Disappearance of interfaces for the porous medium equation with variable density and absorption

Article type: Research Article

Authors: Reyes, Guillermo | Sánchez, Ariel

Affiliations: E.P.S., Universidad Carlos III de Madrid, 28911 Madrid, Spain E‐mail: greyes@math.uc3m.es | E.S.C.E.T., Universidad Rey Juan Carlos, 28933 Madrid, Spain E‐mail: asanchezv@escet.urjc.es

Abstract: We improve previous results on the qualitative properties of solutions to the Cauchy problem for the equation ρ(x)ut=(um)xx−c0up, where 1<m<p and c0>0 with nonnegative, compactly supported initial data. Precisely, we prove that if the density ρ(x) decays faster than |x|−k with k>k*=2(p−1)/(p−m), then the interfaces disappear in finite time for any initial data in the above class. We also weaken conditions previously imposed on the density function.

Keywords: non‐homogeneous porous media, positivity, disappearance of interfaces

Journal: Asymptotic Analysis, vol. 36, no. 1, pp. 13-20, 2003