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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 950132, 7 pages
A Time-Oscillating Hartree-Type Schrödinger Equation
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Received 14 March 2014; Revised 19 May 2014; Accepted 19 May 2014; Published 9 June 2014
Academic Editor: Claudio Cuevas
Copyright © 2014 Xu Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the time-oscillating Hartree-type Schrödinger equation , where is a periodic function. For the mean value of , we show that the solution converges to the solution of for their local well-posedness and global well-posedness.
In this paper, we discuss the following Hartree-type Schrödinger equation: where represents the convolution operator, , , and is a periodic function belonging to . 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, [1–3].
Different from the classical Hartree-type Schrödinger equation, the coefficient of nonlinearity of 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: Then, is a time-oscillating equation and is the corresponding deterministic one. In this paper, our purpose is to discuss the relationship of well-posedness of solutions between and .
The Cauchy problem has been settled by Cazenave and Weissler [4, 5] and Miao et al. [6–8]. 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 , there exists a unique solution of (or ) defined on the maximal life interval with , . Moreover, the following properties hold.(1) for any admissible pair ; (2)(blow-up alternative) if (resp., ), then, for , one has and, for , one has .
As mentioned above, we are concerned with the behavior of solution of , when . Precisely, in the maximal life interval of solution of , we attempt to find the relationship of solutions between and as is sufficiently large. Mimicking the approach of Cazenave and Scialom  and Fang and Han  in the case of the Schrödinger equation with the local nonlinear term, we obtain the following theorems for Hartree-type.
Theorem 2. Assume the initial data and define as the solutions of . Let be the solution of with the maximal life interval . Then, we have(1)for any time satisfying , if is sufficiently large, the solution of exists in ;(2)for any admissible pair and time , in as . In particular, the convergence holds in .
Theorem 2 describes the relationship of local well-posedness of solutions between and . Furthermore, if the solution of is globally existent, that is, , we want to know whether Theorem 2 still holds. The following theorem gives the positive answer if the solution of owns sufficient decay as .
Theorem 3. Under the assumptions of Theorem 2, suppose that and and then it follows that solution of is global; that is, . Moreover, solution of is also global if is sufficiently large, and in as , for all admissible pairs .
The assumption (2) makes sure the solution of owning sufficient decay, by which deduces not only is global but also has scattering state (the details can be referred to in [6–8]). In fact, (2) shows that is global when immediately, according to the blow-up alternative in Proposition 1. And for , the norm of can be controlled by (2), for any , which shows 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  shows (2) is true for defocusing case () when . When , Miao et al. in  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 is global but (2) does not hold, we are not sure the behavior of solution of 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 .
2. 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 (or ): where represents the Schrödinger group.
Definition 4 (admissible pair). A pair is called admissible if and , (if , then ; if , then ).
Lemma 5 (classical Strichartz estimates). The following properties hold.(i)For any and any admissible pair , the function belongs to In addition, there exists a constant such that (ii)Let be an interval in , , and . If is an admissible pair and , then for any admissible pair , the function belongs to . Moreover, there exists a constant independent of such that
We also need the following maximal estimate, which follows immediately from the sharp Hardy inequality (see ).
Lemma 6. Let ; one has
The following lemma is the key to discussing the relationship between and , which shows that when goes to infinity, the nonlinearity of converges to the nonlinearity of . The lemma has been proved by Cazenave and Scialom ; therefore, we only state it here without any detailed proof.
Lemma 7. Let be an admissible pair, and fix a time . Given , it follows that in , for any admissible pair .
Lemma 8. Let the initial data . For any , define as the solution of , and is the solution of with the maximal life interval . Fix a time satisfying , and suppose exists in the interval 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
and then Proposition 1 deduces .
It follows from (3) that , where By Lemma 6, Hardy-Littlewood-Sobolev inequality, Hölder inequality, and Sobolev embedding, we obtain where Therefore, we can obtain from Strichartz estimates and Lemma 7 that It follows from Strichartz estimates, Lemma 6, Hardy-Littlewood-Sobolev inequality, Hölder inequality, and Sobolev embedding that Equations (19) and (18) can deduce that The conclusion (12) can be obtained from the above inequality, if we can show
Divide into subintervals , , with , such that in each subinterval, we have where only depends on and .
In the initial interval , since , (19), (18), and (22) deduce that where we let by the special choice of .
Then by the continuity argument, we have Since and both belong to , we choose and obtain
In the interval , Strichartz estimates and inequalities (18), (22), and (25) deduce that Let and apply the continuity argument; we have Furthermore, let again; we have Therefore, by induction argument, we obtain where .
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 (or ), which is useful for the proof of Theorem 3.
Proof. We assume ; then according to Proposition 1, we obtain and for any , . Since , we can choose sufficiently close to such that
where is sufficiently small.
For any admissible pair , Strichartz estimate deduces that Note that It follows from Hardy-Littlewood-Sobolev inequality and Hölder inequality that From (33) and (35), we obtain By Sobolev embedding and the definition of , we have If we choose , we have , which is uniformly bounded for any . Then let converge to ; we have , which is a contradiction. Now, we know . Then, by (33) and (35), we know . The similar way can show ; thus we finish the proof.
3. The Proof of Theorem 2
Proof. For any , let and ; furthermore, we have , where the norm is defined as (11). Divide the interval into subintervals , , and , , such that in each subinterval , we have
where only depends on and , and is a sufficiently small constant which is chosen later.
In each subinterval , the integral forms (3), (19) (let , and ), and (38) apply where we choose sufficiently small such that and
On , since , then by Strichartz estimate, (19) (let ), and (39), we obtain Then the continuity argument deduces that if is sufficiently small such that , where the second inequality comes from the Strichartz estimate and the definition of .
Therefore, if is sufficiently large, the solution exists on and (10) holds. By Lemma 8, we have .
On , Strichartz estimate and (19) deduce Applying the continuity argument again, we have
Therefore, when is sufficiently large, the solution still exists in , and (10) holds. Furthermore, by Lemma 8, we have .
By induction, on each subinterval , we always have since the number of subintervals is finite, which only depends on and . So, if is sufficiently large, the solution exists in , and the condition (10) holds. Therefore, Theorem 2 follows from Lemma 8; thus we complete the proof.
4. The Proof of Theorem 3
Proof. The global existence of solution of is followed from Lemma 9. For any , Theorem 2 shows that the solution of exists in for sufficiently large and in as with any admissible pair . In particular,
Choose sufficiently large such that (32) holds, where satisfies the smallness in the proof of Lemma 9. According to the proof of Lemma 9, we have For any , triangle inequality deduces that Let go to on both sides; then we obtain from (45) (let ) that The arbitrary value of shows that when is sufficiently large, the solution is global existence by blow-up alternative in Proposition 1. Then Lemma 9 can deduce that
Finally, we show that in as for all admissible pairs . Theorem 2 shows that for any therefore, our attention is focused on
We note that Strichartz estimates and Theorem 2 show that By Lemma 9, we know ; then Lemma 7 deduces that Since then it follows from Strichartz estimates, Hardy-Littlewood-Sobolev inequality, and Hölder inequality that
By Sobolev embedding and interpolation, for any time interval , we have
We set . By (52)–(57), we have where we suppose .
Choose sufficiently large such that Since and both are admissible pairs, then it follows from (58) with that The standard continuity argument shows that if is sufficiently small, that is, large enough, we have Finally, (53), (54), (58), and (61) deduce that
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author thanks the anonymous referees for constructive remarks.
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