Improved lower bounds of analytic radius for the Benjamin-Bona-Mahony equation

TL;DR

This paper establishes an improved lower bound on the analytic radius decay rate for solutions of the BBM equation, using higher order almost conservation laws in analytic spaces.

Contribution

It introduces a higher order almost conservation law and an analytic norm with a smooth symbol to improve the lower bound on the analytic radius for the BBM equation.

Findings

Lower bound of analytic radius is proportional to t^{-2/3} as t→∞

New higher order almost conservation law in analytic spaces

Analytic norm with smooth symbol enhances analysis of solution regularity

Abstract

This paper is devoted to the spatial analyticity of the solution of the BBM equation on the real line with an analytic initial data. It is shown that the analytic radius has a lower bound like $t^{-\frac{2}{3}}$ as time $t$ goes to infinity, which is an improvement of previous results. The main new ingredient is a higher order almost conservation law in analytic spaces. This is proved by introducing an equivalent analytic norm with smooth symbol and establishing some algebra identities of higher order polynomials.