Abstract
We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to the space variables. The case of arbitrarily many space variables is considered. We establish conditions for the existence and uniqueness of a solution of this problem independent of the behavior of the solution as |x| → + ∞. The indicated classes of existence and uniqueness are the spaces of locally integrable functions, and, furthermore, the dimension of the domain does not limit the order of nonlinearity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. G. Petrovskii, Lectures on Partial Differential Equations [in Russian], Gostekhizdat, Moscow (1950).
S. K. Godunov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1971).
A. Carpio, “Existence of global solutions to some nonlinear dissipative wave equations,” J. Math. Pures Appl., 75, No. 5, 471–488 (1994).
B. Rubino, “Weak solutions to quasilinear wave equations of Klein-Gordon or Sine-Gordon type and relaxation to reaction-diffusion equations,” Nonlin. Different. Equat. Appl., No. 4, 439–457 (1997).
E. Vittilaro, “Global nonexistence theorems for a class of evolution equations with dissipation,” Arch. Ration. Mech. Anal., 149, No. 2, 155–182 (1999).
H. Pecher, “Sharp existence results for self-similar solutions of semilinear wave equations,” Nonlin. Different. Equat. Appl., No. 7, 323–341 (2000).
K. Agre and M. A. Rammaha, “Global solutions to boundary-value problems for a nonlinear wave equation in high space dimensions,” Different. Integr. Equat., 14, 1315–1331 (2001).
G. Todorova and B. Yordanov, “Critical exponent for a nonlinear wave equation with damping,” J. Different. Equat., No. 174, 464–489 (2001).
P. Ya. Pukach, “A mixed problem for a weakly nonlinear hyperbolic equation with increasing coefficients in an unbounded domain,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 4, 149–154 (2004).
P. Ya. Pukach, “A mixed problem for a nonlinear hyperbolic system of the second order in a domain unbounded with respect to space variables,” Visn. L’viv Univ., Ser. Mekh.-Mat., Issue 64, 214–231 (2005).
S. P. Lavrenyuk and M. O. Oliskevych, “Galerkin method for hyperbolic systems of the first order with two independent variables,” Ukr. Mat. Zh., 54, No. 10, 1356–1370 (2002).
H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen [Russian translation], Mir, Moscow (1978).
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires [Russian translation], Editorial URSS, Moscow (2002).
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955).
V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow (1976).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1523–1531, November, 2007.
Rights and permissions
About this article
Cite this article
Lavrenyuk, S.P., Pukach, P.Y. Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables. Ukr Math J 59, 1708–1718 (2007). https://doi.org/10.1007/s11253-008-0020-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-008-0020-0