Geodesic
Stabilization of the solutions of the wave equation in unbounded domains
Sbornik. Mathematics, Tome 187 (1996) no. 6, pp. 917-947 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The behaviour of the solution of the first mixed problem for the wave equation in an unbounded domain with smooth boundary is studied for large values of time. Estimates of the solution of the first boundary-value problem for the Helmholtz equation with spectral parameter in the closed upper half-plane are obtained, and uniqueness of its solution for a parameter on the real line is proved. 
@article{SM_1996_187_6_a8,
     author = {A. V. Filinovskii},
     title = {Stabilization of the~solutions of the~wave equation in unbounded domains},
     journal = {Sbornik. Mathematics},
     pages = {917--947},
     year = {1996},
     volume = {187},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_6_a8/}
}
TY  - JOUR
AU  - A. V. Filinovskii
TI  - Stabilization of the solutions of the wave equation in unbounded domains
JO  - Sbornik. Mathematics
PY  - 1996
SP  - 917
EP  - 947
VL  - 187
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_1996_187_6_a8/
LA  - en
ID  - SM_1996_187_6_a8
ER  - 
%0 Journal Article
%A A. V. Filinovskii
%T Stabilization of the solutions of the wave equation in unbounded domains
%J Sbornik. Mathematics
%D 1996
%P 917-947
%V 187
%N 6
%U http://geodesic.mathdoc.fr/item/SM_1996_187_6_a8/
%G en
%F SM_1996_187_6_a8
A. V. Filinovskii. Stabilization of the solutions of the wave equation in unbounded domains. Sbornik. Mathematics, Tome 187 (1996) no. 6, pp. 917-947. http://geodesic.mathdoc.fr/item/SM_1996_187_6_a8/

[1] Lax P. D., Phillips R. S., “The wave equation in exterior domains”, Bull. Amer. Math. Soc., 68:1 (1962), 47–49 | DOI | MR | Zbl

[2] Morawetz C. S., “The limiting amplitude principle”, Comm. Pure Appl. Math., 15:3 (1962), 349–361 | DOI | MR | Zbl

[3] Lax P. D., Morawetz C. S., Phillips R. S., “Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle”, Comm. Pure Appl. Math., 16:4 (1963), 477–486 | DOI | MR | Zbl

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

[5] Ivrii V. Ya., “Eksponentsialnoe ubyvanie reshenii volnovogo uravneniya vo vneshnosti pochti zvezdnoi oblasti”, DAN SSSR, 189:5 (1969), 938–940 | MR | Zbl

[6] Morawetz C. S., “Decay for solutions of the exterior problem for the wave equation”, Comm. Pure Appl. Math., 28:2 (1975), 229–264 | DOI | MR | Zbl

[7] Ralston J. V., “Solutions of the wave equation with localized energy”, Comm. Pure Appl. Math., 22:6 (1969), 807–823 | DOI | MR | Zbl

[8] Morawetz C. S., Ralston J. V., Strauss W. A., “Decay of solutions of the wave equation outside nontrapping obstacles”, Comm. Pure Appl. Math., 30:4 (1977), 447–508 | DOI | MR | Zbl

[9] Muravei L. A., “Ob asimptoticheskom povedenii pri bolshikh znacheniyakh vremeni reshenii vneshnikh kraevykh zadach dlya volnovogo uravneniya s dvumya prostranstvennymi peremennymi”, Matem. sb., 107(149):1 (1978), 84–133 | MR | Zbl

[10] Zachmanoglou E. C., “The decay of solutions of the initial-boundary value problem for the wave equation in unbounded regions”, Arch. Rathional Mech. Anal., 14:4 (1963), 312–325 | MR | Zbl

[11] Zachmanoglou E. C., “An example of slow decay of the solution of the initial-boundary value problem for the wave equation in unbounded regions”, Bull. Amer. Math. Soc., 70:4 (1964), 633–636 | DOI | MR | Zbl

[12] Muravei L. A., “Volnovoe uravnenie i uravnenie Gelmgoltsa v neogranichennoi oblasti so zvezdnoi granitsei”, Tr. MIAN, 185, Nauka, M., 1988, 171–180 | MR

[13] Ramm A. G., “Spektralnye svoistva operatora Shrëdingera v oblastyakh s beskonechnoi granitsei”, Matem. sb., 66(108):3 (1965), 321–343 | MR | Zbl

[14] Odeh F. M., “Uniqueness theorems for Helmholtz equation in domains with infinite boundaries”, J. Math. Mech., 12:6 (1963), 857–867 | MR | Zbl

[15] Vinnik A. A., “Ob usloviyakh izlucheniya dlya oblastei s beskonechnymi granitsami”, Izv. vuzov. Ser. matem., 1977, no. 7(182), 37–45 | MR | Zbl

[16] Mikhailov V. P., “O stabilizatsii resheniya odnoi nestatsionarnoi granichnoi zadachi”, Tr. MIAN, 91, Nauka, M., 1967, 100–112 | MR

[17] Rellich F., “Über das asymptotische Verhalten der Losungen von $\Delta u+\lambda u=0$ in unendlichen Gebieten”, Jahresber. Deutsch. Math.-Verein, 53:1 (1943), 57–65 | MR | Zbl

[18] Tayoshi T., “A remark on the limiting absorbtion principle for elliptic equations”, Repts. Univ. Electro-Communs., 25:2 (1975), 197–202 | MR

[19] Vinnik A. A., Eidus D. M., “Printsip predelnoi amplitudy dlya oblastei tipa paraboloida”, Izv. vuzov. Ser. matem., 1979, no. 3(202), 28–37 | MR | Zbl

[20] Karamata J., “Weiterführung der N. Wienerschen Methode”, Math. Z., 38:5 (1934), 701–708 | DOI | MR | Zbl

[21] Filinovskii A. V., “Ob asimptoticheskom povedenii reshenii odnoi vneshnei nestatsionarnoi zadachi”, Kompleksnye metody v matematicheskoi fizike, Donetsk, 1984, 186

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

[23] Plancherel M., “Sur la couvergence et sur la sommation par les moyennes de Cesaro de $\lim_{z=\infty} \int_a^z f(x) \cos xy\,dx$”, Math. Ann., 76 (1915), 315–326 | DOI | MR | Zbl