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

TL;DR

This paper establishes an improved algebraic lower bound on the radius of spatial analyticity for solutions to the generalized KdV equation, showing it remains positive over time and decays at a controlled rate in the defocusing case.

Contribution

It provides a new, sharper lower bound on the decay rate of the radius of analyticity for the gKdV equation, extending previous results.

Findings

Radius of analyticity remains constant in a local time interval.

In the defocusing case, the decay rate of analyticity radius is at least polynomial in time.

The results improve upon previous bounds established by Bona et al.

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+\mu u^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 $k\geq 4$. 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 $k\geq 4$ even, we prove that when the local solution extends globally in time, then for any $T\geq T_0$, the radius of analyticity cannot decay faster than $cT^{-\left(\frac{2k}{k+4}+\epsilon\right)}$, $\epsilon>0$ 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. Gruji\'c, H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincar\'e, 22 (2005) 783--797].