On the Cauchy problem of fourth-order nonlinear Schrödinger equations

Abstract

In this paper we study the Cauchy problem of the fourth-order Schrödinger equation i ∂ t u + a Δ u + b Δ 2 u = ± u p for dimension ≤ 4 , where p is an integer greater than 1, with initial data in Besov spaces. We prove that for any 4 ( p 2 − 1 ) ( 4 − n ) p + 4 + n ≤ q ≤ ∞ , the Cauchy problem of this equation is locally well-posed in B ̇ 2 , q s p ( R n ) and B 2 , q s ( R n ) , where s p = n 2 − 4 p − 1 and s > s p , and for any 1 ≤ q ≤ ∞ almost global well-posedness holds in these spaces if initial data are small. We also prove that if a = 0 , then global well-posedness holds in these spaces for small initial data.