Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption

TL;DR

This paper investigates the conditions under which solutions to a nonlinear diffusion equation with spatially inhomogeneous strong absorption vanish in finite time, providing new proofs and extending known results on extinction phenomena.

Contribution

It introduces a critical exponent for the absorption term and proves finite time extinction for certain parameter ranges and initial conditions, enhancing understanding of the extinction behavior.

Findings

Finite time extinction occurs for <<^* for >1.

Extinction is proved for ^* when >1 and ^*.

The paper provides an alternative proof for known extinction conditions.

Abstract

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$, $q\in(0,1)$ and $\sigma>0$, is addressed. Introducing the critical exponent $\sigma^* := 2(1-q)/(m-1)$ for $m>1$ and $\sigma_*=\infty$ for $m=1$, extinction in finite time is known to take place for $\sigma\in [0,\sigma^*)$ and an alternative proof is provided therein. When $m>1$ and $\sigma\ge \sigma^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of $\sigma$ and showing their sharpness.