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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extinction and decay estimates for viscous Hamilton-Jacobi equations in ${\mathbb{R}}^N$


Authors: Said Benachour, Philippe Laurençot and Didier Schmitt
Journal: Proc. Amer. Math. Soc. 130 (2002), 1103-1111
MSC (1991): Primary 35B40, 35B05, 35K55
Published electronically: October 1, 2001
MathSciNet review: 1873785
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Abstract: We consider non-negative and integrable classical solutions to the Cauchy problem $u_t-\Delta u+\vert\nabla u\vert^p=0$ when $p\in (0,+\infty)$. For $p\in (0,N/(N+1))$ we prove that any such solution vanishes identically after a finite time. For higher values of $p$temporal decay estimates are obtained.


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Additional Information

Said Benachour
Affiliation: Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandœuvre les Nancy cedex, France
Email: benachou@iecn.u-nancy.fr

Philippe Laurençot
Affiliation: Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandœuvre les Nancy cedex, France
Address at time of publication: Mathématiques pour l’Industrie et la Physique, UNR CNRS 5640, Université Paul Sabatier-Toulouse 3, 118, route de Narbonne, F-31062 Toulouse Cedex 4, France
Email: laurenco@iecn.u-nancy.fr, laurencot@mip.ups-tlse.fr

Didier Schmitt
Affiliation: Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandœuvre les Nancy cedex, France
Email: dschmitt@iecn.u-nancy.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06140-8
PII: S 0002-9939(01)06140-8
Keywords: Extinction in finite time, temporal decay estimates, viscous Hamilton-Jacobi equations
Received by editor(s): March 23, 2000
Received by editor(s) in revised form: October 12, 2000
Published electronically: October 1, 2001
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society