Scattering theory for energy-supercritical Klein-Gordon equation

Abstract

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u{|}^{2}u=0$ in dimension $d\ge 5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u,u_t)\in L_t^{\infty }(I;H_x^{s_c}({\mathbb{R}}^d)\times H_x^{s_c-1}({\mathbb{R}}^d))$ 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 Schrödinger 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.

24pages

References in corpus (5)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Energy-supercritical NLS: critical ${\dot{H}}^s$-bounds imply scattering
• The radial defocusing energy-supercritical cubic nonlinear wave equation in R^{1+5}
• The defocusing energy-supercritical NLS in four space dimensions
• Threshold solutions in the case of mass-shift for the critical Kline-Gordon equation