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

Abstract

In this paper, we prove that the Sobolev norms of solutions for the linear wave equation with unbounded perturbations of order one remain bounded for 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 for the linear wave equation with general quasi-periodic unbounded perturbations on the one-dimensional torus.