On the Flow Map for a Class of Parabolic Equations

Luc Molinet (), Francis Ribaud () and Abdellah Youssfi ()
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Luc Molinet: Université Paris-Nord, L.A.G.A., Institut Galilée
Francis Ribaud: Université de Marne-La-Vallée, Equipe d’Analyse et de Mathématiques Appliquées
Abdellah Youssfi: Université de Marne-La-Vallée, Equipe d’Analyse et de Mathématiques Appliquées

A chapter in Function Spaces, Differential Operators and Nonlinear Analysis, 2003, pp 393-401 from Springer

Abstract: Abstract We consider the Cauchy problem for the one-dimensional parabolic equations $$\partial _t u - \partial _{xx} u \pm \partial _x^d u^k = 0,\;k \in \mathbb{N}^* ,\;d \in \{ 0,1\} ,$$ , with initial data in $$H^s (\mathbb{R}).$$ . We study the flow map corresponding to the integral equation. Our results complete the known results on ill-posedness in $$H^s (\mathbb{R}).$$ and show the particularity of the case (k, d) = (2, 0) for which we prove that the critical space $$H^{s_c } (\mathbb{R}) = H^{ - 3/2} (\mathbb{R})$$ suggesting by standard scaling arguments cannot be reached. Our results hold also in the periodic setting.

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-8035-0_28

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DOI: 10.1007/978-3-0348-8035-0_28

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