In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{ eqnarray*} u_{ t}-|D_{ x}|^{ \alpha}u_{ x}+\partial_{ x}^{ -1}\partial_{ y}^{ 2}u+\frac{ 1}{ 2}\partial_{ x}(u^{ 2})=0,\alpha\geq4. \end{ eqnarray*} The goal of this paper is two-fold. Firstly, we prove that the problem is locally well-posed in anisotropic Sobolev spaces H^{ s_{ 1},\>s_{ 2}}(\R^{ 2}) with s_{ 1}>\frac{ 1}{ 4}-\frac{ 3}{ 8}\alpha, s_{ 2}\geq 0 and \alpha\geq4. Secondly, we prove that the problem is globally well-posed in anisotropic Sobolev spaces H^{ s_{ 1},\>0}(\R^{ 2}) with -\frac{ (3\alpha-4)^{ 2}}{ 28\alpha}<s_{ 1}\leq0. and \alpha\geq4. Thus, our global well-posedness result improves the global well-posedness result of Hadac (Transaction of the American Mathematical Society, 360(2008), 6555-6572.) when 4\leq \alpha\leq6.
Categories: math.AP