Problems in nonlinear diffusion

TLDR: In this paper, the authors consider the analysis of diffusive solutions of the generalized porous media equation and stress-driven diffusion in solids and show that the initial deviations from similarity solution form in these problems are asymptotic beyond all orders.

Abstract: A variety of effects can occur from different forms of nonlinear diffusion or from coupling of diffusion to other physical processes. I consider two such classes of problems; first, the analysis of behavior of diffusive solutions of the generalized porous media equation, and second, the study of stress-driven diffusion in solids. The porous media equation is a nonlinear diffusion equation that has applications to numerous physical problems. By combining classical techniques for the study of similarity solutions with perturbation methods, I have examined some new initial-boundary value problems for the porous media equation, including "stopping" and "merging" problems. Using matched asymptotic expansions and boundary layer analysis, I have shown that the initial deviations from similarity solution form in these problems are asymptotic beyond all orders. Applications of these studies to the Cahn-Hilliard and Fisher's equations are also considered. In my examination of stress-driven diffusion, I consider models for the behavior of systems in the emerging technological field of viscoelastic diffusion in polymer materials. Using asymptotic analysis, I studied some of the non-traditional effects, shock formation in particular, that occur in initial-boundary value problems for these models. Phase-interface traveling waves for "Case II" diffusive transport were also studied, using phase plane techniques.