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Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions

  • Soichiro Katayama
  • Akitaka Matsumura
  • Hideaki SunagawaEmail author
Article

Abstract

We consider the Cauchy problem for systems of semilinear wave equations in two space dimensions. We present a structural condition on the nonlinearity under which the energy decreases to zero as time tends to infinity if the Cauchy data are sufficiently small, smooth and compactly-supported.

Keywords

Nonlinear wave equations Energy decay 

Mathematics Subject Classification

Primary 35L71 Secondary 35B40 

References

  1. 1.
    Agemi, R.: Oral communication.Google Scholar
  2. 2.
    Alinhac S.: The null condition for quasilinear wave equations in two space dimensions, I. Invent. Math. 145, 597–618 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alinhac S.: The null condition for quasilinear wave equations in two space dimensions, II. Am. J. Math. 123, 1071–1101 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alinhac S.: Remarks on energy inequalities for wave and Maxwell equations on a curved background. Math. Ann. 329, 707–722 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Christodoulou D.: Global solutions of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math. 39, 267–282 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Godin P.: Lifespan of solutions of semilinear wave equations in two space dimensions. Comm. Partial Differ. Equ. 18, 895–916 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hörmander, L.: L 1, L estimates for the wave operator. In: Analyse Mathématique et Applications, Contributions en l’Honneur de J. L. Lions. Gauthier–Villars, Paris, pp. 211–234 (1988)Google Scholar
  8. 8.
    Hoshiga A.: The initial value problems for quasi-linear wave equations in two space dimensions with small data. Adv. Math. Sci. Appl. 5, 67–89 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hoshiga A.: The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space. Funkc. Ekvac. 49, 357–384 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hoshiga A.: The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space. Hokkaido Math. J. 37, 669–688 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hoshiga A., Kubo H.: Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition. SIAM J. Math. Anal. 31, 486–513 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoshiga A., Kubo H.: Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions. Differ. Integral Equ. 17, 593–622 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Katayama S.: Global existence for systems of nonlinear wave equations in two space dimensions. Publ. RIMS Kyoto Univ. 29, 1021–1041 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Katayama S.: Global existence for systems of nonlinear wave equations in two space dimensions, II. Publ. RIMS Kyoto Univ. 31, 645–665 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Katayama S.: Global existence and asymptotic behavior of solutions to systems of semilinear wave equations in two space dimensions. Hokkaido Math. J. 37, 689–714 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Katayama S., Li C., Sunagawa H.: A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D. Differ. Integral Equ. 27, 301–312 (2014)MathSciNetGoogle Scholar
  17. 17.
    Katayama, S., Matoba, T., Sunagawa, H.: Semilinear hyperbolic systems violating the null condition. Math. Ann. (2014) (in press) doi: 10.1007/s00208-014-1071-1
  18. 18.
    Katayama S., Murotani D., Sunagawa H.: The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions. J. Evol. Equ. 12, 891–916 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kim D., Sunagawa H.: Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities. Nonlinear Anal. 97, 94–105 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math. 23. AMS, Providence, pp. 293–326 (1986)Google Scholar
  21. 21.
    Kubo, H.: Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Discrete Contin. Dynam. Syst. Supplement Volume, 602–613 (2007)Google Scholar
  22. 22.
    Lindblad H.: On the lifespan of solutions of nonlinear wave equations with small initial data. Commun. Pure Appl. Math. 43, 445–472 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lindblad. H.: Global solutions of quasilinear wave equations. Amer. J. Math. 130:115–157 (2008)Google Scholar
  24. 24.
    Mochizuki K., Motai T.: On energy decay-nondecay problems for wave equations with nonlinear dissipative term in \({\mathbb{R}^N}\). J. Math. Soc. Jpn. 47, 405–421 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sogge, C.D.: Lectures on non-linear wave equations. International Press, Boston (1995)Google Scholar
  26. 26.
    Todorova G., Yordanov B.: The energy decay problem for wave equations with nonlinear dissipative terms in \({\mathbb{R}^n}\). Indiana Univ. Math. J. 56, 389–416 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Soichiro Katayama
    • 1
  • Akitaka Matsumura
    • 2
  • Hideaki Sunagawa
    • 3
    Email author
  1. 1.Department of MathematicsWakayama UniversityWakayamaJapan
  2. 2.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversityToyonakaJapan
  3. 3.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

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