2000 character limit reached
Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions (1512.01510v1)
Published 4 Dec 2015 in math.AP
Abstract: We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H{s+1}(\mathbb{R}d) \times Hs(\mathbb{R}d) \times \dot{H}s(\mathbb{R}d) \times \dot{H}{s-1}(\mathbb{R}d)$. The critical value of $s$ is $s=s_c=d/2-2$. 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 $U2, V2$ type spaces. On the other hand, if the initial datum is non-radial, then we prove the local well-posedness at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge 5$ by applying the $U2, V2$ type spaces.