A damping term for higher-order hyperbolic equations

  • Marcello D’Abbicco
  • Enrico JannelliEmail author


We construct a damping term for general higher-order strictly hyperbolic homogeneous equations with constant coefficients. We derive long-time decay estimates for the solution to the Cauchy problem, and we show that no better dissipative effect can be obtained with a different damping term.


Higher-order equations Damping Decay estimates 

Mathematics Subject Classification



  1. 1.
    Beauchard, K., Zuazua, E.: Large time asymptotics for partially dissipative hyperbolic systems. Arch. Ration. Mech. Anal. 199(1), 177–227 (2011). doi: 10.1007/s00205-010-0321-y MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D’Abbicco, M.: The influence of a nonlinear memory on the damped wave equation. Nonlinear Anal. 95, 130–145 (2014). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D’Abbicco, M.: The threshold of effective damping for semilinear wave equations. Math. Methods Appl. Sci. doi: 10.1002/mma.3126
  4. 4.
    D’Abbicco, M., Lucente, S., Reissig, M.: Semilinear wave equations with effective damping. Chin. Ann. Math. 34B(3), 345–380 (2013). doi: 10.1007/s11401-013-0773-0 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ikehata, R., Mayaoka, Y., Nakatake, T.: Decay estimates of solutions for dissipative wave equations in \({\mathbb{R}}^N\) with lower power nonlinearities. J. Math. Soc. Jpn. 56(2), 365–373 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ikehata, R., Tanizawa, K.: Global existence of solutions for semilinear damped wave equations in \(R^N\) with noncompactly supported initial data. Nonlinear Anal. 61, 1189–1208 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lin, J., Nishihara, K., Zhai, J.: Critical exponent for the semilinear wave equation with time-dependent damping. Discrete Contin. Dyn. Syst. 32(12), 4307–4320 (2012). doi: 10.3934/dcds.2012.32.4307 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Matsumura, A.: On the asymptotic behavior of solutions of semi-linear wave equations. Publ. RIMS. 12, 169–189 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nishihara, K.: Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J. Differ. Equ. 137(2), 384–395 (1997). doi: 10.1006/jdeq.1997.3268 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nishihara, K.: Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping. Tokyo J. Math. 34, 327–343 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ruzhansky, M., Smith, J.: Dispersive and Strichartz Estimates for Hyperbolic Equations with Constant Coefficients. MSJ Memoirs 22. Mathematical Society of Japan, Tokyo (2010)Google Scholar
  12. 12.
    Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14(2), 249–275 (1985). doi: 10.14492/hokmj/1381757663 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Todorova, G., Yordanov, B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174, 464–489 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wakasugi, Y.: Critical exponent for the semilinear wave equation with scale invariant damping, Fourier analysis. In: Ruzhansky, M., Turunen, V. (eds.) Trends in Mathematics, pp. 375–390. Springer, Basel (2014)Google Scholar
  15. 15.
    Wirth, J.: Diffusion phenomena for partially dissipative hyperbolic systems. J. Math. Anal. Appl. 414(2), 666–677 (2014). doi: 10.1016/j.jmaa.2014.01.034 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departamento de Computação e MatemáticaUniversidade de São Paulo (USP)Ribeirão PretoBrazil
  2. 2.Department of MathematicsUniversity of BariBariItaly

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