Abstract
Under consideration is a mixed problem in the half-strip Π = {(x, t): 0 < x < 1, t > 0} for a first order homogeneous linear hyperbolic system with delay in t in the boundary conditions. We study the behavior of the Laplace transform of a solution to this problem for the large values of the complex parameter. The boundary conditions are found under which the smoothness of a solution to the corresponding mixed problem increases with t.
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References
Abolinya V. E. and Myshkis A. D., “A mixed problem for an almost linear hyperbolic system on the plane,” Mat. Sb., 50, No. 4, 423–442 (1960).
Godunov S. K., Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).
Myshkis A. D. and Filimonov A. M., “Continuous solutions of hyperbolic systems of quasilinear equations with two independent variables,” in: Nonlinear Analysis and Nonlinear Differential Equations [in Russian], Fizmatlit, Moscow, 2003, pp. 337–351.
Akramov T. A., Differential Equations and Some of Their Applications in Modeling of Physicochemical Processes [in Russian], Bashkirsk. Univ., Ufa (2000).
Kmit I., “Delta-waves for a strongly singular initial-boundary hyperbolic problem with integral boundary condition,” J. Anal. Appl., 24, No. 1, 29–74 (2005).
Eltysheva N. A., “On qualitative properties of solutions of some hyperbolic systems on the plane,” Mat. Sb., 137, No. 2, 186–209 (1988).
Lavrent’ev M. M. Jr. and Lyul’ko N. A., “Increasing smoothness of solutions to some hyperbolic problems,” Siberian Math. J., 38, No. 1, 92–105 (1997).
Lyul’ko N. A., “Increasing the smoothness of solutions of a mixed problem for the wave equation on the plane,” Mat. Fiz. Anal. Geom., 11, No. 2, 169–176 (2004).
Lyul’ko N. A., “A mixed problem for a hyperbolic system on the plane with delay in the boundary conditions,” Siberian Math. J., 46, No. 5, 879–901 (2005).
Brushlinskiĭ K. V., “On the growth of solutions of a mixed problem in the case of incompleteness of eigenfunctions,” Izv. Akad. Nauk SSSR Ser. Mat., 23, No. 6, 893–912 (1959).
Birkhoff G. D. and Langer R. E., “The boundary problems and developments associated with a system of ordinary linear differential equations of the first order,” Proc. Amer. Acad. Arts Sci., 58, No. 2, 51–128 (1923).
Naĭmark M. A., Linear Differential Operators [in Russian], Nauka, Moscow (1969).
Levin B. Ya., Distribution of Roots of Entire Functions [in Russian], Gostekhizdat, Moscow (1956).
Lavrent’ev M. A. and Shabat B. V., Methods of the Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1986).
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Original Russian Text Copyright © 2008 Lyul’ko N. A.
The author was supported by the Russian Foundation for Basic Research (Grant 06-08-00386), by the Presidium of the Russian Academy of Sciences (Program No. 14, Project No. 115), and the Siberian Branch of the Russian Academy of Sciences (Projects 1.6 and 42).
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Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 6, pp. 1333–1350, November–December, 2008.
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Lyul’ko, N.A. Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions. Sib Math J 49, 1062–1077 (2008). https://doi.org/10.1007/s11202-008-0102-0
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DOI: https://doi.org/10.1007/s11202-008-0102-0