The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 ×...The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 if small initial data （u0 （x）, u1 （x）, n0 （x）, n1 （x）） ∈ （H^1 ×L^2× L^2 × H^-1）. It answers an open problem： Is it globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 for 3D Klein-Gordon- Zakharov 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 F^s and N^3 in one dimension [3] to higher dimension.展开更多
文摘The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 if small initial data （u0 （x）, u1 （x）, n0 （x）, n1 （x）） ∈ （H^1 ×L^2× L^2 × H^-1）. It answers an open problem： Is it globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 for 3D Klein-Gordon- Zakharov 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 F^s and N^3 in one dimension [3] to higher dimension.
