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

Saïd Benachour () and Philippe Laurençot ()
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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

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 27-37 from Springer

Abstract: 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: 35B40; 35B45; 35K55; Viscous Hamilton-Jacobi equation; temporal decay estimates; gradient estimates (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_3

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DOI: 10.1007/978-3-0348-7924-8_3

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