Improved algebraic lower bound for the radius of spatial analyticity for the generalized KdV equation

Abstract

We consider the initial value problema (IVP) for the generalized Korteweg-de Vries (gKdV) equation \begin{equation} \begin{cases} \partial_tu+\partial_x^3u+μu^k\partial_xu=0, \,\;\; x\in \mathbb{R}, \, t \in \mathbb{R},\\ u(x,0)=u_0(x), \end{cases} \end{equation} where $u(x,t)$ is a real valued function, $u_0(x)$ 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 $k\ge 4$ even, 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 $c{T}^{-\left(\frac{2k}{k+4}+\epsilon \right)}$, $\epsilon >0$ arbitrarily small and $c>0$ a constant. The result of this work improves the one obtained by Bona et al. in [ J. L. Bona, Z. Grujić, H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincaré, 22 (2005) 783--797].

15 pages