We study a mixed problem for a nonlinear connected evolution system of equations with integral perturbation in a domain unbounded with respect to the time variable. We prove that a global generalized solution of this problem does not exist.
Similar content being viewed by others
References
R. F. Apolaya, H. R. Clark, and A. J. Feitosa, “On a nonlinear coupled system with internal damping,” Electron. J. Different. Equat., 2000, No. 64, 1–17 (2000).
A. Arosio and S. Spagnolo, “Global solutions of the Cauchy problem for a nonlinear hyperbolic equation,” in: H. Brezis and J.-L. Lions (editors), Nonlinear Partial Differential Equations and Their Applications, Vol. 6, Pitman, Boston (1984), pp. 1–26.
E. Bisignin, “Perturbation of Kirchhoff–Carrier’s operator by Lipschitz functions,” in: Proc. XXXI Bras. Sem. Analysis, Rio de Janeiro (1992).
M. R. Clark and O. A. Lima, “On a mixed problem for a coupled nonlinear system,” Electron. J. Different. Equat., 1997, No. 06, 1–11 (1997).
M. R. Clark and O. A. Lima, “Existence of solutions for a variational unilateral system,” Electron. J. Different. Equat., 2002, No. 22, 1–18 (2002).
P. D’Ancona and S. Spagnolo, Nonlinear Perturbation of the Kirchhoff–Carrier Equations, University of Pisa (1992).
M. Hosoya and Y. Yamada, “On some nonlinear wave equations. I. Local existence and regularity of solutions,” J. Fac. Sci. Univ. Tokyo, Sec. IA, Math., 38, 225–238 (1991).
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris (1969).
M. P. Matos, “Mathematical analysis of the nonlinear model for the vibrations of a string,” Nonlin. Anal., Theory, Meth. Appl., 17, No. 12, 1125–1137 (1991).
L. A. Medeiros and M. Milla Miranda, “Local solutions for a nonlinear degenerate hyperbolic equation,” Nonlin. Anal., 10, 27–40 (1986).
L. A. Medeiros, “On some nonlinear perturbation of Kirchhoff–Carrier’s operator,” Comp. Appl. Math., 13, No. 3, 225–233 (1994).
S. A. Messaoudi, “A blowup result in a multidimensional semilinear thermoelastic system,” Electron. J. Different. Equat., 2001, No. 30, 1–9 (2001).
M. O. Nechepurenko, “The mixed problem for a nonlinear coupled evolution system in a bounded domain,” Visn. L’viv. Univ., Ser. Mekh.-Mat., 67, 207–223 (2007).
M. O. Nechepurenko, “Mixed problem for a nonlinear coupled system in unbounded domains,” Mat. Stud., 32, No. 1, 33–44 (2009).
M. O. Nechepurenko and G. P. Torgan, “On the existence of a generalized solution of a nonlinear evolution system in a domain unbounded with respect to the time variable,” Ukr. Mat. Visn., 7, No. 1, 49–72 (2010).
S. I. Pokhozhaev, “On one class of quasilinear hyperbolic equations,” Mat. Sb., 96(138), No. 1, 152–166 (1975).
H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen [Russian translation], Mir, Moscow (1978).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 14, No. 4, pp. 435–444, October–December, 2011.
Rights and permissions
About this article
Cite this article
Bazylyak, H.R., Nechepurenko, M.O. Nonexistence of a global solution of a mixed problem for a nonlinear system of equations with integral perturbation in a domain unbounded with respect to time variable. Nonlinear Oscill 14, 461–471 (2012). https://doi.org/10.1007/s11072-012-0170-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11072-012-0170-8