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 $$\partial_t u - \Delta_p u + \vert\nabla u\vert^q = 0$$ 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 $$ \vert\nabla u\vert^q$$ 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.