Long-time extinction of solutions of some semilinear parabolic equations

TL;DR

This paper investigates the long-term behavior of solutions to certain semilinear parabolic equations, establishing conditions under which solutions extinguish in finite time using energy methods and semi-classical analysis.

Contribution

It introduces a Dini-like condition on the coefficient function that guarantees finite-time extinction, employing novel analytical approaches.

Findings

Solutions vanish in finite time under specified conditions.

Two different methods confirm the extinction result.

Provides new criteria for solution extinction based on coefficient behavior.

Abstract

We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-\Delta u+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp(\frac{\omega(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $\omega$ a positive continuous radial function. We give a Dini-like condition on the function $\omega$ by two different method which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schr\"odinger operators.