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, Equation required. We consider the problem for x ε ℝN 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 |Δu|p and the diffusive term Δpu have a similar size for large times. The study performed in this paper shows that a delicate asymptotic equilibrium occurs, so that the large-time behaviour of solutions is described by a rescaled version of a suitable self-similar solution of the Hamilton-Jacobi equation |ΔW| 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. © European Mathematical Society 2011.