We consider parabolic variational inequalities in Sobolev spaces and generalized Lebesgue spaces. Sufficient conditions of existence and uniqueness of solution of these inequalities have been established. We have also proved some comparison theorems.
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References
A. Bensaussan and J. L. Lions, Impulsive Control and Quasi-Variational Inequalities, Dunod, Paris (1982).
O. M. Buhrii, “Parabolic variational inequalities in generalized Lebesgue spaces,” Nauk. Zap. Vinnyts. Ped. Univ. im. Kotsyubyns’kyi, Ser. Fiz. Mat., Issue 1, 310–321 (2002).
O. M. Buhrii, “On problems with homogeneous boundary conditions for degenerate nonlinear equations,” Ukr. Math. Bull, 5, No. 4, 425–457 (2008).
O. M. Buhrii, “Finiteness of the stabilization time of solution of a nonlinear parabolic variational inequality with variable degree of nonlinearity,” Mat. Stud., 24, No. 2, 167–172 (2005).
O. M. Buhrii and S. P. Lavrenyuk, “On a parabolic variational inequality that generalizes the equation of polytropic filtration,” Ukr. Mat. Zh., 53, No. 7, 867–878 (2001); English translation: Ukr. Math. J., 53, No. 7, 1027–1042 (2001).
O. M. Buhrii and O. T. Panat, “Some properties of the solutions of parabolic variational inequalities with variable degree of nonlinearity,” Mat. Metody Fiz.-Mekh. Polya, 49, No. 2, 99–107 (2006).
O. Buhrii and S. Lavrenyuk, “Mixed problem for a parabolic equation that generalizes the equation of polytropic filtration,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 56, 33–43 (2000).
H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen [Russian translation], Mir, Moscow (1978).
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications [Russian translation], Mir, Moscow (1983).
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires [Russian translation], Mir, Moscow (1972).
I. I. Sharapudinov, “On the topology of the space L p(t ) ([0,1]) ,” Mat. Zametki, 26, No. 4, 613–632 (1979).
G. E. Shilov, Mathematical Analysis. Second Special Course [in Russian], Moscow University, Moscow (1984).
A. Besenyei, “On nonlinear parabolic variational inequalities containing nonlocal terms,” Acta Math. Hung., 116, No. 1–2, 145–162 (2007).
O. M. Buhrii and R. A. Mashiev, “Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity,” Nonlinear Anal., 70, 2325–2331 (2009).
O. Kováčik and J. Rákosnik, “On spaces L p( x ) and W l , p( x ) ,” Czech. Math. J., 41, No. 4, 592–618 (1991).
J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin (1983).
H. Nagase, “On a few properties of solutions of nonlinear parabolic variational inequalities,” Nonlinear Anal., 47, 1659–1669 (2001).
W. Orlicz, “Über konjugierte Exponentenfolgen,” Stud. Math. (Lwow), 3, 200–211 (1931).
M. Rudd and K. Schmitt, “Variational inequalities of elliptic and parabolic type,” Taiwan. J. Math., 6, No. 3, 287–322 (2002).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 52, No. 4, pp. 42–57, October–December, 2009.
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Buhrii, O.M., Hlynyans’ka, K.P. Some parabolic variational inequalities with variable exponent of nonlinearity: unique solvability and comparison theorems. J Math Sci 174, 169–189 (2011). https://doi.org/10.1007/s10958-011-0288-8
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DOI: https://doi.org/10.1007/s10958-011-0288-8