Periodic solutions of nonlinear wave equations with non-monotone forcing terms

Massimiliano Berti; Luca Biasco

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 2, page 117-124
  • ISSN: 1120-6330

Abstract

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Existence and regularity of periodic solutions of nonlinear, completely resonant, forced wave equations is proved for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. The corresponding infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. This difficulty is overcome finding a-priori estimates for the constrained minimizers of the reduced action functional, through techniques inspired by regularity theory as in [10].

How to cite

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Berti, Massimiliano, and Biasco, Luca. "Periodic solutions of nonlinear wave equations with non-monotone forcing terms." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.2 (2005): 117-124. <http://eudml.org/doc/252299>.

@article{Berti2005,
abstract = {Existence and regularity of periodic solutions of nonlinear, completely resonant, forced wave equations is proved for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. The corresponding infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. This difficulty is overcome finding a-priori estimates for the constrained minimizers of the reduced action functional, through techniques inspired by regularity theory as in [10].},
author = {Berti, Massimiliano, Biasco, Luca},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Wave equation; Periodic solutions; Variational methods; A-priori estimates; Lyapunov-Schmidt reduction; wave equation; periodic solutions; variational methods; -priori estimates; Lyapunov-Schmidt},
language = {eng},
month = {6},
number = {2},
pages = {117-124},
publisher = {Accademia Nazionale dei Lincei},
title = {Periodic solutions of nonlinear wave equations with non-monotone forcing terms},
url = {http://eudml.org/doc/252299},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Berti, Massimiliano
AU - Biasco, Luca
TI - Periodic solutions of nonlinear wave equations with non-monotone forcing terms
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/6//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 2
SP - 117
EP - 124
AB - Existence and regularity of periodic solutions of nonlinear, completely resonant, forced wave equations is proved for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. The corresponding infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. This difficulty is overcome finding a-priori estimates for the constrained minimizers of the reduced action functional, through techniques inspired by regularity theory as in [10].
LA - eng
KW - Wave equation; Periodic solutions; Variational methods; A-priori estimates; Lyapunov-Schmidt reduction; wave equation; periodic solutions; variational methods; -priori estimates; Lyapunov-Schmidt
UR - http://eudml.org/doc/252299
ER -

References

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  1. AMBROSETTI, A. - MALCHIODI, A., Perturbation methods and semilinear elliptic problems on R n . To appear. Zbl1115.35004MR2186962
  2. BERTI, M. - BIASCO, L., Forced vibrations of wave equations with non-monotone nonlinearities. Preprint SISSA 2004. Zbl1103.35076MR2245752DOI10.1016/j.anihpc.2005.05.004
  3. BERTI, M. - BOLLE, P., Periodic solutions of nonlinear wave equations with general nonlinearities. Comm. Math. Phys., 243, n. 2, 2003, 315-328. Zbl1072.35015MR2021909DOI10.1007/s00220-003-0972-8
  4. BERTI, M. - BOLLE, P., Multiplicity of periodic solutions of nonlinear wave equations. Nonlinear Anal., 56, 2004, 1011-1046. Zbl1064.35119MR2038735DOI10.1016/j.na.2003.11.001
  5. BRÉZIS, H. - NIRENBERG, L., Forced vibrations for a nonlinear wave equation. Comm. Pure Appl. Math., 31, n. 1, 1978, 1-30. Zbl0378.35040MR470377DOI10.1002/cpa.3160310102
  6. CORON, J.-M., Periodic solutions of a nonlinear wave equation without assumption of monotonicity. Math. Ann., 262, n. 2, 1983, 273-285. Zbl0489.35061MR690201DOI10.1007/BF01455317
  7. DE SIMON, L. - TORELLI, H., Soluzioni periodiche di equazioni a derivate parziali di tipo iperbolico non lineari. Rend. Sem. Mat. Univ. Padova, 40, 1968, 380-401. Zbl0198.13704MR228836
  8. HOFER, H., On the range of a wave operator with nonmonotone nonlinearity. Math. Nachr., 106, 1982, 327-340. Zbl0505.35058MR675766DOI10.1002/mana.19821060128
  9. PLOTNIKOV, P.I. - YUNGERMAN, L.N., Periodic solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length. Translation in Differential Equations, 24 (1988), n. 9, 1989, 1059-1065. Zbl0675.35058MR965608
  10. RABINOWITZ, P., Periodic solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math., 20, 1967, 145-205. Zbl0152.10003MR206507
  11. RABINOWITZ, P., Time periodic solutions of nonlinear wave equations. Manuscripta Math., 5, 1971, 165-194. Zbl0219.35062MR326179
  12. WILLEM, M., Density of the range of potential operators. Proc. Amer. Math. Soc., 83, n. 2, 1981, 341-344. Zbl0478.49012MR624926DOI10.2307/2043523

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