Journal of Function Spaces

Journal of Function Spaces / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 968603 | https://doi.org/10.1155/2013/968603

Shuxia Wang, "Energy Scattering for Schrödinger Equation with Exponential Nonlinearity in Two Dimensions", Journal of Function Spaces, vol. 2013, Article ID 968603, 13 pages, 2013. https://doi.org/10.1155/2013/968603

Energy Scattering for Schrödinger Equation with Exponential Nonlinearity in Two Dimensions

Academic Editor: Baoxiang Wang
Received09 Jan 2013
Accepted24 Feb 2013
Published09 Apr 2013

Abstract

When the spatial dimensions , the initial data , and the Hamiltonian , we prove that the scattering operator is well defined in the whole energy space for nonlinear Schrödinger equation with exponential nonlinearity , where .

1. Introduction

We consider the Cauchy problem for the following nonlinear Schrödinger equation: in two spatial dimensions with initial data and . Solutions of the above problem satisfy the conservation of mass and Hamiltonian: where

Nakamura and Ozawa [1] showed the existence and uniqueness of the scattering operator of (1) with (2). Then, Wang [2] proved the smoothness of this scattering operator. However, both of these results are based on the assumption of small initial data . In this paper, we remove this assumption and show that for arbitrary initial data and , the scattering operator is always well defined.

Wang et al. [3] proved the energy scattering theory of (1) with , where and the spatial dimension . Ibrahim et al. [4] showed the existence and asymptotic completeness of the wave operators for (1) with when the spatial dimensions , , and . Under the same assumptions as [4], Colliander et al. [5] proved the global well-posedness of (1) with (2).

Theorem 1. Assume that , , and . Then problem (1) with (2) has a unique global solution in the class .

Remark 2. In fact, by the proof in [5], the global well-posedness of (1) with (2) is also true for .

In this paper, we further study the scattering of this problem. Note that . Nakanishi [6] proved the existence of the scattering operators in the whole energy space for (1) with when . Then, Killip et al. [7] and Dodson [8] proved the existence of the scattering operators in for (1) with . Inspired by these two works, we use the concentration compactness method, which was introduced by Kenig and Merle in [9], to prove the existence of the scattering operators for (1) with (2).

For convenience, we write (1) and (2) together; that is, where and . Our main result is as follows.

Theorem 3. Assume that the initial data , , and . Let be a global solution of (5). Then

In Section 2, Lemma 9 will show us that Theorem 3 implies the following scattering result.

Theorem 4. Assume that the initial data , , and . Then the solution of (5) is scattering in the energy space .

We will prove Theorem 3 by contradiction in Section 5. In Section 2, we give some nonlinear estimates. In Section 3, we prove the stability of solutions. In Section 4, we give a new profile decomposition for sequence which will be used to prove concentration compactness.

Now, we introduce some notations:

We define

For Banach space , , or , we denote

When , is abbreviated to  . When or is infinity or when the domain is replaced by , we make the usual modifications. Specially, we denote

For , we split , where

For any two Banach spaces and , . denotes positive constant. If depends upon some parameters, such as , we will indicate this with .

Remark 5. Note that in Theorem 3; we only need to prove the result for , . Hence, we always suppose that in the context.
Moreover, we always suppose that the initial data of (5) satisfies and .

2. Nonlinear Estimates

In order to estimate (2), we need the following Trudinger-type inequality.

Lemma 6 (see [10]). Let . Then for all satisfying , one has

Note that for for all ,

By Lemma 6 and Hölder inequality, for and for all , we have and thus

Lemma 7 (Strichartz estimates). For or , (the pairs were called admissible pairs) we have

Lemma 8 (see [3, Proposition 2.3]). Let be fixed indices. Then for any ,

As shown in [6, 11], to obtain the scattering result, it suffices to show that any finite energy solution has a finite global space-time norm. In fact, if Theorem 3 is true, we have the following theorem.

Lemma 9 (Theorem 3 implies Theorem 4). Let be a global solution of (5), , and . Then, for all admissible pairs, we have
Moreover, there exist such that

Proof. Defining , , by Strichartz estimates, (14) and (15),
Using the same way as in Bourgain [12], one can split into finitely many pairwise disjoint intervals:
By (21),
Since and can be chosen small arbitrarily, by interpolation, for all admissible pairs and . The desired result (19) follows.
Let
By (19) and (21),
Thus, were well defined and belong to . Since we must have (20) was proved.

3. Stability

Lemma 10 (stability). For any and , there exists with the following property: suppose that satisfies for all , and approximately solves (5) in the sense that
Then for any initial data satisfying and , there is a unique global solution to (5) satisfying .

Proof. Denote , then and . Let . By the similar estimates as (21), we have
Then we subdivide the time interval into finite subintervals , , such that for each . Let be small such that
Then by (31) on , we have and
Using the same analysis as above, we can get . Iterating this for , we obtain ; the desired result was obtained.

4. Linear Profile Decomposition

In this section, we will give the linear profile decomposition for Schrödinger equation in . First, we give some definitions and lemmas.

Definition 11 (symmetry group, [13]). For any phase , position , frequency , and scaling parameter , we define the unitary transformation by the formula
We let be the collection of such transformations; this is a group with identity , inverse , and group law
If is a function, we define , where by the formula or equivalently

If , we can easily prove that and .

Definition 12 (enlarged group, [13]). For any phase , position , frequency , scaling parameter , and time , we define the unitary transformation by the formula or in other words
Let be the collection of such transformations. We also let act on global space-time function by defining or equivalently

Lemma 13 (linear profiles for sequence, [14]). Let be a bounded sequence in . Then (after passing to a subsequence if necessary) there exists a family , of functions in and group elements for such that one has the decomposition for all ; here, is such that its linear evolution has asymptotically vanishing scattering size:
Moreover, for any ,
Furthermore, for any , one has the mass decoupling property For any , we have

Remark 14. If the orthogonal condition (45) holds, then (see [14])
Moreover, if , then (see [14, 15]), for any , If , then (see [16, Lemma 5.5])

Remark 15. As each linear profile in Lemma 13 is constructed in the sense that weakly in (see [14]), after passing to a subsequence in , rearrangement, translation, and refining accordingly, we may assume that the parameters satisfy the following properties:(i) as , or for all ;(ii) or as , or for all ;(iii) as , or with ;(iv)when , and , we can let .

Our main result in this section is the following lemma.

Lemma 16 (linear profiles for sequence). Let be a bounded sequence in . Then up to a subsequence, for any , there exists a sequence in and a sequence of group elements such that
Here, for each , and must satisfy is such that
Moreover, for any , one has the same orthogonal conditions as (45). For any , one has the following decoupling properties:

Proof. Let
Then, we have
By Lemma 13, after passing to a subsequence if necessary, we can obtain with the stated properties (i)–(iv) in Remark 15 and (43)–(47). Denote
Step 1. We prove that with and for each fixed , where
By (44) and , (64) holds obviously. For (62), we prove it by induction. For every , suppose that
Case 1. If , we have .
In fact, by (66),
Thus,
Using (47),
By direct calculation,
Let . When , When , When , When ,
By (68)–(74), and thus .
Case 2. If , we can prove
By absorbing the error into , we can suppose . Since for each fixed , we must have .
Now, we begin to prove (75). Let be the characteristic function of the set and , and then where
Note that We have
When , we have . Choosing , then by (79), , the desired result follows.
When and , we have
When and , we denote and . The line (when , we use the line instead) separates the frequency space into two half-planes. We let to be the half-plane which contains the point , and then
By (79), we have . Note that (75) holds.
When and , let be the half-plane which does NOT contain the point ; we can prove (75) similarly as above.
By the proof above, we get and . Denote and suppose