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 Mat. Sb. (N.S.), 1973, Volume 92(134), Number 2(10), Pages 224–241 (Mi msb3345)

On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems

B. R. Vainberg

Abstract: In this paper elliptic problems in exterior domains polynomially depending on a spectral parameter $k$ are considered. These problems are obtained from a mixed problem for hyperbolic equations by substituting $k$ for $id/dt$. For such elliptic problems analytic properties of the resolvent are studied in the neighborhood of the point $k=0$, which permits, for the corresponding nonstationary problem, a complete asymptotic expansion of solutions as $t\to\infty$.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1973, 21:2, 221–239

Bibliographic databases:

UDC: 517.944
MSC: 35J40, 35L30, 35B40

Citation: B. R. Vainberg, “On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems”, Mat. Sb. (N.S.), 92(134):2(10) (1973), 224–241; Math. USSR-Sb., 21:2 (1973), 221–239

Citation in format AMSBIB
\Bibitem{Vai73} \by B.~R.~Vainberg \paper On~exterior elliptic problems polynomially depending on a~spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems \jour Mat. Sb. (N.S.) \yr 1973 \vol 92(134) \issue 2(10) \pages 224--241 \mathnet{http://mi.mathnet.ru/msb3345} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=346319} \zmath{https://zbmath.org/?q=an:0294.35031} \transl \jour Math. USSR-Sb. \yr 1973 \vol 21 \issue 2 \pages 221--239 \crossref{https://doi.org/10.1070/SM1973v021n02ABEH002014} 

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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. B. R. Vainberg, “On a point source in an inhomogeneous medium”, Math. USSR-Sb., 23:1 (1974), 123–148
2. B. R. Vainberg, “On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as $t\to\infty$ of solutions of non-stationary problems”, Russian Math. Surveys, 30:2 (1975), 1–58
3. Minoru Murata, “Asymptotic expansions in time for solutions of Schrödinger-type equations”, Journal of Functional Analysis, 49:1 (1982), 10
4. Arne Jensen, “Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in L2(R4)”, Journal of Mathematical Analysis and Applications, 101:2 (1984), 397
5. Daniel Eidus, “Solutions of external boundary problems for small values of the spectral parameter”, Integr equ oper theory, 9:1 (1986), 47
6. D Eidus, “Asymptotical expansions of solutions of linear parabolic equations as t → ∞”, Journal of Mathematical Analysis and Applications, 130:1 (1988), 155
7. N. Weck, K. J. Witsch, “The low frequency limit of the exterior Dirichlet problem for the reduced wave equation”, Applicable Analysis, 38:1-2 (1990), 33
8. Song Jiang, “Lp-Lq Estimates For Solutions To The Damped Plate Equation In Exterior Domains”, Results. Math, 18:3-4 (1990), 231
9. N. Weck, K. J. witsch, “Exterior Dirichlet problem for the reduced wave equation: asymptotic analysis of low frequencies”, Communications in Partial Differential Equations, 16:2-3 (1991), 173
10. Norbert Weck, Karl J. Witsch, “Complete Low Frequency Analysis for the Reduced Wave Equation with Variable Coefficients in Three Dimensions *”, Communications in Partial Differential Equations, 17:9-10 (1992), 1619
11. N Weck, K.J Witsch, “Exact low frequency analysis for a class of exterior boundary value problems for the reduced wave equation in two dimensions”, Journal of Differential Equations, 100:2 (1992), 312
12. R. Kleinman, B. Vainberg, “Full low-frequency asymptotic expansion for second-order elliptic equations in two dimensions”, Math Meth Appl Sci, 17:12 (1994), 989
13. Wakako Dan, “On the Low-frequency Asymptotic Expansion for some Second-order Elliptic Systems in a Two-dimensional Exterior Domain”, Math Meth Appl Sci, 19:13 (1996), 1073
14. Michael Melgaard, “Spectral Properties at a Threshold for Two-Channel Hamiltonians I. Abstract Theory”, Journal of Mathematical Analysis and Applications, 256:1 (2001), 281
15. ARNE JENSEN, GHEORGHE NENCIU, “A UNIFIED APPROACH TO RESOLVENT EXPANSIONS AT THRESHOLDS”, Rev. Math. Phys, 13:06 (2001), 717
16. Michael Melgaard, “Spectral Properties in the Low- Energy Limit of One -Dimensional Schrödinger OperatorsH = -d2/dx2 +V. The Case 〈1,V1〉 ≠ 0”, Math Nachr, 238:1 (2002), 113
17. Mishio Kawashita, Wakako Kawashita, “Analyticity of the resolvent for elastic waves in a perturbed isotropic half space”, Math Nachr, 278:10 (2005), 1163
18. Lassaad Aloui, Moez Khenissi, “Stabilization of Schrödinger equation in exterior domains”, ESAIM COCV, 13:3 (2007), 570
19. E Lakshtanov, B Vainberg, “Resonance regimes of scattering by small bodies with impedance boundary conditions”, J Phys A Math Theor, 43:41 (2010), 415205
20. Lassaad Aloui, Slim Ibrahim, Moez Khenissi, “Energy decay for linear dissipative wave equations in exterior domains”, Journal of Differential Equations, 2015
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