Time decay for hyperbolic equations with homogeneous symbols

Tokio Matsuyama; Michael Ruzhansky

doi:10.1016/j.crma.2009.06.010

Partial Differential Equations

Time decay for hyperbolic equations with homogeneous symbols
[Estimations en temps pour des équations hyperboliques à symboles homogènes]

Présenté par : Jean-Michel Bony

Tokio Matsuyama ¹ ; Michael Ruzhansky ²

¹ Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
² Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 915-919.

Résumé
Abstract

Dans cette Note, nous établissons des estimations dispersives des solutions d'équations strictement hyperboliques à coefficients dépendant du temps et à dérivées intégrables. Nous estimons le taux de décroissance en temps des normes des solutions en fonction d'indices géométriques associés aux caractéristiques de l'équation limite. Les résultats sont appliqués à la résolvabilité globale des équations de type Kirchhoff à données petites et à des estimations dispersives des solutions.

The aim of this Note is to present dispersive estimates for strictly hyperbolic equations with time dependent coefficients that have integrable derivatives. We will relate the time decay rate of $L^{p} – L^{q}$ norms of solutions to certain geometric indices associated to the characteristics of the limiting equation. Results will be applied to the global solvability of Kirchhoff type equations with small data, and to the dispersive estimates for their solutions.

Reçu le : 2007-08-01
Accepté le : 2009-05-23
Publié le : 2009-07-02

DOI : 10.1016/j.crma.2009.06.010

Affiliations des auteurs :

Tokio Matsuyama ¹ ; Michael Ruzhansky ²

¹ Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
² Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

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     title = {Time decay for hyperbolic equations with homogeneous symbols},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {915--919},
     publisher = {Elsevier},
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     number = {15-16},
     year = {2009},
     doi = {10.1016/j.crma.2009.06.010},
     language = {en},
}

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Tokio Matsuyama; Michael Ruzhansky. Time decay for hyperbolic equations with homogeneous symbols. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 915-919. doi : 10.1016/j.crma.2009.06.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.010/

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