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

Abstract

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

17 pages, no figure