When is the energy of the 1D damped Klein-Gordon equation decaying?

  • Satbir Malhi
  • Milena Stanislavova


We study time decay of the energy for the one dimensional damped Klein-Gordon equation. We give an explicit necessary and sufficient condition on the continuous damping function \(\gamma \ge 0\) for which the energy
$$\begin{aligned} E(t)=\int _{-\infty }^\infty |u_x|^2+|u|^2+ |u_t|^2 dx \end{aligned}$$
decays, whenever \((u(0), u_t(0))\in H^2(\mathbb R)\times H^1(\mathbb R)\).

Mathematics Subject Classification

35B35 35B40 35G30 


  1. 1.
    Anantharaman, N., Léautaud, M.: Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE 7(1), 159–214 (2014). (With an appendix by Stéphane Nonnenmacher) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bardos, C., Lebeau, G., Rauch, J.: Un exemple dutilisation des notions de propagation pour le contrôle et la stabilisation de problemes hyperboliques. Rend. Sem. Mat. Univ. Politec. Torino pp. 11–31 (1988)Google Scholar
  3. 3.
    Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Batty, C., Borichev, A., Tomilov, Y.: \(L^p\)-tauberian theorems and \(L^p\)-rates for energy decay. J. Functional Anal. 270(3), 1153–1201 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burq, N., Hitrik, M.: Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 14(1), 35–47 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burq, N., Joly, R.: Exponential decay for the damped wave equation in unbounded domains. Commun. Contemp. Math. 18(06), 1650012 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gearhart, L.: Spectral theory for contraction semigroups on hilbert space. Trans. Am. Math. Soc. 236, 385–394 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gesztesy, F., Jones, C.K.R.T., Latushkin, Y., Stanislavova, M.: A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. Indiana Univ. Math. J. 49(1), 221–243 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Huang, F.: Characteristic conditions for exponential stability of linear dynamical systems in hilbert spaces. Ann. Differ. Equ. 1(1), 43–56 (1985)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Latushkin, Y., Shvydkoy, R.: Hyperbolicity of semigroups and Fourier multipliers. In: Borichev, A.A., Nikolski, N.K. (eds.) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol. 129, pp. 341–363. Birkhäuser, Basel (2001)CrossRefGoogle Scholar
  11. 11.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations, applied mathematical sciences, vol. 44. Springer-Verlag, New York (1983)CrossRefzbMATHGoogle Scholar
  12. 12.
    Phung, K.: Polynomial decay rate for the dissipative wave equation. J. Differ. Equ. 240(1), 92–124 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Prüss, J.: On the spectrum of \(c_0\)-semigroups. Trans. Am. Amer. Math. Soc. 284(2), 847–857 (1984)CrossRefzbMATHGoogle Scholar
  14. 14.
    Rauch, J., Taylor, M., Phillips, R.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24(1), 79–86 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stahn, R.: Optimal decay rate for the wave equation on a square with constant damping on a strip. Zeitschrift für angewandte Mathematik und Physik 68(2), 36 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wunsch, Jared: Periodic damping gives polynomial energy decay. Math. Res. Lett. 24, 519–528 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA

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