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.
Citation:
A. V. Filinovskii, “Spectrum and stabilization in hyperbolic problems”, Tr. Semim. im. I. G. Petrovskogo, 31, 2016, 231–256; J. Math. Sci. (N. Y.), 234:4 (2018), 531–547
\Bibitem{Fil16}
\by A.~V.~Filinovskii
\paper Spectrum and stabilization in hyperbolic problems
\serial Tr. Semim. im. I.~G.~Petrovskogo
\yr 2016
\vol 31
\pages 231--256
\mathnet{http://mi.mathnet.ru/tsp97}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2018
\vol 234
\issue 4
\pages 531--547
\crossref{https://doi.org/10.1007/s10958-018-4027-2}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85052867807}
Linking options:
https://www.mathnet.ru/eng/tsp97
https://www.mathnet.ru/eng/tsp/v31/p231
This publication is cited in the following 1 articles:
Mussakan Muratbekov, Madi Muratbekov, “Separability and Estimates for Eigenvalues and Singular Numbers of a Class of Hyperbolic‐Type Differential Operators”, Math Methods in App Sciences, 2025
Citation in format AMSBIB
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