Reducibility of Klein-Gordon equations with maximal order perturbations

Abstract

We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $\psi_{tt}-\psi_{xx}+\mathtt{m}\psi+Q(\omega t)\psi=0 $ where $ Q(\omega t) := a^{(2)}(\omega t, x) \partial_{xx} + a^{(1)}(\omega t, x)\partial_x + a^{(0)}(\omega t, x) $ is a differential operator of order $ 2 $, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients $ a^{(2) }, a^{(1) }, a^{(0) } $ are small enough and the forcing frequency $\omega\in {\mathbb R}^{\nu}$ belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation $ Q (\omega t) $ of the second order differential operator $ a^{(2)}(\omega t, x)\partial_{xx} $. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis.