Almost sure scattering for the nonlinear Klein–Gordon equations with Sobolev critical power

Abstract

In this paper, we study the almost sure scattering for the Klein–Gordon equations with Sobolev critical power. We obtain the almost sure scattering with random initial data in Hs×Hs−1,3d−14(d−1)<s<1 for 4≤d≤6. We use the induction on scales and bushes argument in Bringmann (2018) where the model equation is wave equation. For d=5,6, we use the mass term of the Klein–Gordon equation to obtain the control of the increment of energy in the process of induction on scales.