Initial-boundary value problems for degenerate hyperbolic equations

A. I. Kozhanov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 43-53.

Résumé

The aim of the paper is to study solvability in Sobolev spaces initial–boundary value problems for differential equations $$u_{tt}-\varphi(t)Au+c(x,t)u=f(x,t)$$ in which $A$ is an elliptic operator acting in the spatial variables $x_1$,\ldots,$x_n$ and $\varphi(t)$ is a non-negative function on the segment $[0,T]$. Existence theorems of regular solutions are proven. Some generalizations of the results are also described.

Détail

Keywords: hyperbolic equations, degeneration, initial-boundary value problems, regular solutions
Mots-clés : existence.

Bibliographie

[1] A.V. Bitsadze, A.M. Nakhushev, “Theory of degenerate hyperbolic equations in multidimensional domains”, Sov. Math., Dokl., 13 (1972), 807–810 | Zbl

[2] V.N. Vragov, “The problems of Goursat and Darboux for a class of hyperbolic equations”, Diff. Uravn., 8:1 (1972), 7–16 | Zbl

[3] V.N. Vragov, “O Nekotorykh Kraevyh Zadachakh dlya Giperbolo–Parabolicheskikh Uravnenii”, Dinamika Splosh. Sredy, 17, Institut Gidrodinamiki, Novosibirsk, 1974, 5–11

[4] N.I. Popivanov, “A multidimensional analogy to the Darboux problem for degenerating hyperbolic equations”, Diff. Uravn., 14:1 (1978), 80–93 | Zbl

[5] A.M. Nakhushev, “Uniqueness criterion for the solution to a Darboux problem for a certain degenerate hyperbolic equation of moisture transfer”, Diff. Uravn., 16:9 (1980), 1643–1649 | Zbl

[6] S.K. Kumykova, “A problem with non-local conditions on the characteristics for a hyperbolic equation degenerating in the interior of the domain”, Diff. Uravn., 17:1 (1981), 81–90 | Zbl

[7] T. Sh. Kalmenov, O Regulyarnykh Kraevykh Zadachakh dlya Uravnenii Giperbolicheskogo i Smeshannogo Typov, Avtoref. Dissert. D-ra Fiz.-Mat. Nauk, MGU, 1982

[8] M.S. Salakhitdinov, M. Mirsaburov, “O nekotorykh kraevykh zadachakh typa zadachi Bitsadze–Samarskogo dlya uravneniya giperbolicheskogo typa, vyrozhdayuschegosya na granitse oblasti”, Neklassich. Zadach. Matem. Fiziki, Tashkent, 1985, 3–25

[9] O.A. Repin, “Boundary value problem of Bitsadze-Samarksij type for an equation of moisture transfer”, Vychisl. Prikl. Mat., 64, Kiev, 1988, 45–49 | Zbl

[10] N.I. Popivanov, M. Schneider, “The Darboux problems in $\mathbb{R}^3$ for a class of degenerate hyperbolic equations”, J. Math. Anal. Appl., 175:2 (1993), 573–578 | Zbl

[11] A.A. Kilbas, O.A. Repin, M. Saigo, “Solution in closed form of boundary value problem for degenerate equation of hyperbolic type”, Kyungpook Math. J., 36:2 (1996), 261–273 | Zbl

[12] M.M. Smirnov, Degenerating elliptic and hyperbolic equations, Nauka, M., 1966 | Zbl

[13] A.M. Nakhushev, Problems with shifts for partial differential equations, Nauka, M., 2006 | Zbl

[14] A.V. Bitsadze, Bitsadze, Izd. AN SSSR, M., 1959 | Zbl

[15] M.S. Salakhitdinov, Uravneniya Smeshanno–Sostavnogo Typa, FAN, Tashkent, 1974

[16] T.D. Dzhuraev, Boundary value problems for equations of mixed and mixed-composite types, FAN, Tashkent, 1979 | Zbl

[17] M.M. Smirnov, Equations of mixed type, Vyssh. Shkola, M., 1985 | MR

[18] O.A. Repin, Boundary value problems with shift for equations of hyperbolic and mixed type, Izd. Saratov. Un-ta, Samara, 1992 | Zbl

[19] K.B. Sabitov, K Teorii Uravneniy Smeshannogo Typa, Fizmatlit, M., 2014

[20] Yu. K. Sabitova, “Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation”, Russ. Math., 53:12 (2009), 41–49 | Zbl

[21] S.A. Aldashev, “The well-posedness of the Dirichlet and Poincaré problems in a cylindric domain for the multi-dimensional Chaplygin equation”, Vladikavkaz. Mat. Zh., 15:2 (2013), 3–11 | Zbl

[22] N.V. Zaitseva, “Keldysh type problem for $B$–hyperbolic equation with integral boundary value condition of the first kind”, Lobachevskii J. Math., 38:1 (2017), 162–169 | Zbl

[23] K.B. Sabitov, N.V. Zaytseva, “The second initial-boundary value problem for a $B$-hyperbolic equation”, Russ. Math., 63:10 (2019), 66–76 | Zbl

[24] V.N. Vragov, “On a mixed problem for a class of hyperbolic-parabolic equations”, Sov. Math., Dokl., 16 (1975), 1179–1183 | Zbl

[25] V.N. Vragov, “K Teorii Kraevykh Zadach dlya Uravneniy Smeshannogo Typa”, Diff. Uravn., 13:6 (1977), 1098–1105 | Zbl

[26] V.N. Vragov, “O Postanovkakh i Razreshimosti Kraevykh Zadach dlya Uravneniy Smeshanno–Sostavnogo Typa”, Matem. Analiz i Smezh. Vopr. Matem., Nauka, Novosibirsk, 1978, 5–13

[27] N.A. Lar'kin, “On the linearized equation of non-stationary gas dynamics”, Non-classical equations and equations of mixed type, Collect. sci. Works, Izd. In-ta Matem. SO AN SSSR, Novosibirsk, 1983, 107–118 | Zbl

[28] I.E. Egorov, “Solvability of a boundary-value problem for a high-order equation of mixed type”, Differ. Equations, 23:9 (1987), 1075–1081 | Zbl

[29] I.E. Egorov, V.E. Fedorov, Neklassicheskie Uravneniya Matematicheskoy Fiziki Vysokogo Poryadka, Izd. Vychisl. Tsetra SO RAN, Novosibirsk, 1995

[30] M.V. Keldysh, “O Nekotorykh Sluchayakh Vyrozhdeniya Uravneniy Ellipticheskogo Typa na Granitse Oblasti”, Dokl. AN SSSR, 77:2 (1951), 181–183 | Zbl

[31] S.L. Sobolev, Some applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, 90, AMS, Providence, 1991 | Zbl

[32] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Publ., Amsterdam etc., 1978 | Zbl

[33] S. Ya. Yakubov, Linear operator-differential equations and their applications, Ehlm, Baku, 1985 | Zbl

[34] P. Aviles, J. Sandefur, “Nonlinear second order equations with applications to partial differential equations”, J. Differ. Equations, 58:3 (1985), 404–427 | Zbl

[35] A.I. Kozhanov, Composite type equations and inverse problems, VSP, Utrecht, 1999 | Zbl

[36] J.L. Lions, Quelques methods de resolution des problems aux limites non lineares, Dunod Cauthier-Villars, Paris, 1969 | Zbl