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

TL;DR

This paper proves almost sure scattering for the nonlinear Klein-Gordon equations with Sobolev critical power, using probabilistic methods and induction on scales for dimensions 4 and 5.

Contribution

It extends almost sure scattering results to Klein-Gordon equations with Sobolev critical power in higher dimensions, employing novel probabilistic and inductive techniques.

Findings

Almost sure scattering established for d=4 and d=5.

Used induction on scales and bushes argument adapted from wave equations.

Controlled energy increments using the mass term in Klein-Gordon equations.

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.