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

Abstract

We improve previous results on the qualitative properties of solutions to the Cauchy problem for the equation ρ(x)u_t=(u^m)_xx−c₀u^p, 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.