Abstract:

This paper deals with nonlinear diffusion problems involving degenerate parabolic problems, such as the Stefan problem and the porous medium equation, and cross-diffusion systems in population ecology. The degeneracy of the diffusion and the effect of cross-diffusion, that is, nonlinearities of the diffusion, complicate its analysis. In order to avoid the nonlinearities, we propose a reaction-diffusion system with solutions that approximate those of the nonlinear diffusion problems. The reaction-diffusion system includes only a simple reaction and linear diffusion. Resolving semilinear problems is typically easier than dealing with nonlinear diffusion problems. Therefore, our ideas are expected to reveal new and more effective approaches to the study of nonlinear problems.