We consider the generalized two-dimensional Zakharov-Kuznetsov equation
ut+∂xΔu+∂x(uk+1)=0, where k≥3 is an integer
number. For k≥8 we prove local well-posedness in the L2-based Sobolev
spaces Hs(R2), where s is greater than the critical scaling
index sk=1−2/k. For k≥3 we also establish a sharp criteria to obtain
global H1(R2) solutions. A nonlinear scattering result in H1(R2) is
also established assuming the initial data is small and belongs to a suitable
Lebesgue space