Asymptotic Behaviour of a Nonlinear Parabolic Equation with Gradient Absorption and Critical Exponent

We study the large-time behaviour of the solutions 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 $x\in \real^N$ 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.