On the Growth of the Sobolev Norms for the Nonlinear Klein–Gordon Equation

Abstract

In this article we study higher order Sobolev norms’ growth in time for the nonlinear Klein–Gordon equation posed on a three dimensional Riemannian compact manifold with or without boundary. We prove, without any further requirement on the geometry of the manifold, that these norms cannot grow faster than a polynomial, to do so we introduce a modified energy involving time derivatives of the solution which allows us to prove our main result by combining Strichartz estimates and a induction argument.