Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in five and more dimensions

We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d\ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}($\mathbb{R}$^d) \times H^s($\mathbb{R}$^d) \times \dot{H}^s($\mathbb{R}$^d) \times \dot{H}^{s-1}($\mathbb{R}$^d)$. The critical value of $s$ is $s_c=d\mathrm{/}2-2$. By $U^2,V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d\ge 5$.