On the existence of positive solutions of a degenerate reaction diffusion model applied in ecology

Abstract

The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also in- vestigated is the existence of positive maximal and minimal so- lutions of the corresponding quasilinear elliptic system. The el- liptic operators in both systems are allowed to be degenerate in the sense that the density-dependent di¤usion coe¢ cients Di (ui) may have the property Di (0) = 0 for some or all i = 1; ...N, and the boundary condition is ......... Using the method of up- per and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal so- lution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution.