# Extended lifespan of the fractional BBM equation

## TL;DR

This paper demonstrates that solutions to the fractional BBM equation with small initial data have an extended lifespan, increasing the existence time from order 1/ε to 1/ε^2 using a modified energy approach.

## Contribution

It introduces a novel method combining a normal form transformation and ideas from fractional KdV analysis to extend the lifespan of solutions to the fractional BBM equation.

## Key findings

- Existence time extended to 1/ε^2 for small initial data.
- Modified energy method effectively prolongs solution lifespan.
- Techniques from fractional KdV are adapted for the fractional BBM equation.

## Abstract

For $0<\alpha<1$ and with initial data $\vert\vert u_0\vert\vert_{H^{N+\frac{\alpha}{2}}}=\varepsilon$, sufficently small, we show that the existence time for solutions of the fractional BBM equation $\partial_tu+\partial_xu+u\partial_xu+\vert\mathrm{D}\vert^\alpha\partial_tu=0$, can be extended beyond the hyperbolic existence time $\frac{1}{\varepsilon}$, to $\frac{1}{\varepsilon^2}$. For the proof we use a modified energy, based on a normal form transformation as in [Hunter, Ifrim, Tataru, Wong, 2015]. In addition we employ ideas and techniques from [Ehrnstr\"om, Wang, 2018], in which the authors obtain an enhanced existence time for the fractional KdV equation.