor of the parabolic system as well as an one-sided stability of the maximal and minimal solutions. Applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density-dependent and are degenerate. These applications exhibit quite distinct dynamical behavior of the population species between degenerate density-dependent diffusion and constant diffusion.