ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

doi:10.1016/S0252-9602(12)60144-1

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (5): 1835-1850.doi: 10.1016/S0252-9602(12)60144-1

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ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

WU Hai-Gen¹, ZHANG Jun-Yong²

1.School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China；Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China； 2.Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received:2011-03-21 Revised:2011-06-28 Online:2012-09-20 Published:2012-09-20
Supported by:
H.G. Wu was supported by the National Science Foundation of China (11071057, 10801015), China Postdoctoral Science Foundation (20100470570), the Guozhi Xu Posdoctoral Research Foundation, and Doctoral Foundation of Henan Polytechnic University.

Abstract

Abstract:

In this paper, we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d 3. By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation, respectively, we obtain the corresponding theory for energy subcritical and critical cases. The exponent range of the decay estimates is extended to 0 < γ ≤4 and γ < d with Hartree potential V (x) = |x|^−γ.

Key words: Klein-Gordon equation, Hartree nonlinearity, decay estimate, scattering the-ory

CLC Number:

35P25

WU Hai-Gen, ZHANG Jun-Yong. ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA[J].Acta mathematica scientia,Series B, 2012, 32(5): 1835-1850.

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ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

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