Abstract
We study the connection between the stabilization of solutions of a mixed hyperbolic problem and spectral properties of the corresponding elliptic boundary value problem. We consider the first mixed problem for the wave equation in bounded and unbounded domains in ℝn, determine the class of its energy solutions, and represent the solutions in terms of the Bochner–Stieltjes integral. We study how the spectrum of the elliptic operator affects the behavior of local energy of a solution and describe a method which allows us to study the stabilization of solutions with the help of estimates in the spectral parameter for solutions of the stationary problem on the upper half-plane.
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References
P. D. Lax, Hyperbolic Partial Differential Equations, Courant Lect. Notes, Amer. Math. Soc., New York (2006).
A. V. Filinovskii, “Stabilization and spectrum in wave propagation problems,” in: I. V. Astashova, ed., Qualitative Properties of Solutions of Differential Equations and Associated Questions of Spectral Analysis [in Russian], UNITY-DANA, Moscow (2012), pp. 289–463.
A. V. Filinovskii, “Stabilization of solutions of the external Dirichlet problem for the equation of plate vibrations,” Mat. Zametki, 39, No. 4, 586–596 (1986).
S. G. Mikhlin, The Problem of Minimum of a Quadratic Functional, Holden-Day, San Francisco (1965).
A. V. Filinovskii, “Stabilization of solutions of the wave equation in unbounded domains,” Mat. Sb., 187, No. 6, 131–160 (1996).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
C. F. Muckenhoupt, “Almost periodic functions and vibrating systems,” J. Math. Phys., 8, 163–198 (1929).
S. L. Sobolev, “On almost-periodicity of solutions of the wave equation I, II,” Dokl. Akad. Nauk SSSR, 48, No. 8, 570–573 (1945); Dokl. Akad. Nauk SSSR, 48, No. 9, 646–648 (1945).
F. Riesz and B. Nagy, Leçons D’Analyse Fonctionelle, Akadémiai Kiado, Budapest (1972).
N. Wiener, The Fourier Integral and Certain of Its Applications, Dover (1958).
L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence (2010).
L. A. Liusternik and V. J. Sobolev, Elements of Functional Analysis [in Russian], Nauka, Moscow (1965).
D. M. Eidus, “The limit amplitude principle,” Usp. Mat. Nauk, 24, No. 3, 91–156 (1969).
A. A. Vinnik, “Radiation conditions for domains with infinite boundaries,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7 (182), 37–45 (1977).
C. S. Morawetz, “The decay for solutions of the exterior initial boundary-value problem for the wave equation,” Commun. Pure Appl. Math., 14, No. 3, 561–568 (1961).
E. C. Zachmanoglou, “The decay for solutions of the initial boundary-value problem for the wave equation in unbounded regions,” Arch. Ration. Mech. Anal., 14, No. 4, 315–325 (1963).
O. A. Ladyzhenskaya, “On the limit amplitude principle,” Usp. Matem. Nauk, 12, No. 3, 161–164 (1957).
V. P. Mikhailov, “On the limit amplitude principle,” Dokl. Akad. Nauk SSSR, 159, No. 4, 750–752 (1964).
L. A. Muravei, “Large time asymptotic behavior of solutions of exterior boundary-value problems for the wave equation with two spatial variables,” Mat. Sb., 107, No. 1, 84–133 (1978).
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, CRC Press (1989).
Y. M. Chen, “On slow decay of the solution of the initial boundary-value problem for the wave equation in the exterior of a three-dimensional obstacle,” J. Math. Anal. Appl., 15, No. 3, 389–396 (1966).
V. M. Favorin, “On t → ∞ behavior of solutions of mixed problems for the 3d wave equation in a cylindrical domain,” Differ. Uravn., 17, No. 2, 354–365 (1981).
N. Wiener and R. C. Paley, Fourier Transforms in the Complex Domain, Amer. Math. Soc., New York (1954).
A. V. Filinovskii, “Hyperbolic equations with growing coefficients in unbounded domains,” Tr. Semin. Petrovskogo, 29, 455–473 (2013).
A. V. Filinovskii, “Stabilization of solutions of the wave equation in domains with noncompact boundaries,” Mat. Sb., 189, No. 8, 141–160 (1998).
A. V. Filinovskii, “Energy decay of solutions of the first mixed problem for the wave equation in domains with noncompact boundaries,” Mat. Zametki, 67, No. 5, 311–315 (2000).
A. V. Filinovskii, “Stabilization of solutions of the first mixed problem for the wave equation in domains with noncompact boundaries,” Mat. Sb., 193, No. 9, 107–138 (2002).
A. V. Filinovskii, “On the dacay rate of solutions of the wave equation in domains with star-shaped boundaries,” Tr. Semin. Petrovskogo, 26, 391–407 (2007).
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 31, pp. 231–256, 2016.
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Filinovskii, A.V. Spectrum and Stabilization in Hyperbolic Problems. J Math Sci 234, 531–547 (2018). https://doi.org/10.1007/s10958-018-4027-2
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DOI: https://doi.org/10.1007/s10958-018-4027-2