# Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent

### Razvan Gabriel Iagar

Universidad Autónoma de Madrid, Spain### Philippe Laurençot

Université de Toulouse, Toulouse, France### Juan Luis Vázquez

Universidad Autónoma de Madrid, Spain

## Abstract

We study the large-time behaviour of solutions of the evolution equation involving nonlinear diffusion and gradient absorption,

We consider the problem posed for $x∈R_{N}$ and $t>0$ with nonnegative and compactly supported initial data. We take the exponent $p>2$ which corresponds to slow $p$-Laplacian diffusion. The main feature of the paper is that the exponent $q$ takes the critical value $q=p−1$ which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term $∣∇u∣_{q}$ and the diffusive term $Δ_{p}u$ have a similar size for large times. The study performed in this paper shows that a delicate asymptotic equilibrium happens, so that the large-time behaviour of the solutions is described by a rescaled version of a suitable self-similar solution of the Hamilton-Jacobi equation $∣∇W∣_{p−1}=W$, with logarithmic time corrections. The asymptotic rescaled profile is a kind of sandpile with a cusp on top, and it is independent of the space dimension.

## Cite this article

Razvan Gabriel Iagar, Philippe Laurençot, Juan Luis Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent. Interfaces Free Bound. 13 (2011), no. 2, pp. 271–295

DOI 10.4171/IFB/258