Abstract
We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than e −μt, μ > 0, as t → −∞.
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REFERENCES
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlineaires [Russian translation], Mir, Moscow (1972).
A. A. Pankov, Bounded and Almost-Periodic Solutions of Nonlinear Differential Operator Equations [in Russian], Naukova Dumka, Kiev (1985).
O. A. Oleinik and G. A. Iosif'yan, “An analog of the Saint–Venant principle and the uniqueness of solutions of boundary–value problems for parabolic equations in unbounded domains,” Usp. Mat. Nauk, 31, No. 6, 142–166 (1976).
O. A. Oleinik, “On the behavior of solutions of linear parabolic systems of differential equations in unbounded domains,” Usp. Mat. Nauk, 30, No. 2, 219–220 (1975).
O. A. Oleinik and E. V. Radkevich, “Analyticity and the Liouville and Phragmen-Lindelöf theorems for general parabolic systems of differential equations,” Funkts. Anal. Pril., 8, Issue 4, 59–70 (1974).
S. D. Ivasishen, “On parabolic boundary-value problems without initial conditions,” Ukr. Mat. Zh., 34, No. 5, 547–552 (1982).
S. D. Ivasishen, “On the correct solvability of some parabolic boundary-value problems without initial conditions,” Differents. Uravn., 14, No. 2, 361–363 (1978).
S. D. Eidel'man, Parabolic Systems [in Russian], Nauka, Moscow (1964).
N. M. Bokalo, “On a problem without initial conditions for some classes of nonlinear parabolic equations,” Tr. Sem. I. G. Petrovskogo, Issue 14, 3–44 (1989).
M. M. Bokalo and V. M. Sikorsky, “The well–posedness of Fourier problem for quasilinear parabolic equations of arbitrary order in isotropic spaces,” Mat. Studii, 8, No. 1, 53–70 (1997).
S. P. Lavrenyuk, “Systems of parabolic variational inequalities without initial conditions with arbitrary behavior of solutions at infinity,” Dopov. Akad. Nauk Ukr., No. 2, 40–43 (1998).
S. P. Lavrenyuk, “Parabolic variational inequalities without initial conditions,” Differents. Uravn., 32, No. 10, 1–5 (1996).
S. P. Lavrenyuk, “Systems of parabolic variational inequalities without initial conditions,” Ukr. Mat. Zh., 49, No. 4, 540–547 (1997).
O. M. Buhrii, “On some parabolic variational inequalities without initial conditions,” Visn. L'viv. Univ., Issue 49, 113–121 (1998).
S. P. Lavrenyuk, “Parabolic variational inequalities without initial data,” Nelin. Granich. Zadachi, No. 8, 173–178 (1998).
O. Kovacik, Z. Rakosnik, and J. Rakosnik, “On spaces L p(x), W k p (x),” Czech. Math. J., 41, No. 4, 592–618 (1991).
V. N. Samokhin, “On one class of equations that generalize the equations of polytropic filtration,” Differents. Uravn., 32, No. 5, 643–651 (1996).
M. M. Bokalo and V. M. Sikors'kyi, “On properties of solutions of a problem without initial conditions for equations that generalize the equations of polytropic filtration,” Visn. L'viv. Univ., Issue 51, 85–99 (1998).
H. Gajewski, K. Gröger, and K. Zacharias, Nonlinear Operator Equations and Operator Differential Equations [Russian translation], Mir, Moscow (1978).
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Buhrii, O.M., Lavrenyuk, S.P. On a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration. Ukrainian Mathematical Journal 53, 1027–1042 (2001). https://doi.org/10.1023/A:1013368412665
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DOI: https://doi.org/10.1023/A:1013368412665