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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.