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

TL;DR

This paper establishes well-posedness and scattering results for the Klein-Gordon-Zakharov system in four or more spatial dimensions, highlighting differences between radial and non-radial initial data.

Contribution

It proves global well-posedness and scattering for radial data at the critical space, and local well-posedness for non-radial data at specific regularities in higher dimensions.

Findings

Radial initial data leads to global well-posedness and scattering.

Non-radial initial data results in local well-posedness at certain regularities.

Different techniques are used for radial and non-radial cases, including Strichartz estimates and $U^2, V^2$ spaces.

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 $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s=s_c=d/2-2$. 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 $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge 5$ by applying the $U^2, V^2$ type spaces.