Geodesic
Spectrum and stabilization in hyperbolic problems
Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 31 (2016) no. 31, pp. 231-256 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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 $\mathbb R^n$, 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. 
@article{TSP_2016_31_31_a10,
     author = {A. V. Filinovskii},
     title = {Spectrum and stabilization in hyperbolic problems},
     journal = {Trudy Seminara im. I.G. Petrovskogo},
     pages = {231--256},
     year = {2016},
     volume = {31},
     number = {31},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a10/}
}
TY  - JOUR
AU  - A. V. Filinovskii
TI  - Spectrum and stabilization in hyperbolic problems
JO  - Trudy Seminara im. I.G. Petrovskogo
PY  - 2016
SP  - 231
EP  - 256
VL  - 31
IS  - 31
UR  - http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a10/
LA  - ru
ID  - TSP_2016_31_31_a10
ER  - 
%0 Journal Article
%A A. V. Filinovskii
%T Spectrum and stabilization in hyperbolic problems
%J Trudy Seminara im. I.G. Petrovskogo
%D 2016
%P 231-256
%V 31
%N 31
%U http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a10/
%G ru
%F TSP_2016_31_31_a10
A. V. Filinovskii. Spectrum and stabilization in hyperbolic problems. Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 31 (2016) no. 31, pp. 231-256. http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a10/

[1] Laks P. D., Giperbolicheskie differentsialnye uravneniya v chastnykh proizvodnykh, Reg. khaot. dinamika, M.–Izhevsk, 2010

[2] Filinovskii A. V., “Stabilizatsiya i spektr v zadachakh rasprostraneniya voln”, Kachestvennye svoistva reshenii differentsialnykh uravnenii i smezhnye voprosy spektralnogo analiza, ed. I. V. Astashova, YuNITI-DANA, M., 2012, 289–463

[3] Filinovskii A. V., “O stabilizatsii reshenii vneshnei kraevoi zadachi Dirikhle dlya uravneniya kolebanii plastiny”, Matem. zametki, 39:4 (1986), 586–596 | MR

[4] Mikhlin S. G., Problema minimuma kvadratichnogo funktsionala, Gostekhizdat, M., 1952 | MR

[5] Filinovskii A. V., “Stabilizatsiya reshenii volnovogo uravneniya v neogranichennykh oblastyakh”, Matem. sb., 187:6 (1996), 131–160 | DOI | MR

[6] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR

[7] Muckenhoupt C. F., “Almost periodic functions and vibrating systems”, J. Math. Phys., 8 (1929), 163–198 | DOI | MR

[8] Sobolev S. L., “O pochti-periodichnosti reshenii volnovogo uravneniya I, II”, DAN SSSR, 48:8 (1945), 570–573; 48:9, 646–648

[9] Riss F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR

[10] Viner N., Integral Fure i nekotorye ego prilozheniya, Fizmatlit, M., 1963

[11] Evans L. K., Uravneniya s chastnymi proizvodnymi, Nauchnaya kniga, Novosibirsk, 2003

[12] Lyusternik L. A., Sobolev V. I., Elementy funktsionalnogo analiza, Nauka, M., 1965 | MR

[13] Eidus D. M., “Printsip predelnoi amplitudy”, UMN, 24:3 (1969), 91–156 | MR

[14] Vinnik A. A., “Ob usloviyakh izlucheniya dlya oblastei s beskonechnymi granitsami”, Izv. vyssh. uchebn. zaved. Matematika, 1977, no. 7(182), 37–45 | MR | Zbl

[15] Morawetz C. S., “The decay for solutions of the exterior initial-boundary value problem for the wave equation”, Comm. Pure Appl. Math., 14:3 (1961), 561–568 | DOI | MR | Zbl

[16] Zachmanoglou E. C., “The decay for solutions of the initial boundary value problem for the wave equation in unbounded regions”, Arch. Ration. Mech. Anal., 14:4 (1963), 315–325 | MR

[17] Ladyzhenskaya O. A., “O printsipe predelnoi amplitudy”, UMN, 12:3 (1957), 161–164 | MR | Zbl

[18] Mikhailov V. P., “O printsipe predelnoi amplitudy”, DAN SSSR, 159:4 (1964), 750–752 | MR

[19] Muravei L. A., “Ob asimptoticheskom povedenii pri bolshikh znacheniyakh vremeni reshenii vneshnikh kraevykh zadach dlya volnovogo uravneniya s dvumya prostranstvennymi peremennymi”, Matem. sb., 107:1 (1978), 84–133 | MR | Zbl

[20] Vainberg B. R., Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki, Izd-vo Mosk. un-ta, M., 1982 | MR

[21] Chen Y. M., “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., 15:3 (1966), 389–396 | DOI | MR | Zbl

[22] Favorin V. M., “O povedenii pri $t\rightarrow \infty$ reshenii smeshannykh zadach dlya trekhmernogo volnovogo uravneniya v tsilindricheskoi po prostranstvennym peremennym oblasti”, Differents. uravneniya, 17:2 (1981), 354–365 | MR | Zbl

[23] Viner N., Peli R., Preobrazovanie Fure v kompleksnoi oblasti, Nauka, M., 1964 | MR

[24] Filinovskii A. V., “Giperbolicheskie uravneniya s rastuschimi koeffitsientami v neogranichennykh oblastyakh”, Tr. seminara im. I. G. Petrovskogo, 29, 2013, 455–473

[25] Filinovskii A. V., “Stabilizatsiya reshenii volnovogo uravneniya v oblastyakh s nekompaktnymi granitsami”, Matem. sb., 189:8 (1998), 141–160 | DOI | MR

[26] Filinovskii A. V., “Ubyvanie energii reshenii pervoi smeshannoi zadachi dlya volnovogo uravneniya v oblastyakh s nekompaktnymi granitsami”, Matem. zametki, 67:5 (2000), 311–315 | DOI | MR

[27] Filinovskii A. V., “Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya volnovogo uravneniya v oblastyakh s nekompaktnymi granitsami”, Matem. sb., 193:9 (2002), 107–138 | DOI | MR | Zbl

[28] Filinovskii A. V., “O skorosti ubyvaniya reshenii volnovogo uravneniya v oblastyakh so zvezdnoi granitsei”, Tr. seminara im. I. G. Petrovskogo, 26, 2007, 391–407