Bifurcation of free vibrations for completely resonant wave equations

Massimiliano Berti; Philippe Bolle

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 2, page 519-528
  • ISSN: 0392-4041

Abstract

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We prove existence of small amplitude, 2p/v-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.

How to cite

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Berti, Massimiliano, and Bolle, Philippe. "Bifurcation of free vibrations for completely resonant wave equations." Bollettino dell'Unione Matematica Italiana 7-B.2 (2004): 519-528. <http://eudml.org/doc/195783>.

@article{Berti2004,
abstract = {We prove existence of small amplitude, 2p/v-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency $\omega$ belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.},
author = {Berti, Massimiliano, Bolle, Philippe},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {519-528},
publisher = {Unione Matematica Italiana},
title = {Bifurcation of free vibrations for completely resonant wave equations},
url = {http://eudml.org/doc/195783},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Berti, Massimiliano
AU - Bolle, Philippe
TI - Bifurcation of free vibrations for completely resonant wave equations
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/6//
PB - Unione Matematica Italiana
VL - 7-B
IS - 2
SP - 519
EP - 528
AB - We prove existence of small amplitude, 2p/v-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency $\omega$ belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.
LA - eng
UR - http://eudml.org/doc/195783
ER -

References

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  1. BAMBUSI, D.- PALEARI, S., Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69-87. Zbl0994.37040MR1819863
  2. BERTI, M.- BOLLE, P., Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328. Zbl1072.35015MR2021909
  3. BERTI, M.- BOLLE, P., Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Analysis, 56 (2004), 1011-1046. Zbl1064.35119MR2038735
  4. BERTI, M.- BOLLE, P., Cantor families of periodic solutions of completely resonant wave equations and the Nash-Moser theorem, preprint Sissa, 2004. Zbl1160.35476MR2395214
  5. BOURGAIN, J., Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations, 69-97, Chicago Lectures in Math., Univ. Chicago Press, 1999. Zbl0976.35041MR1743856
  6. CRAIG, W.- WAYNE, C. E., Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. Zbl0794.35104MR1239318
  7. GENTILE, G.- MASTROPIETRO, V.- PROCESI, M., Periodic solutions for completely resonant nonlinear wave equations, preprint 2004. Zbl1094.35021
  8. LIDSKIJ, B. V.- SHULMAN, E. I., Periodic solutions of the equation u t ⁢ t - u x ⁢ x + u 3 = 0 , Funct. Anal. Appl., 22 (1988), 332-333. Zbl0837.35012MR977006

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