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 Uspekhi Mat. Nauk, 1975, Volume 30, Issue 2(182), Pages 3–55 (Mi umn3983)

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

B. R. Vainberg

Abstract: In this paper we study the Cauchy problem and boundary-value problem of general form in the exterior of a compact set for hyperbolic operators $L$, whose coefficients depend only on $x$ and are constant near infinity. Assuming that the wave fronts of the Green's matrix for $L$ go off to infinity as $t\to\infty$, we determine the asymptotic behaviour of solutions as $t\to\infty$. For the corresponding stationary problem we obtain the short-wave asymptotic behaviour of solutions for real and complex frequencies.

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English version:
Russian Mathematical Surveys, 1975, 30:2, 1–58

Bibliographic databases:

UDC: 517.4
MSC: 35B40, 35L05, 47F05, 35Exx, 41A60

Citation: 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”, Uspekhi Mat. Nauk, 30:2(182) (1975), 3–55; Russian Math. Surveys, 30:2 (1975), 1–58

Citation in format AMSBIB
\Bibitem{Vai75} \by B.~R.~Vainberg \paper 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 \jour Uspekhi Mat. Nauk \yr 1975 \vol 30 \issue 2(182) \pages 3--55 \mathnet{http://mi.mathnet.ru/umn3983} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=415085} \zmath{https://zbmath.org/?q=an:0308.35011|0318.35006} \transl \jour Russian Math. Surveys \yr 1975 \vol 30 \issue 2 \pages 1--58 \crossref{https://doi.org/10.1070/RM1975v030n02ABEH001406} 

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3. Jeffrey Rauch, “Asymptotic behavior of solutions to hyperbolic partial differential equations with zero speeds”, Comm Pure Appl Math, 31:4 (1978), 431
4. Jeffrey Rauch, “Local decay of scattering solutions to Schrödinger's equation”, Comm Math Phys, 61:2 (1978), 149
5. V. M. Petkóv, “High frequency asymptotics of the scattering amplitude for non-convex bodies”, Communications in Partial Differential Equations, 5:3 (1980), 293
6. A. L. Piatnitski, “A scattering problem in laminar media”, Math. USSR-Sb., 43:3 (1982), 427–441
7. B. Yu. Sternin, V. E. Shatalov, “On a method of solving equations with simple characteristics”, Math. USSR-Sb., 44:1 (1983), 23–59
8. Krzysztof A. Michalski, “BIBLIOGRAPHY OF THE SINGULARITY EXPANSION METHOD AND RELATED TOPICS”, Electromagnetics, 1:4 (1981), 493
9. Minoru Murata, “Asymptotic expansions in time for solutions of Schrödinger-type equations”, Journal of Functional Analysis, 49:1 (1982), 10
10. A.G Ramm, “Mathematical foundations of the singularity and eigenmode expansion methods (SEM and EEM)”, Journal of Mathematical Analysis and Applications, 86:2 (1982), 562
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16. Xue-Ping Wang, “Time-decay of scattering solutions and resolvent estimates for semiclassical Schrödinger operators”, Journal of Differential Equations, 71:2 (1988), 348
17. C.O Bloom, N.D Kazarinoff, “Energy decay for hyperbolic systems of second-order equations”, Journal of Mathematical Analysis and Applications, 132:1 (1988), 13
18. Hirokazu Iwashita, “L q -L r estimates for solutions of the nonstationary stokes equations in an exterior domain and the Navier–Stokes initial value problems inL q spaces”, Math Ann, 285:2 (1989), 265
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20. Yoshihiro Shibata, Zheng Songmu, “On some nonlinear hyperbolic systems with damping boundary conditions”, Nonlinear Analysis: Theory, Methods & Applications, 17:3 (1991), 233
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