Scattering For The (?)^1/2-critical Hartree Equation

In this thesis, we consider the Cauchy problem of nonlinear Schrodinger equation(NLS) of Hartree type in dimension 5, where u is a complex valued function defined on some time-space slab I×R5, d is the dimension.μ=±1 refers to the defocusing and focusing case respectively. In dimension 5, the equation is H1/2-critical, which means that the equation and the H1/2-norm of the initial data is preserved by the scalingThere are four chapters. In ChapterⅠ, some preliminaries and basic es-timates are presented. In ChapterⅡ, we consider the Cauchy problem of the defocusing equation in critical space H1/2. Note that in this critical space, no conservation law is available. And we assume that the solution with maximal lifespan I satisfies (?)‖u(t)‖Hx1/2<∞. The assumption mimic the missing conservation law. Our goal is to obtainTheorem 1 Letμ=1, and let u0∈Hx1/2(R5) be radially symmetric, t0∈R,I is a time interval containing t0. Let u:I×R5→C be a maximal life-span solution to (1.1). Assume (?)‖|▽|1/2u(t)‖2<∞. Then u is global andThe theorem is proved by a contradiction argument. Suppose to the con-trary that there exist blow-up solutions. Then by a concentration compactness process,we will find a minimal H1/2-norm blow-up solution. This minimal H1/2-norm solution has many special properties.It is almost periodic modulo symmetries.Definition 1 (Almost periodic modulo symmetries) Let u be a so-lution to(1) with maximal life-spanⅠ. We call u is almost periodic modulo symmetries if there exist functions N: I→R+, x: I→R5, C: R+→R+ such that for allη>0, t∈I We refer to N(t) as the frequency scale function for the solution, x(t) is the position center of the solution, and C the compactness modulus function. If u is radially symmetric, then x(t)=0. Moreover, there has been a delicate relationship between the frequeficy scale function and the maximal life-span. More precisely, we prove the followingTheorem 2 Suppose Theorem 1 failed for radially symmetric data. Then there exists a maximal life-span solution u:I×R5→C to (1) with sup‖|▽|1/2u‖2 <∞. u is almost periodic modulo scaling, blows up both forward and back-ward. Moreover, the frequency scale function N(t) and the maximal life-span I match one of the following scenariosⅠ. (Finite-time blowup) Either |inf I|<∞or sup I<∝.Ⅱ. (Low-to-high cascade) I=R,Ⅲ. (Soliton-like solution) I=R, N(t)≡1 for all t∈R. Thus,to prove Theorem 1,it suffices to preclude the three scenarios in The-orem 2.To disprove the finite-time blow-up solution,we will first showing the fact that in this case frequency scale function goes to infinity as the time tends to the endpoint of the life-span.Some negative regularity is needed for disproving the rest two scenarios,and our discussions are somewhat involved due to the nonlocal nonlinearity and low regularity. We shall make full use of the frequency localization. Besides, to kill the final enemy, we have to gain additional regularity of at least 1 order differentiability, which means that the soliton-like solution has conserved energy; and thus allows us to apply virial-type argument to disprove it.In ChapterⅢ, we investigate the Cauchy problem of the focusing equation with initial data in the energy space H1. In this case, the solution enjoys two conservation laws: Energy conservation: The Cauchy problem in this case has a stationary solution eit Q, which is global but blows up both forward and backward. Q is the unique radially symmetric positive Swchartz solution to the elliptic equation This Q is expected to be the threshold for scattering versus blow-up. Indeed, we shall show that the threshold is M[Q]E[Q], which is invariant under scaling (2). Our main result in this chapter is the followingTheorem 3 Letμ=-1, u0∈H1(R5), and let u be the corresponding solution to(1)With maximal lifespanⅠ. Assume M[u]E[u]‖Q‖2‖▽Q‖2, assume in addition xu0∈L2 or u0∈H1 be radial. Then u blows up in finite time. The dichotomy of global existence versus blow-up is proved by variational characterization and by using the virial identity. To prove the scattering result, we use the concentration-compactness principle. The main tool is the following linear profile decomposition in H1, i.e.,Lemma 1 (Linear profile decomposition) Letφn∈H1(R5) be a se-quence of functions,‖φn‖H1 are uniformly bounded. Then, up to a subsequence, there exist {ψj}(?)H1, {tnj}(?)R, {xnj}(?)R5 such that for each J≥1, with wnJ∈H1, and for any H1/2-admissible pair (q,r)(i.e. 2/q = d(1/2-1/r)-1/2), Moreover, For 0≤s≤1, J≥1,In ChapterⅣ, we investigate the behavior of the critical Sobolev H1/2-norm of the finite time blow-up solution. By finite time blow-up we it means that there exists T<+∞such that (?)‖▽u‖2=∞. We prove that the finite time blow up solution must also blow up in the critical H1/2-norm, not only blows up in kinetic energy. Moreover, We show that the scaling invariant Lx5/2-norm blows up with a lower bound. Specifically, we have our main result:Theorem 4 Letμ=-1, u0∈Hx1/2∩Hx1(R5), radially symmetric, and let u be the corresponding solution to (1). Assume u blows up in finite time 00 as t close to T. We argue by contradiction.Suppose that the H1/2-norm is uniformlly bounded. Then using the compactness argument we will extract a solution that is global and with non-positive energy.The second step is to prove the following Liou-ville type theorem: Theorem 5 (Reduction to the Liouville theorem) Assume 0≠u0∈Hx1/2∩Hx1(R5), radially symmetric. Also assume E(u0)≤0. Then the corresponding solution u(t) to (1) blows up in finite time 0