One-Parameter Family of Invariant Sets for Nonweakly Coupled Nonlinear Parabolic Systems

This paper is concerned with a system of nonweakly coupled nonlinear parabolic and elliptic equations where the coupling is through the unknown functions as well as their gradients. We discuss the asymptotic behavior of the solution for the parabolic system in the case where there exists a decreasing one-parameter family of invariant sets formed by coupled upper and lower solutions of the corresponding elliptic system. Under fairly general conditions, we show that every solution of the parabolic system starting from the outermost invariant set converges uniformly into the innermost one. The results are used in the study of stability and asymptotic behavior problems of a reaction-diffusion model arising in population genetics.