(?)^S_C-Scattering Of The Solution To The Cauchy Problem Of The Critical Hartree Equation

We consider the focusing mass-supercritical and energy-subcritical biharmonic Hartree equation,and prove the scattering result for the radial data u0∈H2(Rd)satisfying M(u0)2-sc/sc E(u0)<M(Q)2-sc/sc E(Q),||u0||2 2-sc/sc||Δu0||2<||Q||2 2-sc/sc||ΔQ||2,where sc∈(0,2)and Q is the ground-state solution of the elliptic equationΔ2Q+Q-(|·|-γ*|Q|2)Q=0.In Chapter 1,we briefly describe the research background,the main research results and some notations used in this paper.In Chapter 2,we introduce some basic knowledge needed to study the fourth-order Hartree equation and the basic theory used to prove the main results.In Chapter 3,we give the scattering result for mass-supercritical and energy-subcritical biharmonic Hartree equation.Firstly,we prove a dichotomy proposition of global well-posedness versus blow-up results,which yields the comparability of the total energy and the kinetic energy;Then,we prove the existence and compactness of the critical element;Finally,we give the proof of the rigidity theorem which is a contradiction with the critical element established before,then concluding the proof of Theorem 1.3.1.