Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

TL;DR

This paper establishes the existence and uniqueness of exponential self-similar eternal solutions for a specific quasilinear reaction-diffusion equation with critical singular potential, and demonstrates their role in constructing global weak solutions.

Contribution

It introduces new eternal solutions in exponential self-similar form for a reaction-diffusion equation with critical singular potential, extending the understanding of solution behavior.

Findings

Existence and uniqueness of exponential self-similar eternal solutions.

Construction of global weak solutions from initial data in L-infinity.

Persistence of compact support in solutions over time.

Abstract

We prove existence and uniqueness of self-similar solutions with exponential form $$ u(x,t)=e^{\alpha t}f(|x|e^{-\beta t}), \qquad \alpha, \ \beta>0 $$ to the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1<p<m$ and $\sigma=-2(p-1)/(m-1)$ and in dimension $N\geq2$, the same results holding true in dimension $N=1$ under the extra assumption $1<p<(m+1)/2$. Such self-similar solutions are usually known in literature as \emph{eternal solutions} since they exist for any $t\in(-\infty,\infty)$. As an application of the existence of these eternal solutions, we show existence of \emph{global in time weak solutions} with any initial condition $u_0\in L^{\infty}(\real^N)$, and in particular that these weak solutions remain compactly supported at any time $t>0$ if $u_0$ is compactly supported.