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 $H^s\times H^{s-1}$; $11/12<s<1$ for $d=4$, $15/16<s<1$ for $d=5$. We use the induction on scales and bushes argument in [9] where the model equation is wave equation. For d = 5, 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.