Journal of the Mathematical Society of Japan

Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation

Romain JOLY and Julien ROYER

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.

Article information

J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1375-1418.

Received: 31 March 2017
First available in Project Euclid: 6 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L05: Wave equation
Secondary: 35B40: Asymptotic behavior of solutions 47B44: Accretive operators, dissipative operators, etc. 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 47A10: Spectrum, resolvent

damped wave equation energy decay diffusive phenomenon periodic media


JOLY, Romain; ROYER, Julien. Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation. J. Math. Soc. Japan 70 (2018), no. 4, 1375--1418. doi:10.2969/jmsj/77667766.

Export citation


  • L. Aloui, S. Ibrahim and M. Khenissi, Energy decay for linear dissipative wave equations in exterior domains, J. Differ. Equations, 259 (2015), 2061–2079.
  • L. Aloui and M. Khenissi, Stabilisation pour l'équation des ondes dans un domaine extérieur, Rev. Math. Iberoamericana, 18 (2002), 1–16.
  • G. Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, New York, 2002.
  • J.-F. Bony and D. Häfner, Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds, Annales Scientifiques de l' École Normale Supérieure, 45 (2012), 311–335.
  • N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains, Commun. Contemp. Math., 18 (2016), 1650012.
  • A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, Elsevier Science Ltd, 1978.
  • C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024–1065.
  • J.-M. Bouclet, Low frequency estimates and local energy decay for asymptotically Euclidean laplacians, Comm. Part. Diff. Equations, 36 (2011), 1239–1286.
  • J.-M. Bouclet and J. Royer, Local energy decay for the damped wave equation, Jour. Func. Anal., 266 (2014), 4538–4615.
  • N. Burq, 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 (1998), 1–29.
  • R. Chill and A. Haraux, An optimal estimate for the time singular limit of an abstract wave equation, Funkc. Ekvacioj, Ser. Int., 47 (2004), 277–290.
  • C. Conca, R. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM J. Math. Anal., 33 (2002), 1166–1198.
  • C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), 1639–1659.
  • F. Dewez, Asymptotic estimates of oscillatory integrals with general phase and singular amplitude: Applications to dispersive equations, 2016, preprint, Arxiv 1507.00883.
  • K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer, 2000.
  • T. Hosono and T. Ogawa, Large time behavior and $L^{p}$-$L^{q}$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differ. Equations, 203 (2004), 82–118.
  • R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differ. Equations, 186 (2002), 633–651.
  • R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential, J. Math. Soc. Japan, 65 (2013), 183–236.
  • T. Kato, Perturbation Theory for linear operators, Classics in Mathematics, Springer, second edition, 1980.
  • M. Khenissi, Équation des ondes amorties dans un domaine extérieur, Bull. Soc. Math. France, 131 (2003), 211–228.
  • G. Lebeau, Équation des ondes amorties, In: Algebraic and geometric methods in mathematical physics, (eds. A. Boutet de Monvel and V. Marchenko), Kluwer Academic Publishers, 1996, 73–109.
  • G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465–491.
  • A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169–189.
  • R. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J., 46 (1979), 43–59.
  • P. Marcati and K. Nishihara, The $L^{p}$–$L^{q}$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differ. Equations, 191 (2003), 445–469.
  • M. Malloug and J. Royer, Energy decay in a wave guide with damping at infinity, ESAIM: Control, Optimisation and Calculus of Variations, 24 (2018), 519–549.
  • C. S. Morawetz, J. V. Ralston and W. A. Strauss, Decay of the solution of the wave equation outside non-trapping obstacles, Comm. on Pure and Applied Math., 30 (1977), 447–508.
  • T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585–626.
  • K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631–649.
  • H. Nishiyama, Remarks on the asymptotic behavior of the solution to damped wave equations, J. Differ. Equations, 261 (2016), 3893–3940.
  • J. H. Ortega and E. Zuazua, Large time behavior in ${\mathbb{R}}^N$ for linear parabolic equations with periodic coefficients, Asymptotic Anal., 22 (2000), 51–85.
  • R. Orive, E. Zuazua and A. F. Pazoto, Asymptotic expansion for damped wave equations with periodic coefficients, Math. Models Methods Appl. Sci., 11 (2001), 1285–1310.
  • J. Royer, Local energy decay and diffusive phenomenon in a dissipative wave guide, J. Spectral Theory, 8 (2018), 769–841.
  • J. Royer, Local decay for the damped wave equation in the energy space, J. Inst. Math. Jussieu, 17 (2018), 509–540.
  • J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79–86.
  • P. Radu, G. Todorova and B. Yordanov, Decay estimates for wave equations with variable coefficients, Trans. Amer. Math. Soc., 362 (2010), 2279–2299.
  • P. Radu, G. Todorova and B. Yordanov, The generalized diffusion phenomenon and applications, SIAM J. Math. Anal., 48 (2016), 174–203.
  • M. Sobajima and Y. Wakasugi, Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain, J. Differ. Equations, 261 (2016), 5690–5718.
  • L. Tartar, The General Theory of Homogenization, Lecture Notes of the Unione Matematica Italiana, 2009.
  • G. Todorova and B. Yordanov, Weighted $L^2$-estimates for dissipative wave equations with variable coefficients, J. Differ. Equations, 246 (2009), 4497–4518.
  • Y. Wakasugi, On diffusion phenomena for the linear wave equation with space-dependent damping, J. Hyperbolic Differ. Equ., 11 (2014), 795–819.
  • Y. Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452–487.
  • J. Wunsch, Periodic damping gives polynomial energy decay, Math. Res. Lett., 24 (2017), 571–580.
  • M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, 138, Amer. Math. Soc., 2012.