A Reducibility Result for a Class of Linear Wave Equations on T-d

We prove a reducibility result for a class of quasi-periodically forced linear wave equations on the d-dimensional torus Td of the form ∂ttv − v + εP(ωt)[v] = 0, where the perturbation P(ωt) is a second order operator of the form P(ωt) = −a(ωt) − R(ωt), the frequency ω ∈ Rν is in some Borel set of large Lebesgue measure, the function a : Tν → R (independent of the space variable) is sufficiently smooth and R(ωt) is a time-dependent finite rank operator. This is the first reducibility result for linear wave equations with unbounded perturbations on the higher dimensional torus Td. As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible.