Abstract
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.
Similar content being viewed by others
References
J. E. Munoz Rivera and H. Portillo Oquendo, Acta Appl. Math. 60, 1–21 (2000).
R. Dautray and J. L. Lions, Analyse et calcul numerique pour les sciences et les techniques (Masson, Paris, 1984), Vol.1.
W. D. Bastos and C. A. Raposo, Electron. J. Differ. Equations, No. 60, 10 (2007).
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéires (Dunod, Paris, 1969).
J. T. Beale and S. I. Rosencrans, Bull. Am. Math. Soc. 180 (6), 1276–1278 (1974).
C. I. Frota and N. A. Larkin, Progr. Nonlinear Differ. Equations Appl. 66, 297–312 (2005).
P. J. Graber and B. Said-Houari, J. Differ. Equations 252, 4898–4941 (2012).
A. Vicente, Bol. Soc. Parana. Mat. 3 (27(1)), 29–39 (2009).
J. Y. Park and T. G. Ha, J. Math. Phys. 50 (1), 1–18 (2009).
J. M. Jeong, J. Y. Park, and Y. H. Kang, Jeong et al. Boundary Value Problems 2017 (42), 1–10 (2017).
C. L. Frota, L. A. Medeyros, and A. Vicente, Electron. J. Differ. Equations 2014 (243), 1–14 (2014).
S. A. Gabov, New Problems in Mathematical Theory of Waves (Nauka, Moscow, 1998) [in Russian].
P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).
V. Komornik, Int. Ser. Numer. Anal. 118, 253–266 (1994).
E. Vitillaro, Arch. Ration. Mech. Anal. 149, 155–182 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.B. Aliev, S.E. Isayeva, 2018, published in Doklady Akademii Nauk, 2018, Vol. 483, No. 2.
Rights and permissions
About this article
Cite this article
Aliev, A.B., Isayeva, S.E. The Existence and Behavior of Global Solutions to a Mixed Problem with Acoustic Transmission Conditions for Nonlinear Hyperbolic Equations with Nonlinear Dissipation. Dokl. Math. 98, 555–558 (2018). https://doi.org/10.1134/S1064562418070062
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562418070062