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Abstract and Applied Analysis
Volume 2008, Article ID 836124, 9 pages
http://dx.doi.org/10.1155/2008/836124
Research Article

Global Self-similar Solutions of a Class of Nonlinear Schrödinger Equations

Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received 11 November 2007; Accepted 4 March 2008

Academic Editor: Thomas Bartsch

Copyright © 2008 Yaojun YE. 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.

Abstract

For a certain range of the value in the nonlinear term , in this paper we mainly study the global existence and uniqueness of global self-similar solutions to the Cauchy problem for some nonlinear Schrödinger equations using the method of harmonic analysis.

1. Introduction

This paper is devoted to study the initial value problem for the nonlinear Schrödinger equation where are constants, is a positive integer, is a complex-valued function defined in , the initial value is a complex-valued function defined in .

When , (1.1) is a classical nonlinear Schrödinger equation of the second order:

For the Cauchy problem of (1.2), the existence and the scattering theorem of solutions have been studied extensively by many authors with various methods and techniques [15], Cazenave and Weissler [6] (also Ribaud and Youssfi [7]) established existence of global self-similar solutions by introducing new function space. When , Pecher and von Wahl [8] established the existence of classical solution of the Cauchy problem (1.1) employing the related estimate of the elliptic equation and the compact method. Sjölin and Sjögren in [9, 10] recently discussed the local smooth effect of solutions of the Cauchy problem (1.1) applying the Strichartz estimate in the nonhomogeneous Sobolev space. In [11], by constructing a time-weighted space and using the contractive mapping method, the author established global solutions of the problem (1.1) in the possible range of , and further got the continuous dependence of the solution on the initial value together with its strong decay estimate. In addition, there are also much more efforts working for studying the scattering theorem and the existence of global strong solutions of the problem (1.1) [12, 13]. In this paper, we mainly investigate the existence of global self-similar solutions basing on the existence and uniqueness of global solution for the Cauchy problem (1.1).

In the following discussion, we suppose that satisfieswhere is a positive solution of the equation , which also can be interpreted as a positive integer satisfying . In fact, condition (1.3) is equivalent toFor which satisfies (1.3) or (1.4), letthen we may introduce our work space as follows. Let be a space consisting of all Bochner measurable functions:such that

In order to prove our main result, we should transform the Cauchy problem (1.1) into the following equivalent integral equation:where is a free group produced by the free Schrödinger equation . Besides, we denote, respectively, by and the Fourier transformation and the inverse Fourier transformation with respect to the space variables.

For convenience, we provide some useful symbols. denotes the usual Lebesgue space on with the norm , . For any , stands for the dual to , that is, . which may be different when appeared every time is a constant depending on the dimension or any other constant.

In the end, we will review the definition of the homogeneous Besov space, the details on the properties, and the embedding theorems reference [1, 14].

Let be a symmetric Bump function with real values satisfying the conditions , , , , thenare also symmetric Bump functions. Denote by and the convolution operator of and , respectively, that is, If , , , thenis called a homogeneous Besov space and

2. Lemmas and Main Results

The linear Schrödinger group satisfies the following estimate [14, 15]:We first provide two lemmas that may be useful in in the following.

Lemma 2.1. Let , , then .

Proof. According to the property of the Fourier transformation and , we getLet , thenThusSinceIt is easy to see that from (2.4) and (2.5),namely,

Lemma 2.2. Let , , , , thenThe detailed proof can be referred to [16].

In order to prove the main results, we need the following known theorems [11].

Theorem 2.3 (existence of global solutions). Suppose that satisfies (1.3) or (1.4), , if there is , such thatthen the Cauchy problem (1.1) has a unique solution which satisfies .

Theorem 2.4 (the continuous dependence of the solution on the initial value). Suppose that and both satisfy the condition (2.9), are two solutions of the Cauchy problem (1.1) corresponding to the initial value and , then In addition, if then where , .

In this paper, our object is to study the global self-similar solutions of the Cauchy problem (1.1). At first, we introduce the definition of the self-similar solution.

Definition 2.5. Suppose that is a solution of the Cauchy problem (1.1), if then is called the self-similar solution of the problem (1.1).
One easily knows from the above definition that is a solution of the problem (1.1) which satisfies the initial value , provide that is just a solution of the Cauchy problem (1.1).
Now, we give our main result.

Theorem 2.6. Let satisfy (1.3) or (1.4), , , and , thenIn particular, if existing such that , then there exists a unique self-similar solution of (1.1) with the initial value (2.14).

3. The Proof of Main Result

To prove Theorem 2.6, we should provide the following two propositions.

Proposition 3.1. Let then

Proof. By Lemma 2.1, we only illustrate that the following inequality is valid:It follows that from the embedding , it is necessary to prove
Denote , and then can be decomposed as follows:where is referred in the introduction.
Making use of the estimate (2.1) and noting that , then we havewhere .
For , we haveSince , then . Thus, it follows that from the Young inequalityBesides, as , so thatTherefore,By (3.6) together with the Young inequality, we obtainWe know that from the left side of the inequality (1.4),It yields from (3.11) thatOn the other hand, for , thusIt follows that by the Young inequality,We get that from (3.15) and ,
Correspondingly,The right side of (1.4) shows that , consequentlyFrom and the Bernstein inequality, we getCombining (3.13) with (3.19), we have The proof of Proposition 3.1 is finished.

Proposition 3.2. Let , , then

Proof. Since , then . Accordingly, we obtain by Lemma 2.2 that which implies thatThe proof is concluded.

Now, we are ready to prove Theorem 2.6.

Proof. For we have from Proposition 3.1However, noting that as well as Proposition 3.2, we getThen, it follows from (3.25) and (3.26) thatChoosing , then we have for any . From Theorem 2.3, we conclude that there is a unique global solution of the equation in (1.1) with the initial value (2.14). Besides,which gives that by uniquenessThus, is just a self-similar solution of the problem (1.1).
This completes the proof of Theorem 2.6.

Acknowledgment

This work is supported by Natural Science Foundation of Henan Province Education Commission (no. 2007110013) and the Program for Outstanding Young Teacher in Henan Province (2004–2006).

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