We consider the problem posed for \( x\in ℝ^N \) and with nonnegative and compactly supported initial data. We take the exponent which corresponds to slow -Laplacian diffusion. The main feature of the paper is that the exponent takes the critical value which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term and the diffusive term 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 , 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.
Cite this article
Razvan Gabriel Iagar, Philippe Laurençot, Juan Luis Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent. Interfaces Free Bound. 13 (2011), no. 2, pp. 271–295DOI 10.4171/IFB/258