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.

79 pages