[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.
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 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.
Accepté le :
Publié le :
@article{CRMATH_2009__347_15-16_915_0,
author = {Tokio Matsuyama and Michael Ruzhansky},
title = {Time decay for hyperbolic equations with homogeneous symbols},
journal = {Comptes Rendus. Math\'ematique},
pages = {915--919},
publisher = {Elsevier},
volume = {347},
number = {15-16},
year = {2009},
doi = {10.1016/j.crma.2009.06.010},
language = {en},
}
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/
[1] Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985
[2] Global existence for nonlinear hyperbolic systems of Kirchhoff type, J. Differential Equations, Volume 132 (1996), pp. 239-274
[3] A class of nonlinear hyperbolic problems with global solutions, Arch. Rational Mech. Anal., Volume 124 (1993), pp. 201-219
[4] Kirchhoff type equations depending on a small parameter, Chinese Ann. Math. Ser. B, Volume 16 (1995), pp. 413-430
[5] On the global solvability of Kirchhoff equation for non-analytic initial data, J. Differential Equations, Volume 211 (2005), pp. 38-60
[6] Asymptotic behaviour for wave equations with time-dependent coefficients, Ann. Univ. Ferrara, Sez. VII – Sci. Math., Volume 52 (2006) no. 2, pp. 383-393
[7] T. Matsuyama, M. Ruzhansky, Asymptotic integration and dispersion for hyperbolic equations, with applications to Kirchhoff equations, preprint
[8] The Theory of Partial Differential Equations, Cambridge Univ. Press, 1973
[9] decay estimates for wave equations with time-dependent coefficients, J. Nonlinear Math. Phys., Volume 11 (2004), pp. 534-548
[10] estimates for wave equation with bounded time dependent coefficient, Hokkaido Math. J., Volume 34 (2005), pp. 541-586
[11] Singularities of affine fibrations in the theory of regularity of Fourier integral operators, Russian Math. Surveys, Volume 55 (2000), pp. 93-161
[12] A priori estimates for higher order hyperbolic equations, Math. Z., Volume 215 (1994), pp. 519-531
[13] Estimates for hyperbolic equations with non-convex characteristics, Math. Z., Volume 222 (1996), pp. 521-531
[14] Asymptotic integrations of adiabatic oscillator, Amer. J. Math., Volume 69 (1947), pp. 251-272
[15] Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations, Volume 143 (1998), pp. 1-59
Cité par Sources :
Commentaires - Politique
Almost global well-posedness of Kirchhoff equation with Gevrey data
Tokio Matsuyama; Michael Ruzhansky
C. R. Math (2017)
On the boundedness of Fourier integral operators on
Sandro Coriasco; Michael Ruzhansky
C. R. Math (2010)