On the Cauchy problem of fourth-order nonlinear Schrödinger equations

In this paper we study the Cauchy problem of the fourth-order Schrödinger equation i∂tu+aΔu+bΔ2u=±up for dimension ≤4≤4, where pp is an integer greater than 1, with initial data in Besov spaces. We prove that for any 4(p2−1)(4−n)p+4+n≤q≤∞, the Cauchy problem of this equation is locally well-posed in Ḃ2,qsp(Rn) and B2,qs(Rn), where sp=n2−4p−1 and s>sps>sp, and for any 1≤q≤∞1≤q≤∞ almost global well-posedness holds in these spaces if initial data are small. We also prove that if a=0a=0, then global well-posedness holds in these spaces for small initial data.