A reducibility result for a class of linear wave equations on ${\mathbb{T}}^d$

Abstract

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 $${\partial }_{tt}v-\Delta v+\epsilon \mathcal{P}(\omega t)[v]=0$$ where the perturbation $\mathcal{P}(\omega t)$ is a second order operator of the form $\mathcal{P}(\omega t)=-a(\omega t)\Delta -\mathcal{R}(\omega t)$, the frequency $\omega \in {\mathcal{R}}^{\nu }$ is in some Borel set of large Lebesgue measure, the function $a:{\mathbb{T}}^{\nu }\to \mathcal{R}$ (independent of the space variable) is sufficiently smooth and $\mathcal{R}(\omega t)$ 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 ${\mathbb{T}}^d$. As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible.

50 pages. version 2. change of the title and other minor changes with respect to the previous version

References in corpus (2)

• On reducibility of Quantum Harmonic Oscillator on ${\mathbb{R}}^d$ with quasiperiodic in time potential
• Large KAM tori for perturbations of the dNLS equation

Cited by in corpus (4)

• KAM for quasi-linear autonomous NLS
• Reducibility, Lyapunov exponent, pure point spectra property for quasi-periodic wave operator
• A reducibility result for Schrödinger operator with finite smooth and time quasi-periodic potential
• Reducibility for wave equations of finitely smooth potential with periodic boundary conditions