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

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 $x\in {\mathbb{R}}^N$ 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 $|\nabla u{|}^q$ and the diffusive term ${\Delta }_{p}u$ 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 $|\nabla W{|}^{p-1}=W$, 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.