The stability of Sobolev norms for the linear wave equation with unbounded perturbations

TL;DR

This paper proves that solutions to a linear wave equation with unbounded perturbations maintain bounded Sobolev norms over time, using KAM reducibility, marking a novel result for such equations with quasi-periodic potentials.

Contribution

It introduces the first reducibility result for the linear wave equation with general quasi-periodic unbounded potentials on the torus.

Findings

Sobolev norms of solutions remain bounded over time

KAM reducibility technique is effective for unbounded perturbations

First such result for this class of wave equations

Abstract

In this paper, we prove that the Sobolev norm of solutions of the linear wave equation with unbounded perturbations of order one stay bounded for the all time. The main proof is based on the KAM reducibility of the linear wave equation. To the best of our knowledge, this is the first reducibility result of the linear wave equation with general quasi-periodic unbounded potentials on the torus.