Extinction for a singular diffusion equation with strong gradient absorption revisited

TL;DR

This paper investigates finite time extinction of solutions to a singular diffusion equation with gradient absorption, identifying optimal initial decay conditions and extinction rates under specific parameter ranges.

Contribution

It provides the first precise characterization of initial decay rates and extinction rates for solutions of the singular diffusion equation with gradient absorption.

Findings

Identifies optimal initial decay conditions for finite time extinction.

Derives optimal extinction rates near the extinction time.

Establishes parameter ranges for extinction phenomena.

Abstract

When $2N/(N+1)<p<2$ and $0<q<p/2$, non-negative solutions to the singular diffusion equation with gradient absorption $$\partial\_tu-\Delta\_p u + |\nabla u|^q=0 \ \text{ in }\ (0,\infty)\times\mathbb{R}^N$$ vanish after a finite time. This phenomenon is usually referred to as finite time extinction and takes place provided the initial condition $u\_0$ decays sufficiently rapidly as $|x|\to\infty$. On the one hand, the optimal decay of $u\_0$ at infinity guaranteeing the occurence of finite time extinction is identified. On the other hand, assuming further that $p-1<q<p/2$, optimal extinction rates near the extinction time are derived.