Abstract

We prove the global in time existence and the stability of the solutions of a class of quasi-periodically forced linear wave equations on the circle of the form $$u_{tt}−u_{xx}+mu+Q(\omega t)u=0$$ where $Q$ is an unbounded pseudo-differential operator of order 2, parity preserving and reversible, and the forcing frequency $\omega$ belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to constant coefficients, applying first a pseudo-differential normal form reduction and then a KAM diagonalization scheme. A central point is that the equation is equivalent to a first order pseudo-differential system which, at the highest order, is the sum of two backward/forward transport equations, with non-constant coefficients, respectively on the subspaces of functions supported on positive/negative Fourier modes. The key idea is to straighten such operator through a novel quantitative Egorov analysis.

A main point of interest, in view of applications to a nonlinear setting, is that the change of variables that reduce the equation satisfies tame bounds.

This is a joint work with M. Berti, R. Feola and M. Procesi.

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