Local well-posedness and spatial analyticity for the generalized Zakharov–Kuznetsov equation

Abstract

The initial value problem for the generalized Zakharov–Kuznetsov equation on R2 is shown to be local well-posed in spaces of functions which are analytic on a strip without shrinking the width of the strip in time. The proof mainly relies upon the local smoothing effect, a multi-dimensional maximal function estimate and Bourgain-type spaces. These techniques are significant for derivative nonlinear dispersive equation in low regularity spaces. In addition, under the boundedness assumption of a suitable Sobolev norm the generalized Zakharov–Kuznetsov equation is proved to be Gevrey-class analysis. Especially, we obtain an explicit lower bound on the possible decreasing rate of the uniform radius of analyticity of a solution starting from analytic initial data.