Combining Linear and Nonlinear Diffusion
Manuel Delgado, Julián López-Gómez, António Suárez · Advanced Nonlinear Studies · 2004
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, Ω 1 and slow nonlinear, m(x) > 1, in its complement, Ω m := Ω \ Ω̄ 1 . More 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 = Ω 1 ) and of the completely nonlinear case (Ω m = 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.