Boundary value problems for composite type equations with a quasiparabolic operator in the leading part having the variable direction of evolution and discontinuous coefficients

A. I. Grigorieva ; A. I. Kozhanov

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 24 (2018) no. 2, pp. 7-17

Résumé

The solvability of boundary value problems for non-classical Sobolev type differential equations with an alternating function is studied. This function has a discontinuity of the first kind at the point zero and changes its sign depending on the sign of the variable $x$. The existence and uniqueness of regular solutions having generalized derivatives are proved. To this end we derived a priori estimates.

Détail

Mots-clés : Sobolev–type equation, variable direction of evolution, existence
Keywords: boundary value problem, differential operator, regular solution, uniqueness, a priory estimate.

Bibliographie

[1] Ladyzhenskaya O.A., “On the solution of a general problem of diffraction”, Doklady Physics, 96:3 (1954), 433–436 (in Russian) | Zbl

[2] Oleinik O.A., “Boundary-value problems for linear equations of the elliptic and parabolic types with discontinuous coefficients”, Izvestiya: Mathematics, 25 (1961), 3–20 (in Russian) | Zbl

[3] Il'in V.A., “On the solvability of the Dirichlet and Neumann problems for a linear elliptic operator with discontinuous coefficients”, Doklady Physics, 137:1 (1961), 28–30 (in Russian) | Zbl

[4] Il'in V.A., “The Fourier method for a hyperbolic equation with discontinuous coefficients”, Doklady Physics, 142:1 (1962), 21–24 (in Russian) | Zbl

[5] Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N., Linear and quasilinear equations of parabolic type, Nauka, M., 1967 (in Russian) | MR

[6] Il'in V.A., Luferenko P.V., “Mixed problems describing longitudinal vibrations of a rod consisting of two sections with different densities and elasticity moduli, but with the same impedance”, Doklady Physics, 428:1 (2009), 12–15 (in Russian) | Zbl

[7] Il'in V.A., Luferenko P.V., “Generalized solutions of mixed problems for the discontinuous wave equation under the condition of equality of impedances”, Doklady Physics, 429:3 (2009), 317–321 (in Russian) | Zbl

[8] Andropova O.A., “Spectral problems of conjugation with surface dissipation of the energy”, Transactions of the Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, 2009, no. 19, 10–22 (in Russian)

[9] Nikolsky D.N., “Three-dimensional evolution of the pollution boundary in a limited piecewise-homogeneous porous medium”, Computational Mathematics and Mathematical, 51:5 (2011), 913–919 (in Russian) | MR | Zbl

[10] Rogozhnikov A.M., “Study of mixed problems describing the process of vibrations of a rod consisting of several sections under the condition of coincidence of times of passage of waves on each of these sections”, Doklady Physics, 441:4 (2012), 449–451 (in Russian)

[11] Kuleshov A.A., “Mixed problems for the equation of longitudinal vibrations of a inhomogeneous rod with free or fixed right end consisting of two sections with different densities and elasticity moduli”, Doklady Physics, 442:4 (2012), 451–454 (in Russian) | Zbl

[12] Rogozhnikov A.M., “Study of a mixed problem describing the process of vibrations of a rod consisting of several sections with arbitrary lengths”, Doklady Physics, 444:5 (2012), 488–491 (in Russian) | MR | Zbl

[13] Smirnov I.N., “On vibrations described by the telegraph equation in the case of a system consisting of several sections with different densities and elasticity moduli”, Differential Equations, 49:5 (2013), 643–648 (in Russian) | DOI | Zbl

[14] Bitsadze A.V., Equations of a mixed type, Iz-vo AN SSSR, M., 1959 (in Russian)

[15] Ladyzhenskaya O.A., Stupyalis L., “On equations of a mixed type”, Vestnik of Pushkin Leningrad State University, 1967, no. 18, 38–46 (in Russian)

[16] Smirnov M.M., Equations of a mixed type, Nauka, M., 1970 (in Russian)

[17] Ladyzhenskaya O.A., Stupyalis L., “Boundary-value problems for equations of the mixed type”, Proceedings of the Steklov Institute of Mathematics, 116, no. 16, 1971, 101–136 (in Russian) | Zbl

[18] Stupyalis L., “Boundary-value problems for elliptichyperbolic equations”, Proceedings of the Steklov Institute of Mathematics, 125, 1973, 211–229 (in Russian) | Zbl

[19] Tersenov S.A. Introduction to the theory of equation of the parabolic type with varying time direction, Sib. otd-nie AN SSSR. In-t matematiki, Novosibirsk, 1982 (in Russian)

[20] Dzhuraev T.D., Boundary-value problems for equations of the mixed and mixed-composite types, Fan, Tashkent, 1986 (in Russian)

[21] Moiseev E.I., Equations of the Mixed Type with Spectral Parameter, MGU, M., 1988 (in Russian)

[22] Nakhushev A.M., Problems with displacement for partial differential equations, Nauka, M., 2006 (in Russian)

[23] Marichev O.I., Kilbas A.A., Repin O.A., Boundary-Value Problems for Equations with Partial Derivatives with Discontinuous Coefficients, Samarskii gosudarstvennyi ekonomicheskii universitet, Samara, 2008 (in Russian)

[24] Moiseev E.I., Likhomanenko T.N., “On a nonlocal problem for the Lavrent'ev-Bitsadze equation”, Doklady Physics, 446:3 (2012), 256–258 (in Russian) | Zbl

[25] Demidenko G.V., Uspensky S.V., Equations and systems that are unresolved relative to the highest derivative, Nauchnaia kniga, Novosibirsk, 1998 (in Russian)

[26] Gaevsky Kh., Greger K., Zacharias K., Nonlinear Operator Equations and Operator Differential Equations, Mir, M., 1978 (in Russian)

[27] Corpusov M.O., Destruction in nonclassical wave equations, Librokom, M., 2010 (in Russian)

[28] Dzhenaliev M.T., On the theory of linear boundary value problems for loaded differential equations, In-t teoreticheskoi i prikladnoi matematiki, Almaty, 1995 (in Russian)

[29] Nakhushev A.M., Loaded equations and their applications, Nauka, M., 2012 (in Russian)

[30] Kozhanov A. I., Sharin E. F., “The conjugation problem for some nonclassical high-order differential equations”, J. of Mathematical Sciences, 204:3 (2015), 298–314 (in English) | DOI | MR | Zbl

[31] Kozhanov A.I., Potapova S.V., “Conjugation problem for oddorder differential equations with two time variables and with the changing direction of evolution”, Doklady Physics, 474:6 (2017), 661–664 (in Russian) | DOI | Zbl

[32] Grigorieva A.I., “Initial-boundary problem with conjugation conditions for composite-type equations with two breakdown coefficients”, Mathematical Notes of NEFU, 25:2 (2018), 12–26 (in Russian) | DOI | MR

[33] Kozhanov A.I., Pinigina N.R., “Boundary value problems for certain nonclassical differential equations of a high order”, Mathematical notes, 101:3 (2017), 403–412 (in Russian) | DOI | MR | Zbl

[34] Trenogin V.A., Functional Analysis, Nauka, M., 1980 (in Russian)