Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.
Si dimostra l'esistenza di soluzioni quasi periodiche con due frequenze per una classe di equazioni delle onde non lineari completamente risonanti aventi un termine forzante periodico. Consideriamo che la frequenza forzante sia un numero razionale (Caso A), sia irrazionale (Caso B).
@article{RLIN_2005_9_16_2_109_0, author = {Massimiliano Berti and Michela Procesi}, title = {Quasi-periodic oscillations for wave equations under periodic forcing}, journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni}, volume = {16}, year = {2005}, pages = {109-116}, zbl = {1225.35146}, mrnumber = {2225504}, language = {en}, url = {http://dml.mathdoc.fr/item/RLIN_2005_9_16_2_109_0} }
Berti, Massimiliano; Procesi, Michela. Quasi-periodic oscillations for wave equations under periodic forcing. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005) pp. 109-116. http://gdmltest.u-ga.fr/item/RLIN_2005_9_16_2_109_0/
[1] Homoclinics: Poincaré-Melnikov type results via a variational method. Annales I.H.P. - Analyse nonlin., v. 15, n. 2, 1998, 233-252. | MR 1614571 | Zbl 1004.37043
- ,[2] A Birkhoof-Lewis type theorem for some Hamiltonian PDE's. SIAM Journal on Mathematical Analysis, to appear. | MR 2176924 | Zbl 1105.37045
- ,[3] Critical point theorems for indefinite functionals. Invent. Math., 52, n. 3, 1979, 241-273. | MR 537061 | Zbl 0465.49006
- ,[4] Periodic solutions of nonlinear wave equations with non-monotone forcing terms. Rend. Mat. Acc. Lincei, s. 9, v. 16, 2005, 117-124. | MR 2225505 | Zbl 1225.35147
- ,[5] Periodic solutions of nonlinear wave equations with general nonlinearities. Comm. Math. Phys., 243, n. 2, 2003, 315-328. | MR 2021909 | Zbl 1072.35015
- ,[6] Cantor families of periodic solutions for completely resonant non linear wave equations. Preprint SISSA 2004. | Zbl 1103.35077
- ,[7] Quasi-periodic solutions of completely resonant forced wave equations. Preprint SISSA. | MR 2233048 | Zbl 1100.35011
- ,[8] Periodic solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length. Transl. in Diff. Eq., 24, n. 9, 1988, 1059-1065. | MR 965608
- ,[9] Quasi-periodic solutions for completely resonant wave equations in 1D and 2D. Discr. Cont. Dyn. Syst., 13(3), August 2005, 541-552. | MR 2152330 | Zbl 1086.35007
,[10] Periodic solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math., 20, 1967, 145-205. | MR 206507 | Zbl 0152.10003
,