2022/10/13 by Arora, Anudeep K., Roudenko, Svetlana
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.07344
We consider the focusing 5d Hartree equation, which is L2-supercritical, with finite energy initial data, and investigate the solutions at the mass-energy threshold. We establish the existence of special solutions following the work of Duyckaerts-Roudenko [11] for the 3d focusing cubic nonlinear Schrödinger equation (NLS). In particular, apart from the ground state solution Q, which is global but non-scattering, there exist special solutions Q+ and Q-, which in one time direction approach Q exponentially, and in the other time direction Q+ blows up in finite time and Q- exists for all time, exhibiting scattering behavior. We then characterize all radial threshold solutions either as scattering and blow up solutions in both time directions (similar to the solutions under the mass-energy threshold, see Arora-Roudenko [3]), or as the special solutions described above. To obtain the existence and classification result, in this paper we perform a thorough and meticulous investigation of the spectral properties of the linearized operator associated to the Hartree equation.