A reducibility result for a class of linear wave equations with unbounded perturbations on $\mathbb T^d$

Riccardo Montalto
University of Zurich
Tuesday, June 20, 2017 - 14:30 to 16:00

We prove a reducibility result for a class of quasi-periodically forced linear wave equations on the $d$-dimensional torus $\mathbb T^d$ of the form $$\partial_{tt} v - \Delta v + \varepsilon {\cal P}(\omega t)[v] = 0$$where the perturbation ${\cal P}(\omega t)$ is a second order operator of the form ${\cal P}(\omega t) = - a(\omega t) \Delta  -  {\cal R}(\omega t)$, the frequency $\omega \in \mathbb R^\nu$ is in some Borel set of large Lebesgue measure, the function $a : \mathbb T^\nu \to \mathbb R$ (independent of the space variable) is sufficiently smooth and ${\cal R}(\omega t)$ is a time-dependent finite rank operator of the form $${\cal R}(\omega t)[v] = \sum_{k = 1}^N q_k(\omega t, x) \langle p_k(\omega t, \cdot), v \rangle_{L^2_x}\,. $$ The proof relies on a normal form procedure which reduces the equation to a time independent block-diagonal system. Thanks to the special structure of the perturbation, we are able to transform the equation into another one, which is a constant coefficients space-diagonal equation plus an arbitrarily regularizing remainder. Then we implement an iterative KAM reducibility scheme which requires to impose very weak second order Melnikov non resonance conditions (with loss of derivatives in space). Such non resonance conditions are fulfilled for most values of the frequencies $\omega$. 

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