On the fifth-order KdV equation: Local well-posedness and lack of uniform continuity of the solution map

In this paper we prove that the following fifth-order equation arising from the KdV hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,u:Rt×Rx→R,u(0,x)=u0(x),u0∈Hs(R) is locally well-posed in Hs(R)Hs(R) for s>52. Also, we prove the solution map of the equation is not uniformly continuous on a bounded set.