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 $\partial_t u - \Delta u + |\nabla u|^q = 0$ 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.