Time decay for hyperbolic equations with homogeneous symbols

Abstract

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. To cite this article: T. Matsuyama, M. Ruzhansky, C. R. Acad. Sci. Paris, Ser. I 347 (2009). Résumé Estimations en temps pour des équations hyperboliques à symboles homogènes. 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. Pour citer cet article : T. Matsuyama, M. Ruzhansky, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Asymptotic Behaviour of Solutions to Hyperbolic Partial Differential Equations

“…Equations with homogeneous symbols and time-dependent coefficients appear naturally also in the analysis of the Kirchhoff equations. Higher order equations of Kirchhoff type were discussed by Matsuyama and Ruzhansky in [29].…”

Dispersive type estimates for Fourier integrals and applications to hyperbolic systems

“…only that a k,α ∈ C 1 in terms of the regularity in time, which is crucial in applications to the Kirchhoff equations, see e.g. [4].…”

Dispersive estimates for hyperbolic systems with time-dependent coefficients

“…In [6], the asymptotic integration method was developed for scalar hyperbolic equations with time dependent homogeneous symbols. The emphasis there was placed on minimising the time-differentiability assumptions on the symbol, required by applications to the Kirchhoff equations ( [5]). Indeed, the representation of solutions and dispersive estimates have been derived in [6] for equations of any order, by making assumptions on one time-derivative of the coefficients only, see Remark 21.…”