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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up


Authors: Nejla Nouaili and Hatem Zaag
Journal: Trans. Amer. Math. Soc. 362 (2010), 3391-3434
MSC (2000): Primary 35B05, 35K05, 35K55, 74H35
Published electronically: February 17, 2010
MathSciNet review: 2601595
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Abstract: We prove a Liouville theorem for a vector valued semilinear heat equation with no gradient structure. Classical tools such as the maximum principle or energy techniques break down and have to be replaced by a new approach. We then derive from this theorem uniform estimates for blow-up solutions of that equation.


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Additional Information

Nejla Nouaili
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm 75230, Paris Cedex 05, France
Address at time of publication: Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France
Email: nouaili@math.univ-paris13.fr

Hatem Zaag
Affiliation: Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France
Email: zaag@math.univ-paris13.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04902-0
PII: S 0002-9947(10)04902-0
Keywords: Blow-up, Liouville theorem, uniform estimates, heat equation, vector valued
Received by editor(s): October 24, 2007
Published electronically: February 17, 2010
Additional Notes: The authors would like to thank the referee for his valuable suggestions which (we hope) made our paper much clearer and reader friendly.
The second author was supported by a grant from the French Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.
Article copyright: © Copyright 2010 American Mathematical Society