TY - JOUR
T1 - Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential
JF - Nonlinearity
Y1 - 2012
A1 - Massimiliano Berti
A1 - Philippe Bolle
AB - We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T d , d ≥ 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the 'separation properties' of the small divisors assuming weaker non-resonance conditions than in [11]. © 2012 IOP Publishing Ltd.
VL - 25
N1 - cited By (since 1996)3
ER -
TY - JOUR
T1 - Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions
JF - Archive for Rational Mechanics and Analysis
Y1 - 2010
A1 - Massimiliano Berti
A1 - Philippe Bolle
AB - We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash-Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large "clusters of small divisors". © Springer-Verlag (2009).
VL - 195
N1 - cited By (since 1996)6
ER -