Existence and maximal regularity of solutions in $L_2(\mathbb{R}^2)$ for a hyperbolic type differential equation with quickly growing coefficients

M. B. Muratbekov ; Ye. N. Bayandiyev

Eurasian mathematical journal, Tome 11 (2020) no. 1, pp. 95-100.

Résumé

In this paper the problem of the existence of solutions is studied for a hyperbolic type differential equation defined in an unbounded domain. The problem of the smoothness of solutions is also considered here. Such problems are of particular interest when the coefficients are unbounded. The novelty of the work is that the weighted coercive estimate is obtained for the solutions of a hyperbolic type differential equation with quickly growing coefficients.

Bibliographie

[1] A. V. Bitsadze, Some classes of partial differential equations, Nauka, M., 1981 (in Russian) | MR | Zbl

[2] K. Kh. Boimatov, “Coercive estimates and separability for second-order elliptic differential equations”, Sov. Math. Dokl., 305:5 (1988), 1033–1036 | MR

[3] K. Kh. Boimatov, “Separability theorems, weighted spacees and their applications”, Trudy MIAN, 170, 1984, 37–76 (in Russian) | MR

[4] W. N. Everitt, M. Giertz, “An example conserning property for differential operators”, Proc. Soc. Edinburgh, 71 (1972), 159–165 | MR

[5] W. N. Everitt, M. Giertz, “On properties of the powers of a formally self-adjoint differential equations”, Proc. London Math. Soc. (2), 24:1 (1972), 149–170 | DOI | MR | Zbl

[6] W. N. Everitt, M. Giertz, “On some properties of the domains of power of certain differential operators”, Proc. London Math. Soc. (2), 24:4 (1972), 756–768 | DOI | MR | Zbl

[7] A. V. Filinovskii, “Spectrum and stabilization in hyperbolic problems”, J. Math. Sci., 234:4 (2018) | DOI | MR | Zbl

[8] K. Friedrichs, “Symmetric hyperbolic linear differential equations”, Comm. Pure and Appl. Math., 7 (1954), 345–392 | DOI | MR | Zbl

[9] L. Gording, “Linear hyperbolic partial diiferential equations with constant coefficients”, Acta Math., 85 (1951), 1–62 | DOI | MR

[10] J. Hadamard, The Cauchy problem for hyperbolic linear partial differential equations, Nauka, M., 1978 (in Russian) | MR

[11] R. Haller, H. Heck, M. Hieber, “Muckenhoupt weights and maximal $L^p$-regularity”, Arch. Math., 81 (2003), 422–430 | DOI | MR | Zbl

[12] T. Sh. Kalmenov, On the theory of initial boundary value problems for differential equations, Almaty, 2013 (in Russian)

[13] T. I. Kiguradze, “On bounded and periodic in the strip solutions of quasilinear hyperbolic systems”, Differential equations, 30:10 (1994), 1760–1773 | MR

[14] T. I. Kiguradze, “On periodic boundary value problems for linear hyperbolic equations”, Differential equations, 29:2 (1993), 281–297 | MR

[15] P. C. Kunstmann, L. Weis, “Maximal $L^p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus”: G. Da Prato et al., Functional Analytic Methods for Evolution Equations, LNM, 1855, eds. M. Iannelli et al., 2004, 65–311 | MR | Zbl

[16] O. A. Ladyzhenskaya, “On the solvability of the basic boundary value problems for the equations of hyperbolic and parabolic types”, DAN USSR, 97:3 (1951), 395–398 | MR

[17] O. A. Ladyzhenskaya, N. N. Uraltseva, Linear and quasilinear equations of elliptic type, Nauka, M., 1973 (in Russian) | MR | Zbl

[18] J. Leray, Hyperbolic differential equations, Nauka, M., 1984 (in Russian) | MR | Zbl

[19] M. Muratbekov, “Separability of an operator of mixed type and the completeness of its root vector system”, Dif. Equat., 27 (1991), 2128–2137 | MR

[20] M. Muratbekov, M. Otelbaev, “On the existence of a resolvent and separability for a class of singular hyperbolic type differential operators on an unbounded domain”, Eurasian Math. J., 7:1 (2016), 50–67 | MR

[21] M. B. Muratbekov, M. M. Muratbekov, “Sturm-Liouville operator with a parameter and its usaqe to spectrum research of some differential operators”, Comp. Var. Ell. Eq., 64:9 (2019), 1457–1476 | MR | Zbl

[22] M. Nagumo, Lectures on modern theory of partial differential equations, M., 1967 (in Russian) | MR

[23] A. M. Nakhushev, Offset problems for partial differential equations, Nauka, M., 2006 (in Russian) | MR

[24] M. Otelbaev, “On the separability of elliptic operators”, DAN SSSR, 234:3 (1977), 540–543 (in Russian) | MR | Zbl

[25] M. Otelbaev, “Coercive estimates and separability theorems for elliptic equations in $R^n$”, Tr. MIAN, 161, 1983, 195–217 (in Russian) | MR | Zbl

[26] S. L. Sobolev, Selected works, v. 2, Publishing institute of mathematics, Novosibirsk, 2006 | MR

[27] L. Weis, “Operator-valued Fourier multiplier theorems and maximal $L^p$-regularity”, Math. Ann., 319 (2001), 735–758 | DOI | MR | Zbl