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On bounded and time-periodic solutions of nonlinear wave equations. (English) Zbl 0997.35004

A nonlinear wave equation of the form: \[ u_{tt} - u_{ss} = g(t,s,u) + g_1(u) u_t \] on an infinite strip \(\mathbb{R} \times [0,a]\) is studied. Suffient conditions are given in terms of the structural properties of the functions \(g\) and \(g_1\) for the problem to possess at least one time-periodic solution provided \(g\) is periodic in \(t\). Uniqueness of these solutions is also studied.
Reviewer: E.Feireisl (Praha)

MSC:

35B10 Periodic solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
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[1] Ben-Naoum, A. K.; Mawhin, J., The periodic-Dirichlet problem for some semilinear wave equations, J. Differential Equations, 96, 340-354 (1992) · Zbl 0779.35068
[2] Brézis, H.; Nirenberg, L., Characterizations of the range of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa (4), 5, 225-326 (1978) · Zbl 0386.47035
[3] Brézis, H.; Nirenberg, L., Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31, 225-326 (1978) · Zbl 0386.47035
[4] Brézis, H., Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8, 409-426 (1983) · Zbl 0515.35060
[5] Ding, Y.; Li, S.; Willem, M., Periodic solutions of symmetric wave equations, Rapp., Semin. Math., Louvain, Nouv. Ser., 245-260 (1995)
[6] Feireisl, E., Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations, Appl. Mat., 35, 192-208 (1990) · Zbl 0737.35040
[7] Feireisl, E., Time-periodic solutions to quasilinear telegraph equations with large data, Arch. Rational Mech. Anal., 112, 45-62 (1990) · Zbl 0721.35042
[8] Ficken, F. A.; Fleishman, B. A., Initial value problems and time-periodic solutions for a nonlinear wave solution, Comm. Pure Appl. Math., 10, 331-356 (1957) · Zbl 0078.27602
[9] Hale, J. K., Periodic solutions of a class of hyperbolic equations containing a small parameter, Arch. Rational Mech. Anal., 23, 380-398 (1967) · Zbl 0152.10002
[10] Nakao, M., Periodic solutions of linear and nonlinear wave equations, Arch. Rational Mech. Anal., 62, 87-98 (1976) · Zbl 0353.35005
[11] Prodi, G., Soluzioni periodiche di equazioni a derivative parziali di tipo iperbolico non lineari, Ann. Mat. Pura Appl., 42, 25-49 (1956) · Zbl 0072.10101
[12] Prodi, G., Soluzioni periodiche dell’ equazione delle onde con termine dissipativo non lineare, Rend. Sem. Mat. Univ. Padova, 36, 37-49 (1966) · Zbl 0145.35601
[13] Prouse, G., Soluzioni periodiche dell’ equazione delle onde non omogenea con termine dissipativo quadratico, Rend. Sem. Mat. Fis. Milano, 36, 1-19 (1966) · Zbl 0143.13502
[14] Rabinowitz, P. H., Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20, 145-205 (1967) · Zbl 0152.10003
[15] Rabinowitz, P. H., Periodic solutions of nonlinear hyperbolic partial differential equations, II, Comm. Pure Appl. Math., 22, 15-40 (1969) · Zbl 0157.17301
[16] Rabinowitz, P. H., Time periodic solutions of nonlinear wave equations, Manuscripta Math., 5, 165-194 (1971) · Zbl 0219.35062
[17] Štědry, M.; Vejvoda, O., Periodic solutions to weakly nonlinear autonomous wave equations, Czechoslovak Math. J., 25, 536-555 (1975) · Zbl 0319.35052
[18] Tanaka, K., Multiple periodic solutions of a superlinear forced wave equation, Ann. Mat. Pura Appl. (4), 162, 43-76 (1992) · Zbl 0801.35087
[19] Vejvoda, O., Periodic solutions of a linear and weakly nonlinear wave equation in one dimension, I, Czechoslovak Math. J., 14, 341-382 (1964) · Zbl 0178.45302
[20] Vejvoda, O., Partial Differential Equations: Time-Periodic Solutions (1981), Sijthoff & Noordhoff: Sijthoff & Noordhoff Alpeen aan den Rijn · Zbl 0183.10401
[21] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York/London/Sydney · Zbl 0125.32102
[22] Edwards, R. E., Functional Analysis (1965), Holt, Rinehart & Winston: Holt, Rinehart & Winston New York · Zbl 0182.16101
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