Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles

Abstract

Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $${\partial }_{t}u-\Delta u^m+|x{|}^{\sigma }{u}^q=0\text{ in }(0,\infty )\times {\mathbb{R}}^N,$$ with $m>1$, $q\in (0,1)$ and $\sigma =2(1-q)/(m-1)$. Looking for solutions of the form $$u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}),\alpha =\frac{2}{m-1}\beta ,$$ it is shown that, for $m+q>2$, there exists a unique exponent ${\beta }_{\ast }\in (0,\infty )$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.