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Ill-posedness issues for a class of parabolic equations

Published online by Cambridge University Press:  12 July 2007

Luc Molinet ,
Francis Ribaud  and
Abdellah Youssfi
Luc Molinet
Affiliation:
LAGA, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France ([email protected])
Francis Ribaud
Affiliation:
Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne-La-Vallée, 5 bd Descartes, Cité Descartes, Champs-sur-Marne, 77454 Marne-La-Vallée Cedex 2, [email protected]
Abdellah Youssfi
Affiliation:
Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne-La-Vallée, 5 bd Descartes, Cité Descartes, Champs-sur-Marne, 77454 Marne-La-Vallée Cedex 2, [email protected]
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Abstract

We prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in Hs(R), cannot be solved by an iterative scheme based on the Duhamel formula for s < −1 if (k, d) = (2, 0) and s < sc(k, d) = ½ − (2 − d)/(k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in Hs(R) for these equations and shows the particularity of the case (k, d) = (2, 0), for which we prove that the critical space Hsc(R) = H−3/2(R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 132 , Issue 6 , December 2002 , pp. 1407 - 1416
Copyright
Copyright © Royal Society of Edinburgh 2002

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