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

  • Saïd Benachour
  • Philippe Laurençot


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.


Viscous Hamilton-Jacobi equation temporal decay estimates gradient estimates 

Mathematics Subject Classification (2000)

35B40 35B45 35K55 


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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

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