Acta mathematica scientia, Series B >
ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA
Received date: 2011-03-21
Revised date: 2011-06-28
Online 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.
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.
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 . DOI: 10.1016/S0252-9602(12)60144-1
[1] Bergh J, L¨ofstr¨om J. Interpolation Spaces, An Introduciton. Grundlehren der Mathematischen Wissenschaften. Berlin-New York: Springer-Verlag, 1976
[2] Bourgain J. Global well-posedness of defocusing 3D critical NLS in the radial case. J Amer Math Soc, 1999, 12: 145–171
[3] Brenner P. On space-time means and everywhere defined scattering operators for tnonlinear Klein-Gordon equations. Math Z, 1984, 186: 383–391
[4] Brenner P. On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations. J Differ Equ, 1985, 56: 310–344
[5] Colliander J, Keel M, Staffilani G, Takaoka H, Tao T. Viriel, Morawetz and interaction Morawetz inequalities, preprint
[6] Ginibre J, Velo G. Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schr¨odinger equations. Ann Inst H Poincar´e Phys Th´eor, 1985, 43: 399–442
[7] Ginibre J, Velo G. Generalized Strichartz inequalities for the wave equation. J Funct Anal, 1995, 133: 50–68
[8] Ginibre J, Velo G. Scattering Theory in the Energy Space for a Class of Hartree Equations, Nonlinear Wave Equations (Providence, RI, 1998)//Contemp Math, 263. Providence, RI: Amer Math Soc, 2000
[9] Keel M, Tao T. Endpoint Strichartz estimates. Amer J Math, 1998, 120(5): 955–980
[10] Menzala G P, Strauss W A. On a wave equation with a cubic convolution. J Differ Equ, 1982, 43: 93–105
[11] Miao C. The Modern Method of Nonlinear Wave Equations, Lectures in Contemporary Mathematics. 2nd ed. Monographs on Modern Pure Mathematics, No 133. Beijing: Science Press, 2010
[12] Miao C, Xu G, Zhao L. Global well-posedness and scattering for the energy-critical, defousing Hartree equation for radial data. J Funct Anal, 2007, 253: 605–627
[13] Miao C, Zhang B, Fang D. Global well-posedness for the Klein-Gordon equations below the energy norm. J Partial Differ Equ, 2004, 17(2): 97–121
[14] Mochizuki K. On small data scattering with cubic convolution nonlinearity. J Math Soc Japan, 1989, 41: 143–160
[15] Morawetz C, Strauss W A. Decay and scattering of solutions of a nonlinear relativistic wave equation. Comm Pure Appl Math, 1972, 25: 1–31
[16] Nakanishi K. Energy scattering for nonlinear Klein-Gordon and Schr¨oinger equations in spatial dimensions 1 and 2. J Funct Anal, 1999, 169(1): 201–225
[17] Nakanishi K. Energy scattering for Hartree equations. Math Res Lett, 1999, 6: 107–118
[18] Nakanishi K. Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity. Comm Partial Differ Equ, 1999, 24: 185–221
[19] Pecher H. Low energy scattering for nonlinear Klein-Gordon equations. J Funct Anal, 1985, 63: 101–122
[20] Strauss W A. Nonlinear scattering theory at low energy. J Funct Anal, 1981, 41: 110–133
[21] Strauss W A. Nonlinear scattering theory at low energy sequel. J Funct Anal, 1981, 43: 281–293
[22] Tao T. Nonlinear Dispersive Equations, Local and Global Analysis. CBMS Vol 106. Providence, RI: Amer Math Soc, 2006
[23] Visciglia N. On the decay of solutions to a class of defocusing NLS. Math Res Lett, 2009, 16: 919–926
[24] Wu H G, Yang H, Liu J. Global existence for semilinear wave equation in exterior domain. Acta Math Sci, 2007, 27A(6): 1089–1097
/
| 〈 |
|
〉 |