Convergence to steady states for a one-dimensional viscous Hamilton–Jacobi equation with Dirichlet boundary conditions

Philippe Laurençot

Vol. 230 (2007), No. 2, 347–364

DOI: 10.2140/pjm.2007.230.347

Abstract

We investigate the convergence to steady states of the solutions to the one-dimensional viscous Hamilton–Jacobi equation ∂_tu − ∂_x²u = |∂_xu|^p, where (t,x) ∈ (0,∞) × (−1,1) and p ∈ (0,1), with homogeneous Dirichlet boundary conditions. For that purpose, a Liapunov functional is constructed by the approach of Zelenyak (1968). Instantaneous extinction of ∂_xu on a subinterval of (−1,1) is shown for suitable initial data.

Keywords

diffusive Hamilton–Jacobi equation, convergence to steady states, gradient extinction, Liapunov functional

Mathematical Subject Classification 2000

Primary: 35B40

Secondary: 35K55, 37B25

Milestones

Received: 24 October 2005

Accepted: 2 March 2006

Published: 1 April 2007