We consider the initial value problem (IVP) for the generalized Korteweg–de Vries (gKdV) equation $\left\{\begin{array}{c}{\partial }_{t}u+{\partial }_x^{3}u+\mu u^k{\partial }_{x}u=0,x\in \mathbb{R},t\in \mathbb{R},\hfill \\ u(x,0)=u_0\left(x\right),\hfill \end{array}\right.$where $u(x,t)$ is a real valued function, $u_0\left(x\right)$ is a real analytic function, $\mu =\pm 1$ and $k\ge 4$. We prove that if the initial data $u_0$ has radius of analyticity ${\sigma }_0$, then there exists $T_0>0$ such that the radius of spatial analyticity of the solution remains the same in the time interval $[-T_0,T_0]$. In the defocusing case, for $k\in 2\mathbb{N}$, $k\ge 4$, we prove that when the local solution extends globally in time, then for any $T\ge T_0$, the radius of analyticity cannot decay faster than $c{T}^{-\left(\frac{2k}{k+4}+ϵ\right)}$, $ϵ>0$ arbitrarily small and $c>0$ a constant. The result of this work improves the one by Bona et al. (2005) where the authors proved the decay rate is no faster than $c{T}^{-(k^2+3k+2)}$.

我们考虑广义 Korteweg–de Vries (gKdV) 方程的初值问题 (IVP)$\left\{\begin{array}{c}{\partial }_t你+{\partial }_X^3你+\mu {你}^k{\partial }_X你=0,X\epsilon \mathbb{右},t\epsilon \mathbb{右},\hfill \\ 你（X,0）={你}_0（X）,\hfill \end{array}\right.$在哪里$你（X,t）$是一个实值函数，${你}_0（X）$是实解析函数，$\mu =\pm 1$和$k\ge 4$。我们证明如果初始数据${你}_0$具有解析半径${\sigma }_0$，那么存在${\mathrm{时间}}_0>0$使得解的空间解析半径在时间间隔内保持相同$[-{\mathrm{时间}}_0,{\mathrm{时间}}_0]$。在散焦情况下，对于$k\epsilon 2\mathbb{氮}$,$k\ge 4$，我们证明当局部解及时扩展到全局时，那么对于任何$\mathrm{时间}\ge {\mathrm{时间}}_0$，解析性半径的衰减速度不能快于$C{\mathrm{时间}}^{-\left(\frac{2k}{k+4}+\epsilon \right)}$,$\epsilon >0$任意小且$C>0$一个常数。这项工作的结果改进了 Bona 等人的工作结果。（2005）作者证明衰减率不快于$C{\mathrm{时间}}^{-（k^2+3k+2）}$。