## Duke Mathematical Journal

### Cantor families of periodic solutions for completely resonant nonlinear wave equations

#### Abstract

We prove the existence of small amplitude, ($2\pi/\omega$)-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 asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite-dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity

#### Article information

Source
Duke Math. J., Volume 134, Number 2 (2006), 359-419.

Dates
First available in Project Euclid: 8 August 2006

https://projecteuclid.org/euclid.dmj/1155045505

Digital Object Identifier
doi:10.1215/S0012-7094-06-13424-5

Mathematical Reviews number (MathSciNet)
MR2248834

Zentralblatt MATH identifier
1103.35077

#### Citation

Berti, Massimiliano; Bolle, Philippe. Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math. J. 134 (2006), no. 2, 359--419. doi:10.1215/S0012-7094-06-13424-5. https://projecteuclid.org/euclid.dmj/1155045505

#### References

• A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349--381.
• P. Baldi and M. Berti, Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies, Rend. Lincei Mat. Appl. (9) 17 (2006), 257--277.
• D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci. 11 (2001), 69--87.
• M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. 243 (2003), 2, 315--328.
• —, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Anal. 56 (2004), 1011--1046.
• J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices, 1994, no. 11, 475--497.
• —, Quasi-periodic solutions of Hamiltonian perturbations of $2D$ linear Schrödinger equations, Ann. of Math. (2) 148 (1998), 363--439.
• —, Periodic solutions of nonlinear wave equations'' in Harmonic Analysis and Partial Differential Equations (Chicago, 1996), Chicago Lectures in Math., Univ. Chicago Press, Chicago, 1999, 69--97.
• L. Chierchia and J. You, KAM tori for $1$D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211 (2000), 497--525.
• W. Craig, Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panor. Synthèses 9, Soc. Math. France, Montrouge, 2000.
• W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math. 46 (1993), 1409--1498.
• —, Nonlinear waves and the $1:1:2$ resonance'' in Singular Limits of Dispersive Waves (Lyon, 1991), NATO Adv. Sci. Inst. Ser. B Phys. 320, Plenum, New York, 1994, 297--313.
• E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139--174.
• J. FröHlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151--184.
• G. Gentile, V. Mastropietro, and M. Procesi, Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions, Comm. Math. Phys. 256 (2005), 437--490.
• G. Iooss, P. I. Plotnikov, and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Anal. Mech. 177 (2005), 367--478.
• S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum (in Russian), Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 22--37.; English translation in Functional Anal. Appl. 21 (1987), 192--205.
• —, Analysis of Hamiltonian PDEs, Oxford Lecture Ser. Math. Appl. 19, Oxford Univ. Press, New York, 2000.
• S. Kuksin and J. PöSchel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2) 143 (1996), 149--179.
• B. V. LidskĭI and E. I. Shulman, Periodic solutions of the equation $u_tt-u_xx+u^3=0$, Funct. Anal. Appl. 22 (1988), 332--333.
• J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math. 29 (1976), 724--747.
• S. Paleari, D. Bambusi, and S. Cacciatori, Normal form and exponential stability for some nonlinear string equations, Z. Angew. Math. Phys. 52 (2001), 1033--1052.
• J. PöSchel, A KAM-theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 119--148.
• —, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv. 71 (1996), 269--296.
• H.-W. Su, Periodic solutions of finite regularity for the nonlinear Klein-Gordon equation, Ph.D. dissertation, Brown University, Providence, 1998.
• C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479--528.
• A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math. 20 (1973), 47--57.