# Lower bounds on the radius of spatial analyticity for the Kawahara equation

## TL;DR

This paper establishes that solutions to the Kawahara equation with initially analytic data maintain a spatial analyticity radius that decays no faster than inversely proportional to time, providing lower bounds on analyticity over time.

## Contribution

It provides the first lower bounds on the decay rate of the spatial analyticity radius for solutions to the Kawahara equation with analytic initial data.

## Key findings

- The analyticity radius decays at most as 1/|t| over time.
- Solutions preserve a minimal analyticity radius inversely proportional to time.
- The results apply to solutions with initial data having a fixed radius of analyticity.

## Abstract

In this paper we obtain lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation $u_t + uu_x + \alpha u_{xxx} + \beta u_{xxxxx} = 0$, $\beta\neq0$, given initial data which is analytic with a fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ can decay no faster than $1/|t|$ as $|t|\rightarrow\infty$.