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Mat. Sb., 2002, Volume 193, Number 9, Pages 107–138 (Mi msb681)  

This article is cited in 2 scientific papers (total in 2 papers)

Stabilization of solutions of the first mixed problem for the wave equation in domains with non-compact boundaries

A. V. Filinovskii

M. V. Lomonosov Moscow State University

Abstract: In this paper we study the rate of decay, for large values of time, of the local energy of solutions of the first mixed problem for the wave equation in unbounded domains $\Omega\subset\mathbb R^n$, $n\geqslant 2$, with smooth non-compact boundaries. Under the assumption that the boundary surface satisfies a condition generalizing the condition of star-shapedness with respect to the origin we establish a power estimate of the rate of decay of the local energy as $t\to\infty$.
The proof is based on uniform estimates in the half-plane $\{\operatorname{Im} k>0\}$ of solutions of the corresponding spectral problem– the first boundary-value problem for the Helmholtz equation–obtained in the paper.

DOI: https://doi.org/10.4213/sm681

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English version:
Sbornik: Mathematics, 2002, 193:9, 1349–1380

Bibliographic databases:

UDC: 517.956.3
MSC: Primary 35L05; Secondary 35L20, 35B40
Received: 23.07.2001

Citation: A. V. Filinovskii, “Stabilization of solutions of the first mixed problem for the wave equation in domains with non-compact boundaries”, Mat. Sb., 193:9 (2002), 107–138; Sb. Math., 193:9 (2002), 1349–1380

Citation in format AMSBIB
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\paper Stabilization of solutions of the~first mixed problem for the~wave
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  • https://doi.org/10.4213/sm681
  • http://mi.mathnet.ru/eng/msb/v193/i9/p107

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Filinovskii, “Hyperbolic equations with growing coefficients in unbounded domains”, J. Math. Sci. (N. Y.), 197:3 (2014), 435–446  mathnet  crossref  elib
    2. A. V. Filinovskii, “Spectrum and stabilization in hyperbolic problems”, J. Math. Sci. (N. Y.), 234:4 (2018), 531–547  mathnet  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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