Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation ut+∂x2n+1u=∂x(u∂xnu)+∂xn-1(ux2),n≥2,n∈N+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H(-n2+34,-12n)(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H(s,a)(R) with s<-n2+34 and all a∈R.