数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (4): 1117-1152.doi: 10.1016/S0252-9602(16)30059-5
GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION
霍朝辉
- Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION
Zhaohui HUO
- Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
摘要：
The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space
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utt-△u+u=-nu, (x,t)∈R3×R+,
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ntt-△n=△|u|2, (x,t)∈R3×R+,(0.1)
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u(x, 0)=u0(x), ∂tu(x,0)=u1(x), n(x,0)=n0(x), ∂tn(x,0)=n1(x),
is considered. It is shown that it is globally well-posed in energy space H 1×L2×L2×H-1 if small initial data (u0(x), u1(x), n0(x),n1(x))∈(H1×L2×L2×H-1). It answers an open problem:Is it globally well-posed in energy space H1×L2×L2×H-1 for 3D Klein-GordonZakharov equation with small initial data[1, 2]? The method in this article combines the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequalities). We mainly extend the spaces Fs and Ns in one dimension[3] to higher dimension.
中图分类号:
- 35E15