Abstract
We study a boundary value problem for a system of two pseudoparabolic equations when for one of the functions the initial data and its values at the final time are given. Existence and uniqueness theorems for this problem are proven with the help of the properties of solutions to some nonlocal and inverse problems for a pseudoparabolic equation which are established in this article.
References
Damaskin B. B. and Petrii O. A., Introduction to Electrochemical Kinetics [in Russian], Vysshaya Shkola, Moscow (1983).
Vakhrushev A. V. and Molchanov E. K., “Modeling electrochemical deposition of AL2O3 nanoparticles into a Cu matrix. Part 1: A mathematical model,” Nanomechanics Science and Technology, 3, No. 4, 353–371 (2012).
Barenblatt G. I., Zheltov Yu. P., and Kochina I. N., “Basic concepts in the theory of seepage of homogeneous liquids in the theory of seepage of homogeneous liquids in fissured rocks [strata],” J. Appl. Math. Mech., 24, No. 5, 1286–1303 (1961).
Ting T. W., “Parabolic and pseudo-parabolic partial differential equations,” J. Math. Soc. Japan., 21, 440–453 (1969).
Showalter R. E. and Ting T. W., “Pseudoparabolic partial differential equations,” SIAM J. Math. Anal., 1, 1–26 (1970).
Lyubanova A. Sh., “On the approximation of a parabolic inverse problem by pseudoparabolic one,” J. Siberian Federal Univ. Math. Phys., 5, No. 3, 326–336 (2012).
Dzhumabaev D. S. and Bakirova E. A., “Criteria for the unique solvability of a linear two-point boundary value problem for systems of integro-differential equations,” Differential Equations, 49, No. 9, 1087–1102 (2013).
Nakhushev A. M., Equations of Mathematical Physics [in Russian], Vysshaya Shkola, Moscow (1995).
Kozhanov A. I., “A nonlinear loaded parabolic equation and a related inverse problem,” Math. Notes, 76, No. 6, 784–795 (2004).
Kozhanov A. I., “Linear inverse problems for a class of degenerate equations of Sobolev type,” Vestnik South Ural State Univ., No. 5, 33–42 (2012).
Rundell W., “Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data,” Appl. Anal., 10, 231–242 (1980).
Dzhenaliev M. T. and Ramazanov M. I., “On the boundary value problem for the spectrally loaded heat conduction operator,” Siberian Math. J., 47, No. 3, 433–451 (2006).
Sveshnikov A. G., Al’shin A. B., Korpusov M. O., and Pletner Yu. D., Linear and Nonlinear Equations of Sobolev Type [in Russian], Fizmatlit, Moscow (2007).
Vlasii O. D. and Ptashnik B. I., “A problem with nonlocal conditions for partial differential equations unsolved with respect to the leading derivative,” Ukrainian Math. J., 55, No. 8, 1238–1253 (2003).
Galpern S. A., “The Cauchy problem for the general systems of linear partial differential equations,” Tr. Mosk. Mat. Obshch., 9, 401–423 (1960).
Demidenko G. V., “The Cauchy problem for pseudoparabolic systems,” Siberian Math. J., 38, No. 6, 1084–1098 (1997).
Lyubanova A. Sh., “On nonlocal problems for systems of parabolic equations,” J. Math. Anal. Appl., 421, 1767–1778 (2015).
Ladyzhenskaya O. A., Solonnikov V. A., and Ural’tseva N. N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).
Shelukhin V. V., “A problem with time-averaged data for nonlinear parabolic equations,” Siberian Math. J., 32, No. 2, 309–320 (1991).
Lieberman G. M., “Nonlocal problems for quasilinear parabolic equations,” in: Birman M. Sh. (ed.) et al. Nonlinear Problems in Mathematical Physics and Related Topics. I. In honour of Professor O. A. Ladyzhenskaya, Kluwer Academic Publishers and T. Rozhkovskaya Ser. Int. Math., Novosibirsk and New York, 2002, pp. 233–254.
Lions J.-L. and Magenes E., Problémes aux Limites Non Homogénes et Applications. Vol. 1, Dunod, Paris (1968) (Trav. Rech. Math.; 17).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2015 Lyubanova A.Sh.
The author was supported by the Government of the Russian Federation (Grant 14.Y26.31.0006). Krasnoyarsk.
Krasnoyarsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 835–852, July–August, 2015; DOI: 10.17377/smzh.2015.56.410.
Rights and permissions
About this article
Cite this article
Lyubanova, A.S. On some boundary value problems for systems of pseudoparabolic equations. Sib Math J 56, 662–677 (2015). https://doi.org/10.1134/S0037446615040102
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446615040102