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On the dissipative Boussinesq equation in a non-cylindrical domain. (English) Zbl 1117.35307
Summary: We investigate the initial-boundary value problem for the one-dimensional nonlinear Boussinesq equation inside domains with moving ends having both small increasing and decreasing displacements. Global solvability, uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense.

35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
32Q35 Complex manifolds as subdomains of Euclidean space
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
[1] Aubin, J.P., Un theorème de compacité, C. R. acad. sci. Paris, 256, 5042-5044, (1963) · Zbl 0195.13002
[2] Bona, J.; Sachs, R., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. math. phys., 118, 15-29, (1988) · Zbl 0654.35018
[3] Boussinesq, M.J., Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses senciblement pareilles de la surface au fond, J. math. pures appl., 17, 55-108, (1872) · JFM 04.0493.04
[4] Boussinesq, M.J., Essai sur la théorie des eaux courantes, Mém. présentés par divers savants à l’académie des sciences inst. France (séries 2), 17, 1-680, (1877) · JFM 09.0680.04
[5] Clark, H.R.; Rincon, M.A.; Rodrigues, R., Beam equation with weak-internal damping in domain with moving boundary, Appl. numer. math., 47, 139-157, (2003) · Zbl 1068.74035
[6] Cooper, J.; Bardos, C., Nonlinear wave equation in time dependent domain, J. math. anal. appl., 42, 29-60, (1983)
[7] Craig, W., An existence theory for water waves and the Boussinesq and Korteweg de Vries scaling limits, Comm. partial differential equations, 10, 787-1003, (1985) · Zbl 0577.76030
[8] Límaco, J.; Clark, H.R.; Medeiros, L.A., On equations of benjamin – bona – mahony type, Nonlinear anal., 59, 1243-1265, (2004) · Zbl 1066.35002
[9] Límaco, J.; Medeiros, L.A., Kirchhoff – carrier elastic strings in noncylindrical domains, Port. math., 56, 465-500, (1999), Fasc. 4 · Zbl 0943.35001
[10] Lions, J.L., Quelques methodes de resolution des problèmes aux limites non lineaires, (1969), Dunod Paris · Zbl 0189.40603
[11] Lions, J.L., Une remarque sur LES problèmes d’evolution non lineaires dans des domaines non cylindriques, Rev. roumaine math. pure appl., 9, 11-18, (1964) · Zbl 0178.12302
[12] Lions, J.L.; Magenes, E., Problèmes aux limites non homogènes at applications, vol. 1, (1968), Dunod Paris
[13] Liu, F.L.; Russell, D., Solutions of the Boussinesq equation on a periodic domain, J. math. anal. appl., 192, 194-219, (1995) · Zbl 0829.35097
[14] Medeiros, L.A., Nonlinear wave equations in domains with variable boundary, Arch. ration. mech. anal., 47, 47-58, (1972) · Zbl 0239.35066
[15] Medeiros, L.A.; Límaco, J.; Menezes, S.B., Vibrations of elastic strings: mathematical aspects, part two, J. comput. anal. appl., 4, 3, 211-263, (2002) · Zbl 1118.35336
[16] Miles, J.W., Solitary waves, Annu. rev. fluid mech., 12, 11-43, (1980)
[17] Tsutsumi, M.; Matahshi, T., On the Cauchy problem for the Boussinesq-type equation, Math. japonica, 36, 371-379, (1991) · Zbl 0734.35082
[18] Nakao, M.; Narazaki, T., Existence and decay of solutions of some nonlinear wave equations in noncylindrical domains, Math. rep. XI - 2, 117-125, (1978)
[19] Varlamov, V.V., On the initial boundary value problem for the damped Boussinesq equation, Discrete contin. dyn. syst., 4, 3, 431-444, (1998) · Zbl 0952.35103
[20] Varlamov, V.V., Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions, Int. J. math. math. sci., 22, 1, 131-145, (1999) · Zbl 0919.35111
[21] Varlamov, V.V., On the damped Boussinesq equation in a circle, Nonlinear anal., 38, 447-470, (1999) · Zbl 0938.35146
[22] Varlamov, V.V., On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear anal., 46, 699-725, (2001) · Zbl 1007.35073
[23] Zabusky, N.J., Nonlinear partial differential equations, (1967), Academic Press New York · Zbl 0183.18104
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