A Time-Oscillating Hartree-Type Schrödinger Equation

and Applied Analysis 3 Lemma 7. Let (γ, ρ) be an admissible pair, and fix a time t 0 . Given f ∈ Lγ 󸀠 (R, Lρ 󸀠 (R)), it follows that


Introduction
In this paper, we discuss the following Hartree-type Schrödinger equation: where * represents the convolution operator,  ∈ (0, 4] ∩ (0, ),  ∈ R, and  is a periodic function belonging to  1 (R, R).People are interested in Hartree equation since it has many applications in the quantum theory of large systems of nonrelativistic bosonic atoms and molecules.The numbers of bosons in such systems are very large, but the interactions between them are weak.Hartree equation arises in the study of the mean-field limit of such systems; see, for example, [1][2][3].
Different from the classical Hartree-type Schrödinger equation, the coefficient of nonlinearity of (OHS) is a function, especially a periodic function, not some constant, although its  ∞ norm is finite.We assume  is the period of ; then we can define the mean value One can take such mean value as the coefficient of nonlinearity of Hartree-type Schrödinger equation:   + Δ +  () (|| − * || 2 )  = 0,  (0) =  ∈  1 (R  ) .

(HS)
Then, (OHS) is a time-oscillating equation and (HS) is the corresponding deterministic one.In this paper, our purpose is to discuss the relationship of well-posedness of solutions between (OHS) and (HS).
The Cauchy problem has been settled by Cazenave and Weissler [4,5] and Miao et al. [6][7][8].For the sake of conciseness, we only state the results without any detailed proof.The definition of admissible pair is arranged in Section 2, although we use it here.Proposition 1.For any initial data  ∈  1 (R  ), there exists a unique  1 solution of (OHS) (or (HS)) defined on the maximal life interval (− min ,  max ) with 0 <  max ,  min ≤ ∞.Moreover, the following properties hold.

Abstract and Applied Analysis
As mentioned above, we are concerned with the behavior of solution of (OHS), when || → +∞.Precisely, in the maximal life interval of solution of (HS), we attempt to find the relationship of solutions between (OHS) and (HS) as || is sufficiently large.Mimicking the approach of Cazenave and Scialom [9] and Fang and Han [10] in the case of the  1 Schrödinger equation with the local nonlinear term, we obtain the following theorems for Hartree-type.Theorem 2. Assume the initial data  ∈  1 (R  ) and define   as the solutions of (OHS).Let  be the solution of (HS) with the maximal life interval [0,  max ).Then, we have (1) for any time  satisfying 0 <  <  max , if || is sufficiently large, the solution   of (OHS) exists in [0, ]; (2) for any admissible pair (, ) and time 0 <  <  max , In particular, the convergence holds in ([0, ],  1 (R  )).
The assumption (2) makes sure the solution  of (HS) owning sufficient decay, by which deduces  not only is global but also has scattering state (the details can be referred to in [6][7][8]).In fact, (2) shows that  is global when  = 4 immediately, according to the blow-up alternative in Proposition 1.And for  < 4, the norm of ‖∇()‖  2 (R  ) can be controlled by (2), for any  ∈ [0,  max ), which shows  max = ∞ by the blow-up alternative in Proposition 1.The details can be found in Lemma 9.
Many people show that the condition (2) holds in different cases.Cazenave in [4] shows (2) is true for defocusing case (()) when 2 <  < 4. When  = 4, Miao et al. in [6] show (2) is true for defocusing case with the radial initial data and for focusing case with the radial initial data and its energy and kinetic energy smaller than the ground state's.
When solution  of (HS) is global but (2) does not hold, we are not sure the behavior of solution   of (OHS) even  is sufficiently large.In order to have a good understanding of the development of   , we think that we should understand the development of  firstly, especially the blow-up rate of .
In Section 2, we introduce some notations and some useful lemmas.Theorem 2 is proved in Section 3, and Section 4 is devoted to proving Theorem 3.

Notations and Some Tools
In this section, we introduce some notations and useful lemmas.In order to discuss nonlinear Schrödinger equation conveniently, we always consider the equivalence of (OHS) (or (HS)): where ( Δ ) ∈R represents the Schrödinger group.

Lemma 5 (classical Strichartz estimates). The following properties hold.
(i) For any  ∈  2 (R  ) and any admissible pair (, ), the function   →  Δ  belongs to In addition, there exists a constant  such that       Δ        (R,  ) ≤           2 . ( (ii) Let  be an interval in R,  = , and  0 ∈ .If (, ) is an admissible pair and  ∈    (,    (R  )), then for any admissible pair (, ), the function We also need the following maximal estimate, which follows immediately from the sharp Hardy inequality (see [15]).
The following lemma is the key to discussing the relationship between (OHS) and (HS), which shows that when || goes to infinity, the nonlinearity of (OHS) converges to the nonlinearity of (HS).The lemma has been proved by Cazenave and Scialom [9]; therefore, we only state it here without any detailed proof.
Lemma 8. Let the initial data  ∈  1 (R  ).For any  ∈ R, define   as the solution of (OHS), and  is the solution of (HS) with the maximal life interval [0,  max ).Fix a time  satisfying 0 <  <  max , and suppose   exists in the interval [0, ] when || is sufficiently large.Suppose the following conditions hold: where Then, for any admissible pair (, ), one has Proof.From the conditions (10), we can choose two constants  and  such that when || ≥ , we have sup Set and then Proposition 1 deduces  < ∞.
Finally, put all estimates in each subinterval together; we have which shows (21) is true and finishes the proof of lemma.
At the end of section, we give a blow-up alternative for (HS) (or (OHS)), which is useful for the proof of Theorem 3.

The Proof of Theorem 2
In this section, we prove the Theorem 2. In view of Lemma 8, we only need to show that the solution   of (OHS) exists in the interval [0, ] for sufficiently large  and the condition (10) holds.

The Proof of Theorem 3
The last section is devoted to the proof of Theorem 3. By blowup alternative in Proposition 1, the key point is to show the boundness of ‖  ‖ (0,∞) as  being sufficiently large.