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

2006/09/27 by Thierry Gallay, Gallay, Thierry, Philippe Laurençot +1
Physics and Astronomy · Engineering · Mathematics · #Quantum chaos and dynamical systems #Control and Stability of Dynamical Systems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.math/0609750

Abstract

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation u_t - Δu + |∇ 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.

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