Advertisement

Decay estimates for “anisotropic” viscous Hamilton-Jacobi equations in ℝN

  • Saïd Benachour
  • Philippe Laurençot

Abstract

The large time behaviour of the L q -norm of nonnegative solutions to the “anisotropic” viscous Hamilton-Jacobi equation
$$ {u_{t}} - \Delta u + {\sum\limits_{{i = 1}}^{m} {|{u_{{xi}}}|} ^{{Pi}}} = 0 in {\mathbb{R}_{ + }} x {\mathbb{R}^{N}}, $$
is studied for q = 1 and q = ∞, where m ∈ {1,...,N} and p i for i ∈ {1,...,m}. The limit of theL 1-norm is identified, and temporal decay estimates for the L -norm are obtained, according to the values of the p i ’s. The main tool in our approach is the derivation of L-decay estimates for \( \nabla ({u^{\alpha }}),\alpha \in (0,1] \), by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.

Keywords

Viscous Hamilton-Jacobi equation temporal decay estimates gradient estimates 

Mathematics Subject Classification (2000)

35B40 35B45 35K55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Amour, L. and Ben-Artzi, M.Global existence and decay for viscous Hamilton-Jacobi equationsNonlinear Anal. 31 (1998), 621–628.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Ben-Artzi, M. and Koch, H.Decay of mass for a semilinear parabolic equationComm. Partial Differential Equations24(1999), 869–881.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Ben-Artzi, M., Souplet, PH. and Weissler, F.B.The local theory for viscous Hamilton-Jacobi equations in Lebesgue spacesJ. Math. Pures Appl.81(2002), 343–378.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Benachour, S. and LaurenÇot, PH.Global solutions to viscous Hamilton-Jacobi equations with irregular initial dataComm. Partial Differential Equations24(1999), 1999–2021.zbMATHCrossRefGoogle Scholar
  5. [5]
    Benachour, S., LaurenÇot, PH. and Schmitt, D.Extinction and decay estimates for viscous Hamilton-Jacobi equations in IC AProc. Amer. Math. Soc.130(2002), 1103–1111.zbMATHGoogle Scholar
  6. [6]
    Bénilan, PH.Evolution equations and accretive operatorsLecture notes taken by S. Lenhardt, University of Kentucky, Spring 1981.Google Scholar
  7. [7]
    Biler, P., Guedda, M. and Karcii, G.Asymptotic properties of solutions of the viscous Hamilton-Jacobi equationprépublication LAMIFA, Université de Picardie, 2000.Google Scholar
  8. [8]
    Dautray, R. and Lions, J. L.Analyse mathématique et calcul numérique pour les sciences et les techniquesvol. 3, with the collaboration of Philippe Bénilan, Michel Cessenat, Bertrand Mercier and Claude Zuily, Masson, Paris, 1987.Google Scholar
  9. [9]
    Gilding, B. H., Guedda, M. and Kersner, R.The Cauchy problem for \( {u_t} = \Delta u + {\left| {\nabla u} \right|^q} \), prépublication LAMFA28Université de Picardie, 1998.Google Scholar
  10. [10]
    LaurenÇot, PH. and Souplet, P1í.On the growth of mass for a viscous Hamilton-Jacobi equationJ. Anal. Math., to appear.Google Scholar

Copyright information

© Springer Basel AG 2003

Authors and Affiliations

  • Saïd Benachour
    • 1
  • Philippe Laurençot
    • 2
  1. 1.Institut Elie Cartan — NancyUniversité de Nancy 1Vandœuvre-lès-Nancy cedexFrance
  2. 2.Mathématiques pour l’Industrie et la Physique CNRS UMR 5640Université Paul Sabatier—Toulouse 3 118 route de NarbonneFrance

Personalised recommendations