Combining linear and nonlinear diffusion

Abstract

In this paper we study a generalized porous medium equation where the diffusion rate, say m(x) —spatially heterogeneous—, is assumed to be linear, m = 1, on a piece of the support domain, Ω1, and slow nonlinear, m(x) > 1, in its complement, Ωm := Ω \ Ω¯1. Most precisely, we characterize the existence of positive solutions and construct the corresponding global bifurcation diagram as one of the parameters of the model changes, showing that a continuous transition occurs between the diagrams of the completely linear case (Ω = Ω1) and of the completely nonlinear case (Ωm = Ω). As a result, the effect of a localized slow diffusion rate with varying support is completely characterized. Our analysis is imperative in order to design porous media multi-components systems with changing diffusion rates.