Self-similar extinction for a fast diffusion equation with weighted absorption

TL;DR

This paper proves finite time extinction for solutions of a weighted fast diffusion equation with inhomogeneous absorption, and constructs self-similar solutions demonstrating this extinction behavior, contrasting with the classical case without weights.

Contribution

It establishes finite time extinction for a class of weighted fast diffusion equations and constructs corresponding self-similar solutions, revealing new extinction phenomena due to spatial weights.

Findings

Finite time extinction occurs for solutions with certain parameters.

Existence of self-similar solutions with specific profiles.

Contrast with standard fast diffusion where solutions remain positive.

Abstract

Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $N\geq1$ and exponents $$ p>1, \quad m_c=\frac{(N-2)_+}{N}<m<1, \quad \sigma>\sigma_*:=\frac{2(p-1)}{1-m}, $$ is established. Moreover, the existence of self-similar solutions of the form $$ U(x,t)=(T-t)^{\alpha}f(|x|(T-t)^{\beta}), \quad \alpha=\frac{\sigma+2}{(1-m)(\sigma-\sigma_*)}, \ \beta=\frac{p-m}{(1-m)(\sigma-\sigma_*)}, $$ with $f(0)>0$, $f'(0)=0$ and $$ \lim\limits_{\xi\to\infty}\xi^{(\sigma+2)/(p-m)}f(\xi)=L\in(0,\infty). $$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $\sigma=0$), where the strict positivity of solutions for any $t\in(0,\infty)$ is well-known.