IUMJ

Title: Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent

Authors: Thierry Gallay and Philippe Laurencot

Issue: Volume 56 (2007), Issue 1, 459-480

Abstract:

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $\partial_t u - \Delta u + |\nabla u|^q = 0$ in $(0,\infty) \times \mathbb{R}^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a rescaled self-similar solution to the linear heat equation is shown, the rescaling factor being $(\ln{t})^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.