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In this paper, we investigate the Cauchy problem for a higher order shallowwater type equation \begin{eqnarray*}u_{t}-u_{txx}+\partial_{x}^{2j+1}u-\partial_{x}^{2j+3}u+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0,\end{eqnarray*} where $x\in \mathbf{T}=\mathbf{R}/2\pi$ and $j\in N^{+}.$Firstly, we prove that the Cauchy problem for the shallow water type equationis locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -\frac{j-2}{2}$ forarbitrary initial data. By using the $I$-method, we prove that the Cauchyproblem for the shallow water type equation is globally well-posed in$H^{s}(\mathbf{T})$ with $\frac{2j+1-j^{2}}{2j+1}<s\leq 1.$ Our results improvethe result of A. A. Himonas, G. Misiolek (Communications in partialDifferential Equations, 23(1998), 123-139;Journal of Differential Equations,161(2000), 479-495.)

The Cauchy problem for a higher order shallow water type equation on the circle