Well-posedness and ill-posedness for a fifth-order shallow water wave equation

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the k;Z] multiplier norm method of Tao (2001) 2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in with is obtained by the Fourier restriction norm method. And some ill-posedness in with is derived from a general principle of Bejenaru and Tao.