Wednesday, November 21, 2012

1211.4666 (Changxing Miao et al.)

Scattering Theory for Energy-Supercritical Klein-Gordon Equation    [PDF]

Changxing Miao, Jiqiang Zheng
In this paper, we study the global well-posedness and scattering problem for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We prove 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 strategy derived from 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 solution. This is precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.
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