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

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