Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear parabolic equation with gradient absorption, revealing localized, conical asymptotic patterns driven by a Hamilton-Jacobi equation in the superlinear range.

Contribution

It introduces a new analysis of the large-time asymptotics for nonlinear diffusion with gradient absorption, highlighting the emergence of localized conical patterns.

Findings

Solutions exhibit localized, conical asymptotic shapes.

The large-time behavior is governed by a Hamilton-Jacobi equation.

Localization phenomenon occurs for specific exponent ranges.

Abstract

We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^q=0$. We consider the problem posed for $x\in {\mathbb R}^N $ and $t>0$ with non-negative and compactly supported initial data. We take the exponent $p>2$ which corresponds to slow $p$-Laplacian diffusion, and the exponent $q$ in the superlinear range $1<q<p-1$. In this range the influence of the Hamilton-Jacobi term $ |\nabla u|^q$ is determinant, and gives rise to the phenomenon of localization. The large time behaviour is described in terms of a suitable self-similar solution that solves a Hamilton-Jacobi equation. The shape of the corresponding spatial pattern is rather conical instead of bell-shaped or parabolic.