Extinction and non‐extinction for viscous Hamilton–Jacobi equations in $\mathbb{R}$^N

Article type: Research Article

Authors: Benachour, S. | Laurençot, Ph.^; | Schmitt, D. | Souplet, Ph.^;

Affiliations: Institut Elie Cartan‐Nancy, Université de Nancy 1, BP 239, F‐54506 Vandœuvre‐lès‐Nancy cedex, France | Mathématiques pour l'Industrie et la Physique, UMR CNRS 5640, Université Paul Sabatier‐Toulouse 3, 118, route de Narbonne, F‐31062 Toulouse cedex 4, France E‐mail: [email protected]‐tlse.fr | Département de Mathématiques, INSSET, Université de Picardie, F‐02109 Saint‐Quentin, France | Laboratoire de Mathématiques Appliquées, Université de Versailles‐Saint Quentin, 45, avenue des Etats‐Unis, F‐78035 Versailles, France

Note: [] Corresponding author.

Abstract: Extinction in finite time and non‐compactness of the support are investigated for non‐negative classical solutions to the Cauchy problem ut−Δu+|∇u|p=0 when p∈(0,1). The occurrence of these phenomena is shown to depend on the behaviour of the initial data u0 for large values of x. In particular, we obtain the optimal decay rate of u0 at infinity to ensure finite‐time extinction of u. The case of periodic boundary conditions is also considered. Finally the optimality of temporal L∞‐decay estimates obtained previously is discussed.

Keywords: extinction in finite time, non‐extinction, temporal decay estimates, viscous Hamilton–Jacobi equations

Journal: Asymptotic Analysis, vol. 31, no. 3-4, pp. 229-246, 2002