Scattering theory for energy-supercritical Klein-Gordon equation

TL;DR

This paper proves that solutions to the energy-supercritical cubic Klein-Gordon equation in dimensions five and higher are global and scatter if they are a priori bounded in the critical Sobolev space, using concentration compactness and Morawetz estimates.

Contribution

It establishes global well-posedness and scattering for the energy-supercritical Klein-Gordon equation under a boundedness assumption in the critical Sobolev space, extending techniques from wave and Schrödinger equations.

Findings

Solutions are global and scatter under the boundedness condition.

Soliton-like solutions are ruled out using Morawetz inequality.

The approach adapts concentration compactness to Klein-Gordon context.

Abstract

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\R^d)\times H_x^{s_c-1}(\R^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schr\"odinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.