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

In: Nonlinear Evolution Equations and Related Topics

Author

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  • Saïd Benachour

    (Université de Nancy 1, Institut Elie Cartan — Nancy)

  • Philippe Laurençot

    (Université Paul Sabatier—Toulouse 3 118 route de Narbonne, Mathématiques pour l’Industrie et la Physique CNRS UMR 5640)

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.

Suggested Citation

  • Saïd Benachour & Philippe Laurençot, 2003. "Decay estimates for “anisotropic” viscous Hamilton-Jacobi equations in ℝ N," Springer Books, in: Wolfgang Arendt & Haïm Brézis & Michel Pierre (ed.), Nonlinear Evolution Equations and Related Topics, pages 27-37, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7924-8_3
    DOI: 10.1007/978-3-0348-7924-8_3
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