Well-posednessfor the Cauchy problem ofthe Klein-Gordon-Zakharov systemin four and more spatial dimensions

Isao  Kato

doi:10.3934/cpaa.2016036

Article Contents

2016, Volume 15, Issue 6: 2247-2280. Doi: 10.3934/cpaa.2016036

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

Isao Kato^1,

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

Received: December 31, 2015

Revised: June 30, 2016

Published: October 2016

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Abstract

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 $d \ge 5$.

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Mathematics Subject Classification: Primary: 35Q55, 35B40; Secondary: 35A01, 35A02.

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