Extended lifespan of the fractional BBM equation

Abstract

For 0 < α < 1, N ⩾ 2 and with initial data ‖ u 0 ‖ H N + α 2 = ε, sufficiently small, we show that the existence time for solutions of the fractional BBM equation ∂ t u + ∂ x u + u ∂ x u + | D | α ∂ t u = 0, can be extended from the hyperbolic existence time 1 ε to 1 ε 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).