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

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.