Global existence of solutions for a fourth-order nonlinear Schrödinger equation in n+1n+1 dimensions

We study the global well-posedness for the Cauchy problem of the fourth-order nonlinear Schrödinger equation iut+λΔu+μΔ2u+ν|u|2mu=0,x∈Rn,t∈R. By using the I-method, we prove that if s>1+mn−9+(4m−mn+7)2+164m, then the Cauchy problem is globally well-posed in Hs(Rn)Hs(Rn) for either the case λ<0,μ>0,ν>0λ<0,μ>0,ν>0 or the case λ>0,μ<0,ν<0λ>0,μ<0,ν<0 with m,nm,n satisfying some conditions.