Abstract
We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable t. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as t → ∞.
Similar content being viewed by others
References
R. E. Showalter, “Pseudoparabolic partial differential equations,” Diss. Abstrs. B, 29, No. 8, 29–94 (1969).
H. Gajewski, K. Gröger, and K. Zacharias, Nightlinear Operatorgleichungen and Operatordifferentialgleichungen, Academie-Verlag, Berlin (1974).
T. W. Ting, “Parabolic and pseudoparabolic partial differential equations,” J. Math. Soc. Jap., 21, No. 3, 440–453 (1954).
S. L. Sobolev, “On one new problem in mathematical physics,” Izv. Akad. Nauk SSSR, Ser. Mat., 18, No. 1, 3–50 (1954).
S. A. Gal’pern, “The Cauchy problem for general systems of linear partial differential equations,” Dokl. Akad. Nauk SSSR, 119, No. 4, 640–643 (1958).
W. Rundell, “The uniqueness class for the Cauchy problem for pseudoparabolic equations,” Proc. Amer. Math. Soc., 76, No. 2, 253–257 (1979).
I. V. Suveika, “On the asymptotic behavior of a solution of a mixed problem for a pseudoparabolic equation with the third boundary condition in a half-plane,” Differents. Uravn., 22, No. 8, 1416–1424 (1986).
W. H. Ford, “Galerkin approximations to nonlinear pseudoparabolic partial differential equations,” Aequent. Math., 14, No. 3, 271–291 (1976).
M. O. Bas and S. P. Lavrenyuk, “On the uniqueness of a solution of the Fourier problem for a system of the type of Sobolev-Gal’pern,” Ukr. Mat. Th., 48, No. 1, 124–128 (1996).
M. O. Bas and S. P. Lavrenyuk, “The Fourier problem for one nonlinear pseudoparabolic system,” Dep. Ukr. DNTB, No. 2017-Uk 95 (1985).
S. P. Lavrenyuk, “Parabolic variational inequalities without initial conditions,” Differents. Uravn., 32, No. 10, 1–5 (1996).
A. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Differential Operator Equations [in Russian], Naukova Dumka, Kiev (1985).
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris (1969).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 919–929, July, 1998.
Rights and permissions
About this article
Cite this article
Lavrenyuk, S.P., Ptashnyk, M.B. Pseudoparabolic variational inequalities without initial conditions. Ukr Math J 50, 1045–1057 (1998). https://doi.org/10.1007/BF02528833
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02528833