Positivity, decay, and extinction for a singular diffusion equation with gradient absorption

TL;DR

This paper investigates the long-term behavior of solutions to a singular diffusion equation with gradient absorption, classifying conditions for positivity, decay, or extinction based on parameters, and highlighting the diffusion's role in preventing finite-time extinction.

Contribution

It provides a comprehensive classification of solution behaviors for a singular diffusion equation with gradient absorption, including conditions for positivity, decay, and extinction, and compares diffusion effects to the non-diffusive case.

Findings

Solutions are positive as t→∞ for q>p−N/(N+1).

Optimal decay estimates are obtained for p/2≤q≤p−N/(N+1).

Finite-time extinction occurs for 0<q<p/2, but diffusion can prevent extinction in some cases.

Abstract

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -\Delta_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where $N\ge 1$, $p\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\to\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\to\infty$ for $p/2\le q\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton-Jacobi equation.