| The dissertation is devoted to study the global well-posedness and scattering theory of the energy-supercritical nonlinear Schrodinger equation and energy-subcritical or energy-critical Klein-Gordon-Hartree equations by making use of some modern harmonic analysis methods, such as Littlewood-Paley theory, the concentration-compactness approach and so on. The re-search of scattering theory originates from Segal’s conjecture[81]. Since the1970s and eighties, the study of scattering theory for the nonlinear dispersive equations became a hot issues in par-tial differential equations and physics, see Cazenave’s and Tao’s monographs. Prom a physical point of view, research on scattering theory is a very effective method for scientists to prob-ing the microscopic natureof. And it plays an important role in research on quantum physics, chemistry and biology. Prom a mathematical point of view, scattering theory is to study the long time behavior of the solution for the Cauchy problem of nonlinear dispersive equations.In Chapter2, we use concentration-compactness approach[38], Long-time Strichartz estimate[21] and frequency localized interaction Morawetz estimate[19] to study the Cauchy problem of the defocusing energy-supercritical Schrodinger equation (NLS) where u:Rt×Rc4â†'C and sc=2-2/p.Since sc>1, we call (6) energy-supercritical. The main result is:Assume that1<sc<3/2; let u:I×R4â†'C be a maximal-lifespan solution to (6) such that||u||Lt∞(I;Hxsc(R4))<+∞; then the solution u is global and scatters, i.e. there exists unique u±∈Hxsc(R4) such thatIn Chapter3, we utilize concentration-compactness method, interaction Morawetz estimate and ’double Duhamel trick’ to study the Cauchy problem of the defocusing energy-supercritical Hartree equation where u:Rt×Rxdâ†'C and sc:=r/2-1>1,i.e.γ>4,d>7. The main result is: Assume that d>7>4; let u:I×Rdâ†'C be a maximal-lifespan solution to (8) such that||u||Lt∞(I;Hxsc(Rd))<+∞; then the solution u is global and scatters. The outline of proof: We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.In Chapter4, we use concentration-compactness method and Morawetz estimate to study the Cauchy problem of the defocusing energy-critical wave-Hartree where u(t,x): R×Rdâ†'R. The main result is: Assume that d>5and (u0,u1)∈H1(Rd)×L2(Rd), then the solution of (9) u(t) is global and scatters. The main difficulty is the absence of the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone).In Chapter5, we utilize concentration-compactness approach-. Morawetz estimate and variational method to study the Cauchy problem of Klein-Gordon-Hartree equation where u(t, x): R×Rdâ†'R,2<γ<min{4, d}, or7=4and d>5; μ∈{-1,1} with μ=1known as the defocusing case and μ=-1as the focusing case. For defocusing case: μ=1, we obtain: Assume that μ=1,(u0,u1)∈H1(Rd)×L2(Rd) d≥3,2<γ<min{d,4}, or γ=4and d>5. Then there exists a unique global solution u(t) of (10) which scatters.For focusing case: μ=-1, we get: Assume d≥3, μ=-1, and2<γ<min{d,4}. Let (u0,u1) G H1×L2satisfy E(u0,u1)<E(W,0)(W is the ground state of the elliptic equation Q-â–³Q=(|x|-γ*|Q|2)Q) and u be the corresponding solution of (10) with maximal interval of existence I=(-T-(u0,u1),T+(u0,u1)).(â…°) If||â–½u0|||+||u0||22<||â–½W||22+||W||22, and the initial datum (u0,u1) are radial, then u is global and scatters.(â…±) If||â–½u0||22+||u0||22>||â–½W||22+||W||22, then u blows up both forward and backward in finite time, i.e. T-(u0,u1),T+(u0,u1)<∞. |
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