Global well-posedness and blow-up for the hartree equation

Abstract

© 2017 Wuhan Institute of Physics and Mathematics For 2 < ? < min {4,n}, we consider the focusing Hartree equation iu t +?u+(|x| -? *|u| 2 )u=0, x ? R n .Let M[u] and E[u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of - ? Q + Q = (|x| -? *|Q| 2 )Q. Guo and Wang [Z. Angew. Math. Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of (0.1) ifM[u] 1-s c E[u] s c < M[Q] 1-s c E[Q] s c (s c =[Formula presented]). In this paper, we consider the complementary caseM[u] 1-s c E[u] s c = M[Q] 1-s c E[Q] s c and obtain a criteria on blow-up and global existence for the Hartree equation (0.1).