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

We study the large-time behaviour of the solutions u of the evolution equation involving nonlinear diffusion and gradient absorption ∂tu−∆pu+ |∇u| q = 0. We consider the problem posed for x ∈ R 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 |∇u| 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.