Acta mathematica scientia,Series B ›› 2016, Vol. 36 ›› Issue (4): 1117-1152.doi: 10.1016/S0252-9602(16)30059-5

• Articles • Previous Articles     Next Articles

GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION

Zhaohui HUO   

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
  • Received:2015-10-30 Revised:2016-01-29 Online:2016-08-25 Published:2016-08-25
  • Supported by:

    The author is supported by the NSF of China (11471323).

Abstract:

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space
 utt-△u+u=-nu, (x,t)∈R3×R+,
ntt-△n=△|u|2, (x,t)∈R3×R+,(0.1)
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.

Key words: Global well-posedness, 3D Klein-Gordon-Zakharov equation, dyadic Xs,b

CLC Number: 

  • 35E15
Trendmd