Doklady Mathematics

, Volume 98, Issue 3, pp 555–558 | Cite as

The Existence and Behavior of Global Solutions to a Mixed Problem with Acoustic Transmission Conditions for Nonlinear Hyperbolic Equations with Nonlinear Dissipation

  • A. B. AlievEmail author
  • S. E. Isayeva


A mixed problem with acoustic transmission conditions for nonlinear hyperbolic equations with nonlinear dissipation is considered. The existence, uniqueness, and exponential decay of global solutions to this problem with focusing nonlinear sources are proved Additionally, the existence of global solutions and the solution blow-up in a finite time are proved for the case of defocusing nonlinear sources.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. E. Munoz Rivera and H. Portillo Oquendo, Acta Appl. Math. 60, 1–21 (2000).MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Dautray and J. L. Lions, Analyse et calcul numerique pour les sciences et les techniques (Masson, Paris, 1984), Vol.1.Google Scholar
  3. 3.
    W. D. Bastos and C. A. Raposo, Electron. J. Differ. Equations, No. 60, 10 (2007).Google Scholar
  4. 4.
    J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéires (Dunod, Paris, 1969).zbMATHGoogle Scholar
  5. 5.
    J. T. Beale and S. I. Rosencrans, Bull. Am. Math. Soc. 180 (6), 1276–1278 (1974).CrossRefGoogle Scholar
  6. 6.
    C. I. Frota and N. A. Larkin, Progr. Nonlinear Differ. Equations Appl. 66, 297–312 (2005).CrossRefGoogle Scholar
  7. 7.
    P. J. Graber and B. Said-Houari, J. Differ. Equations 252, 4898–4941 (2012).CrossRefGoogle Scholar
  8. 8.
    A. Vicente, Bol. Soc. Parana. Mat. 3 (27(1)), 29–39 (2009).Google Scholar
  9. 9.
    J. Y. Park and T. G. Ha, J. Math. Phys. 50 (1), 1–18 (2009).CrossRefGoogle Scholar
  10. 10.
    J. M. Jeong, J. Y. Park, and Y. H. Kang, Jeong et al. Boundary Value Problems 2017 (42), 1–10 (2017).Google Scholar
  11. 11.
    C. L. Frota, L. A. Medeyros, and A. Vicente, Electron. J. Differ. Equations 2014 (243), 1–14 (2014).Google Scholar
  12. 12.
    S. A. Gabov, New Problems in Mathematical Theory of Waves (Nauka, Moscow, 1998) [in Russian].zbMATHGoogle Scholar
  13. 13.
    P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).Google Scholar
  14. 14.
    V. Komornik, Int. Ser. Numer. Anal. 118, 253–266 (1994).Google Scholar
  15. 15.
    E. Vitillaro, Arch. Ration. Mech. Anal. 149, 155–182 (1999).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsNational Academy of Sciences of AzerbaijanBakuAzerbaijan
  2. 2.Azerbaijan Technical UniversityBakuAzerbaijan
  3. 3.Baku State UniversityBakuAzerbaijan

Personalised recommendations