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

TL;DR

This paper constructs a family of eternal, self-similar solutions with dead cores for a porous medium equation with strong absorption, revealing their behavior and implications for finite-time extinction.

Contribution

It proves the existence and uniqueness of a family of eternal solutions with dead cores for the specified porous medium equation, including their precise interface behavior.

Findings

Existence of a unique exponent for self-similar solutions with dead cores.

Characterization of the solutions' behavior at their interfaces.

Implications for finite-time extinction properties of solutions.

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 $$ u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}), \qquad \alpha=\frac{2}{m-1}\beta,$$ it is shown that, for $m+q>2$, there exists a unique exponent $\beta_*\in(0,\infty)$ 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.