Journal d’Analyse Mathématique

, Volume 89, Issue 1, pp 367–383

On the growth of mass for a viscous Hamilton-Jacobi equation

Article

DOI: 10.1007/BF02893088

Cite this article as:
Laurençot, P. & Souplet, P. J. Anal. Math. (2003) 89: 367. doi:10.1007/BF02893088

Abstract

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationut = Δu + |Δu|p,t>0,x ∈ ℝN, wherep≥1 andu(0,.)=u0≥0,u0≢0,u0L1. DenotingI=limt→∞u(t)1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:
  • • ifp≤(N+2)/(N+1), thenI=∞ for allu0;

  • • if (N+2)/(N+1)<p<2, then bothI=∞ andI<∞ occur;

  • • ifp≥2, thenI<∞ for allu0.

We also consider a similar question for the equationutu+up.

Copyright information

© Hebrew University of Jerusalem 2003

Authors and Affiliations

  1. 1.Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640Université Paul Sabatier-Toulouse 3Toulouse cedex 4France
  2. 2.Département de Mathématiques, INSSETUniversité de PicardieSt-QuentinFrance
  3. 3.Laboratoire de Mathématiques Appliquées, CNRS UMR 7641Université de VersaillesVersaillesFrance