Extended lifespan of the fractional BBM equation

Abstract

For $0\mathrm{<}\mathit{\alpha }\mathrm{<}1$ , $\mathit{N}⩾2$ and with initial data $\Vert {\mathit{u}}_0{\Vert }_{{\mathit{H}}^{\mathit{N}+\frac{\mathit{\alpha }}{2}}}=\mathit{\epsilon }$ , sufficiently small, we show that the existence time for solutions of the fractional BBM equation ${\partial }_{\mathit{t}}\mathit{u}+{\partial }_{\mathit{x}}\mathit{u}+\mathit{u}{\partial }_{\mathit{x}}\mathit{u}+|\mathrm{D}{|}^{\mathit{\alpha }}{\partial }_{\mathit{t}}\mathit{u}=0$ , can be extended from the hyperbolic existence time $\frac{1}{\mathit{\epsilon }}$ to $\frac{1}{{\mathit{\epsilon }}^2}$ . For the proof we use a quasilinear modified energy method, based on a normal form transformation as in Hunter, Ifrim, Tataru, Wong (Proc. Am. Math. Soc., 143(8) (2015) 3407–3412).