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November  2016, 15(6): 2247-2280. doi: 10.3934/cpaa.2016036

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

 1 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

Received  January 2016 Revised  July 2016 Published  September 2016

We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n,$ $\partial_t n)|_{t=0} \in H^{s+1}(R^d) \times H^s(R^d) \times\dot{H}^s(R^d) \times \dot{H}^{s-1}(R^d)$. The critical value of $s$ is $s_c=d/2-2$. By the radial Strichartz estimates and $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 4$ for radial initial datum. For non-radial initial datum, we prove that the local well-posedness hold at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge 5$.
Citation: Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036
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