Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions

Article type: Research Article

Authors: Benachour, Saïd | Dăbuleanu-Hapca, Simona | Laurençot, Philippe

Affiliations: Institut Elie Cartan, Université Henri Poincaré-Nancy I, BP 239, F-54506 Vandoeuvre-lès-Nancy, France E-mail: Said.Benachour@iecn.u-nancy.fr | SIMBIOS Centre, University of Abertay Dundee, DD1 1HG Dundee, UK E-mail: simona.hapca@abertay.ac.uk | Mathématiques pour l'Industrie et la Physique, Université Paul Sabatier-Toulouse 3, 118 route de Narbonne, F-31062 Toulouse cedex 9, France E-mail: laurenco@mip.ups-tlse.fr

Abstract: Global classical solutions to the viscous Hamilton–Jacobi equation ut−Δu=a|∇u|p in (0,∞)×Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W1,∞(Ω) at the same speed as the linear heat semigroup when p>1. For p=1, an exponential decay to zero is also obtained but the rate depends on a and differs from that of the linear heat equation. Finally, if p∈(0,1) and a<0, finite time extinction occurs for non-negative solutions.

Journal: Asymptotic Analysis, vol. 51, no. 3-4, pp. 209-229, 2007