The Cauchy problem for the generalized KdV equation in the Sobolev space $H^s(\mathbf{R})$

In this paper, we are concerned with the Cauchy problem for the generalized KdV equation with random data and rough data. Firstly, when $s\in \mathbf{R}$, by using the initial value randomization technique introduced by Shen et al. (arXiv:2111.11935) and the construction of appropriate auxiliary spaces, we establish the almost sure local well-posedness of the generalized KdV equation in $H^s(\mathbf{R})$, which improves Theorem 1.3 of Hwang and Kwak (Proc. Amer. Math. Soc. 146(2018), 267-280.) and Theorem 1.5 of Yan et al.(arXiv:2011.07128.). Secondly, by using the well-posedness results proved in Theorem 1.1, for $f\in H^s(\mathbf{R}),s\in \mathbf{R}$, we obtain \begin{eqnarray*} &&$\mathbb{P}$\left(\left{ω:\lim_{t\rightarrow0}|u(t,x)-U(t)f^ω(x)|{L{x}^{\infty}}=0\right}\right)=1, \end{eqnarray*} which improves Theorem 1.6 of Yan et al.(arXiv:2011.07128.). Thirdly, by using the dyadic decomposition and constructing appropriate function spaces, we establish nonlinear smoothing for the generalized KdV equation with rough data. Furthermore, by using this estimate, when data $f\in H^s(\mathbf{R})\cap {\widehat{L}}^{\mathrm{\infty }}(\mathbf{R}),s>\frac{1}{2}-\frac{2}{k+1},k\ge 4$, we obtain \begin{eqnarray*} &&\lim_{|x|\rightarrow \infty}u(t,x)=0,\quad t\in[0, T]. \end{eqnarray*} In particular, for $f(x)\in H^s(\mathbf{R}),s>\frac{1}{2}-\frac{2}{k+1},k\ge 4$, we prove \begin{eqnarray*} &&\lim_{|x|\rightarrow \infty}(u(t,x)-U(t)f(x))=0. \end{eqnarray*} Finally, by using Theorem 1.1, when $f\in H^s(\mathbf{R}),s\in \mathbf{R}$, we obtain \begin{eqnarray*} &&$\mathbb{P}$\left(\left{ω: \forall t\in I_ω, \lim_{|x|\rightarrow \infty}\left(u(t,x)-U(t)f^ω(x)\right)=0\right}\right)=1. \end{eqnarray*}