Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles

TL;DR

This paper constructs and analyzes a family of radially symmetric eternal solutions with exponential self-similarity for a porous medium equation with strong nonhomogeneous absorption, demonstrating their boundedness and non-vanishing properties.

Contribution

It proves the existence and uniqueness of a family of eternal solutions with specific self-similar profiles for the equation with critical absorption.

Findings

Existence of a unique exponent 2 for self-similar solutions.

Construction of a one-parameter family of solutions with compact support.

Solutions are bounded and do not vanish in finite time.

Abstract

Existence of specific \emph{eternal solutions} in exponential self-similar form to the following quasilinear diffusion equation with strong absorption$$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$posed for $(t,x)\in(0,\infty)\times\mathbb{R}^N$, with $m>1$, $q\in(0,1)$ and $\sigma=\sigma_c:=2(1-q)/(m-1)$ is proved. Looking for radially symmetric solutions of the form$$u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}), \qquad \alpha=\frac{2}{m-1}\beta,$$we show that there exists a unique exponent $\beta^*\in(0,\infty)$ for which there exists a one-parameter family $(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $\sigma\in (0,\sigma_c)$.