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Reducibility for a fast-driven linear Klein-Gordon Equation. (English) Zbl 1416.35152

Summary: We prove a reducibility result for a linear Klein-Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving; however, we require it to be fast oscillating. In particular, provided that the external frequency is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, diagonal one. We achieve this result in two steps. First, we perform a preliminary transformation, adapted to fast oscillating systems, which moves the original equation in a perturbative setting. Then, we show that this new equation can be put to constant coefficients by applying a KAM reducibility scheme, whose convergence requires a new type of Melnikov conditions.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35L10 Second-order hyperbolic equations
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35B15 Almost and pseudo-almost periodic solutions to PDEs
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