In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{ eqnarray*} u_{ t}+(-1)^{ j+1}\partial_{ x}^{ 2j+1}u + \frac{ 1}{ 2}\partial_{ x}(u^{ 2}) = 0,j\in N^{ +},x\in\mathbf{ T}= [0,2\pi \lambda) \end{ eqnarray*} with low regularity data and $\lambda\geq 1$. Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{ s}(\mathbf{ T})$ with $s\geq -j+\frac{ 1}{ 2},j\geq2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in $H^{ s}(\mathbf{ T})$ with $s<-j+\frac{ 1}{ 2},j\geq2$ in the sense that the solution map is $C^{ 3}.$ The result of this paper improves the result of \cite{ H} with $j\geq2$.
Categories: math.AP