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

We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d\ge 4$ with radial or non-radial 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 $s=s_c=d\mathrm{/}2-2$. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in $d\ge 4$ by applying the radial Strichartz estimates and $U^2,V^2$ type spaces. On the other hand, if the initial datum is non-radial, then we prove the local well-posedness at $s=1\mathrm{/}4$ when $d=4$ and $s=s_c+1\mathrm{/}(d+1)$ when $d \ge 5$byapplyingthe$U^2, V^2$ type spaces.