Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions (Q1298128)

We show the time local well-posedness in the energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations in three space dimensions under the assumption that the propagation speeds of the electric field raised by the electrons and of the ion sound wave are different. On the basis of this result and the energy conservation law, we also prove the unique global existence of solutions for small initial data in the energy space. Our result is closely related to the bilinear estimates of null forms by \textit{S. Klainerman} and \textit{M. Machedon} [Commun. Pure Appl. Math. 46, No. 9, 1221-1268 (1993; Zbl 0803.35095) and Duke Math. J. 81, No. 1, 99-133 (1995; Zbl 0909.35094)].