Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:11:44.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Improved local energy decay for the wave equation on asymptotically Euclidean odd dimensional manifolds in the short range case

Part of: Hyperbolic equations and systems Partial differential equations on manifolds; differential operators General theory of linear operators Scattering theory Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  13 July 2012

Jean-François Bony  and
Dietrich Häfner
Jean-François Bony
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France (bony@math.u-bordeaux1.fr)
Dietrich Häfner
Affiliation:
Université de Grenoble 1, Institut Fourier, UMR 5582 du CNRS, BP 74, 38402 St Martin d’Hères, France (Dietrich.Hafner@ujf-grenoble.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show improved local energy decay for the wave equation on asymptotically Euclidean manifolds in odd dimensions in the short range case. The precise decay rate depends on the decay of the metric towards the Euclidean metric. We also give estimates of powers of the resolvent of the wave propagator between weighted spaces.

Keywords

local energy decayresolvent smoothnesswave equationodd dimensionslow frequenciesasymptotically Euclidean manifolds

MSC classification

Secondary: 35L05: Wave equation 35P25: Scattering theory 47A10: Spectrum, resolvent 58J45: Hyperbolic equations 81U30: Dispersion theory, dispersion relations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 3 , July 2013 , pp. 635 - 650
Copyright
©Cambridge University Press 2012 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aguilar, J. and Combes, J. M., A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279.Google Scholar
Alinhac, S., Geometric analysis of hyperbolic differential equations: an introduction, London Mathematical Society Lecture Note Series, Volume 374 (Cambridge University Press, 2010).Google Scholar
Bony, J.-F. and Häfner, D., The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 35 (2010), 23–67.Google Scholar
Bony, J.-F. and Häfner, D., Low frequency resolvent estimates for long range perturbations of the Euclidean Laplacian, Math. Res. Lett. 17 (2) (2010), 301–306.CrossRefGoogle Scholar
Bony, J.-F. and Häfner, D., Local energy decay for several evolution equations on asymptotically Euclidean manifolds, Ann. Sci. Éc. Norm. Supér. 45 (2) (2012), 311–335.Google Scholar
Bouclet, J.-M., Low frequency estimates and local energy decay for asymptotically euclidean Laplacians, Comm. Partial Differential Equations 36 (2011), 1239–1286.Google Scholar
Burq, N., Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math. 180 (1) (1998), 1–29.Google Scholar
Dolph, C. L., McLeod, J. B. and Thoe, D., The analytic continuation of the resolvent kernel and scattering operator associated with the Schroedinger operator, J. Math. Anal. Appl. 16 (1966), 311–332.Google Scholar
Fröhlich, J., Griesemer, M. and Sigal, I. M., Spectral theory for the standard model of non-relativistic QED, Comm. Math. Phys. 283 (3) (2008), 613–646.Google Scholar
Gérard, C. and Sigal, I. M., Space-time picture of semiclassical resonances, Comm. Math. Phys. 145 (2) (1992), 281–328.Google Scholar
Hörmander, L., The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften, Volume 256 (Springer, 1983).Google Scholar
Hörmander, L., Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications, Volume 26 (Springer, 1997).Google Scholar
Hunziker, W., Distortion analyticity and molecular resonance curves, Ann. Inst. H. Poincaré Phys. Théor. 45 (4) (1986), 339–358.Google Scholar
Jensen, A. and Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (3) (1979), 583–611.Google Scholar
Kian, Y., On the meromorphic continuation of the resolvent for the wave equation with time-periodic perturbations and applications, preprint, 2011 (arXiv:1103.2530).Google Scholar
Klainerman, S., A commuting vectorfields approach to Strichartz-type inequalities and applications to quasi-linear wave equations, Int. Math. Res. Not. (5) (2001), 221–274.CrossRefGoogle Scholar
Lax, P. and Phillips, R., Scattering theory, second ed., Pure and Applied Mathematics, Volume 26 (Academic Press Inc., 1989), With appendices by C. Morawetz and G. Schmidt.Google Scholar
Ralston, J., Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), 807–823.CrossRefGoogle Scholar
Rauch, J., Local decay of scattering solutions to Schrödinger’s equation, Comm. Math. Phys. 61 (2) (1978), 149–168.Google Scholar
Sá Barreto, A. and Zworski, M., Existence of resonances in three dimensions, Comm. Math. Phys. 173 (2) (1995), 401–415.Google Scholar
Sjöstrand, J., Lectures on resonances, preprint on http://math.u-bourgogne.fr/IMB/sjostrand/index.html, 2007, 1–169.Google Scholar
Sjöstrand, J. and Zworski, M., Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (4) (1991), 729–769.Google Scholar
Tang, S.-H. and Zworski, M., Resonance expansions of scattered waves, Comm. Pure Appl. Math. 53 (10) (2000), 1305–1334.Google Scholar
Vaĭnberg, B., Asymptotic methods in equations of mathematical physics. (Gordon & Breach Science Publishers, 1989).Google Scholar