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

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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 ∈ ℝN, wherep≥1 andu(0,.)=u 0≥0,u 0≢0,u 0∈L 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 t=Δu+u p.

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Authors and Affiliations

  1. Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, Université Paul Sabatier-Toulouse 3, 118 route de Narbonne, 31062, Toulouse cedex 4, France

    Philippe Laurençot

  2. Département de Mathématiques, INSSET, Université de Picardie, 02109, St-Quentin, France

    Philippe Souplet

  3. Laboratoire de Mathématiques Appliquées, CNRS UMR 7641, Université de Versailles, 45 avenue des Etats-Unis, 78035, Versailles, France

    Philippe Souplet

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  1. Philippe Laurençot
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  2. Philippe Souplet
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Correspondence to Philippe Laurençot.

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Laurençot, P., Souplet, P. On the growth of mass for a viscous Hamilton-Jacobi equation. J. Anal. Math. 89, 367–383 (2003). https://doi.org/10.1007/BF02893088

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  • DOI: https://doi.org/10.1007/BF02893088

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