The Cauchy problem for a Korteweg–deVries equation with dispersion of order $m=2j+1$, where $j$ is a positive integer, (KdVm), is studied with data in Sobolev and analytic spaces. First, optimal bilinear estimates in Bourgain spaces are proved and using them well-posedness in Sobolev spaces $H^s$, $s>-j+\frac{1}{4}$, is established. Then, well-posedness in analytic Gevrey spaces $G^{\delta ,s}$, $\delta >0$, is proved by using an analytic version of the bilinear estimates. This implies that the uniform radius of analyticity persist for some time. For the later times a lower bound for the radius of spacial analyticity is derived, which is given by $\delta \left(t\right)\ge c{t}^{-\alpha }$, with $\alpha =\frac{4}{3}+\epsilon$, for any $\epsilon >0$, when $j=1$, and $\alpha =1$ when $j\ge 2$. Finally, it is shown that the regularity of the solution in the time variable is Gevrey of order $m$, and this is optimal.

具有阶分散的Korteweg-deVries方程的柯西问题 $米=2Ĵ+\mathrm{1个}$，在哪里 $Ĵ$是一个正整数（KdVm），使用Sobolev和分析空间中的数据进行研究。首先，证明了布尔加因空间中的最佳双线性估计，并利用它们在Sobolev空间中的适定性$H^s$， $s>-Ĵ+\frac{\mathrm{1个}}{4}$， 成立。然后，解析Gevrey空间中的适定性$G^{\delta ，s}$， $\delta >0$通过使用双线性估计的解析版本来证明。这意味着统一的分析半径会持续一段时间。在以后的时间中，得出空间分析半径的下限，由下式给出：$\delta （Ť）\ge C{Ť}^{-\alpha }$，带有 $\alpha =\frac{4}{3}+\epsilon$，对于任何 $\epsilon >0$， 什么时候 $Ĵ=\mathrm{1个}$和 $\alpha =\mathrm{1个}$ 什么时候 $Ĵ\ge 2$。最后，证明了时间变量中解的规律性是阶的Gevrey$米$，这是最佳选择。