Journal d’Analyse Mathématique

, Volume 89, Issue 1, pp 367–383 | Cite as

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

Article

Abstract

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu t = Δu + |Δu| p ,t>0,x ∈ ℝ N , wherep≥1 andu(0,.)=u 0≥0,u 0≢0,u 0L 1. DenotingI =lim t→∞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 allu 0;

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

  • • ifp≥2, thenI <∞ for allu 0.

We also consider a similar question for the equationu tu+u p .

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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

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