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Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $

Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $ The large time behaviour of the $ L^q $ -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ $u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,$ ¶¶is studied for $ q=1 $ and $ q=\infty $ , where $ m\in\{1,\ldots,N\} $ and $ p_i\in [1,+\infty) $ for $ i\in\{1,\ldots,m\} $ . The limit of the $ L^1 $ -norm is identified, and temporal decay estimates for the $ L^\infty $ -norm are obtained, according to the values of the $ p_i $'s. The main tool in our approach is the derivation of $ L^\infty $ -decay estimates for $ \nabla\left(u^\alpha \right), \alpha\in (0,1] $ , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $

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References (4)

Publisher
Springer Journals
Copyright
Copyright © 2003 by Birkhäuser Verlag Basel,
Subject
Mathematics; Analysis
ISSN
1424-3199
DOI
10.1007/s000280300002
Publisher site
See Article on Publisher Site

Abstract

The large time behaviour of the $ L^q $ -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ $u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,$ ¶¶is studied for $ q=1 $ and $ q=\infty $ , where $ m\in\{1,\ldots,N\} $ and $ p_i\in [1,+\infty) $ for $ i\in\{1,\ldots,m\} $ . The limit of the $ L^1 $ -norm is identified, and temporal decay estimates for the $ L^\infty $ -norm are obtained, according to the values of the $ p_i $'s. The main tool in our approach is the derivation of $ L^\infty $ -decay estimates for $ \nabla\left(u^\alpha \right), \alpha\in (0,1] $ , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Feb 1, 2003

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