Extended lifespan of the fractional BBM equation

Article type: Research Article

Authors: Nilsson, Dag^{; *}

Affiliations: Department of Mathematics, Saarland University, 66041, Saarbrucken, Germany. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: For 0<α<1, N⩾2 and with initial data ‖u0‖HN+α2=ε, sufficiently small, we show that the existence time for solutions of the fractional BBM equation ∂tu+∂xu+u∂xu+|D|α∂tu=0, can be extended from the hyperbolic existence time 1ε to 1ε2. 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).

Keywords: fBBM, enhanced life span, dispersive equations, nonlinear equations

DOI: 10.3233/ASY-211727

Journal: Asymptotic Analysis, vol. 129, no. 2, pp. 239-259, 2022