Large time behavior for a quasilinear diffusion equation with critical gradient absorption

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear diffusion equation with critical gradient absorption, revealing self-similar asymptotic profiles, decay rates, and support expansion improvements.

Contribution

It introduces a detailed analysis of the asymptotic behavior of solutions, identifying self-similar profiles and refining decay and support expansion estimates for the first time.

Findings

Asymptotic profile is a self-similar solution of the p-Laplacian.

Established optimal decay rates for solutions.

Determined optimal support expansion rates.

Abstract

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption $$\partial\_t u - \Delta\_{p}u + |\nabla u|^{q\_*} = 0 \quad \hbox{in} (0,\infty)\times\mathbb{R}^N\ ,$$ for $p\in(2,\infty)$ and $q\_*:=p-N/(N+1)$. We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the $p$-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.