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Mathematical Problems in Engineering
Volume 2009, Article ID 298980, 15 pages
http://dx.doi.org/10.1155/2009/298980
Research Article

Self-Similar Solutions for Nonlinear Schrödinger Equations

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

Received 19 March 2009; Accepted 22 August 2009

Academic Editor: Ben T. Nohara

Copyright © 2009 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

We study the self-similar solutions for nonlinear Schrödinger type equations of higher order with nonlinear term by a scaling technique and the contractive mapping method. For some admissible value , we establish the global well-posedness of the Cauchy problem for nonlinear Schrödinger equations of higher order in some nonstandard function spaces which contain many homogeneous functions. we do this by establishing some nonlinear estimates in the Lorentz spaces or Besov spaces. These new global solutions to nonlinear Schrödinger equations with small data admit a class of self-similar solutions.

1. Introduction

This paper is concerned with the following Cauchy problem for the nonlinear Schrödinger type equation:

where is a constant, is an integer, is a complex-valued function defined on , and the initial data is a complex-valued function defined in . Pecher and Wahl [1] have established the existence of the classical solution to the Cauchy problem for the higher-order Schrödinger equation (1.1) by making use of -estimates of the associated elliptic equation in conjunction with the compactness method. Recently Sjögren and Sjölin studied the local smoothing effect of the solutions to the Cauchy problem (1.1) by means of the Strichartz estimates in nonhomogeneous spaces ([2, 3]). Moreover, there are some work ([46]) which is devoted to the investigation of the global well-posedness and the scattering theory of the problem (1.1). However, little attention is paid to the self-similar solutions of the Cauchy problem (1.1).

Our goal is to prove the existence of the global self-similar solutions to the Cauchy problem (1.1) for some admissible parameter . From the scaling principle, it is easy to see that if is a solution of the Cauchy problem (1.1), then with is also a solution of equation in (1.1) with the initial value . We thus have the following definition.

Definition 1.1. is said to be a self-similar solution to the higher-order Schrödinger equation in (1.1) if

By Definition 1.1, we know that the self-similar solution to (1.1) is of the form

where is called profile of the solution, and the initial value is of the form

where and is defined on the unit sphere of . Therefore the problem (1.1) can be studied through a nonlinear higher-order elliptic equation on . However, this is usually very complicated. By virtue of this method, Kavian and Weissler [7] have dealt with the radially symmetric solutions of (1.1) in the case .

Another important way of looking for self-similar solutions for the nonlinear Schrödinger equation in (1.1) is to study the small global well-posedness of associated Cauchy problem (1.1) in some suitable function spaces. These global solutions admit a class of self-similar solutions. As a consequence, if is the unique solution of the Cauchy problem (1.1) with the initial data given by (1.4), then is a self-similar solution of the problem.

On the other hand, if is a self-similar solution to the problem (1.1), then the initial function is . So is homogeneous of degree . In general, such homogeneous functions do not belong to the usual Lebesgue spaces and Sobolev spaces.

To do our work, several definitions and notations are required. Denote by and the Schwartz space and the space of Schwartz distribution functions, respectively. denotes the usual Lebesgue space on with the norm for . For and , let , the inhomogeneous Sobolev space in terms of Bessel potentials; let , the homogeneous Sobolev space in terms of Riesz potentials, and write and . We will omit from spaces and norms. For any interval (or ) and for any Banach space , we denote by the space of strongly continuous functions from to and by the space of strongly measurable functions from to with . Finally, let stands for the dual to , that is, ; denotes the largest integer less or equal to .

When , the equation in (1.1) becomes the classical Schrödinger equation

which describes many physical phenomena, and the well-posedness as well as the scattering theory for the Cauchy problem (1.5) has been extensively studied by many authors ([811]). Cazenave and Weissler [12, 13] (also Ribaud and Youssfi [14]) have studied the self-similar solutions of the equation in (1.5) with initial value as (1.4). Their common ideas are to introduce the new function space which consists of all Bochner measurable functions such that , where and . They then established the existence of global self-similar solutions in for the problem (1.5) under the condition that .

This paper is organized as follows. In the next section, we will recall the definition and basic properties of function spaces that we require. Then in Section 3 we state the main results and the related propositions. The last section is devoted to the proof of main results.

2. Function Spaces

2.1. Lorentz Spaces

Definition 2.1. Let , be the nonincreasing rearrangement of a measurable function, then is said to be inif and only if when , and when , where is the quasinorm of space .

We refer the reader to [15, 16] for the definitions and detailed properties of the nonincreasing rearrangement functions and Lorentz spaces. In fact, Lorentz space is a generalization of Lebesgue space . We have as , and as . Meanwhile, a lot of properties of Lebesgue spaces are still valid in Lorentz spaces.

We may prove the following results according to Definition 2.1.

Proposition 2.2. Suppose that, then

The inequalities (2.3) and (2.4) are essentially the Hölder and Minkowski inequality in Lorentz spaces, respectively, and they can be proved by using Definition 2.1. Furthermore, noting that is a real interpolation of Lebesgue space, we immediately obtain the following proposition.

Proposition 2.3. Let and , then

2.2. Besov Spaces

We first recall briefly the definition of Besov spaces. For detailed properties and embedding theorems, we are referred to [15, 17].

Let satisfy as and as ,

then we have the Littlewood-Paley decomposition

For convenience, we introduce the following notions:

where and stand for Fourier and inverse Fourier transforms, respectively.

Definition 2.4. Assume that, then is called Besov space, and is homogeneous Besov space.

In particular, we have

Besides the classical Besov spaces, we also need the so-called generalized Besov spaces.

Definition 2.5. Letbe a Banach space, then, forand, definesas where is the Littlewood-Paley operator on defined as above.

Remark 2.6. If is the Lorentz space , then This space is useful in the study of self-similar solutions.

Remark 2.7. Let with or being an interval, then we have where .

Remark 2.8. In addition to the Besov spaces norm in Definition 2.4, we usually use the following equivalent norms for the Besov spaces and : where ; and with a nonnegative integer and . When is not an integer, (2.16) is also equivalent to the following norm: where . In the case when , the above norm should be modified as follows:

3. Main Results

To solve our problems, we may rewrite (1.1) in the equivalent integral equation of the form

where is the free group generated by the free equation of Schrödinger type .

Definition 3.1. One calls a classical admissible pair with respect to the -order Schrödinger operator if where for ; for .

To prove Theorem 3.3 we need the following generalized Strichartz estimates which follow directly from the stationary phase method, the Strichartz estimates, and interpolation theorems (see [5, 15, 18] for details).

Proposition 3.2. Let , and satisfy (3.2); then Moreover, if , then where .

Our main results state as follows.

Theorem 3.3. (i) Let for; for . There exists an such that if with , then the Cauchy problem (1.1) (or (3.1)) has a unique global solution with
(ii), and . There exists an such that if with , then (1.1) has a unique global solution

(iii), and let the condition (a) for ; or (b) for , be satisfied, where and are two positive roots of equation and . There exists an such that if with , then the problem (1.1) has a unique global solution:

Corollary 3.4 (see [19]). Let , where is a positive constant, satisfies the assumptions in Theorem 3.3; then there exists a unique global self-similar solution for the Cauchy problem (1.1) with the initial value .

Theorem 3.5. Let satisfy the conditions of Theorem 3.3; then the global solution obtained in Theorem 3.3 satisfies .

4. The Proof of Main Results

To prove the main results, we need the following lemmas.

Lemma 4.1 (see [20]). Let and with . Suppose that If , then there exists a constant such that for all .

Lemma 4.2. Let , where for ; for , then

Proof. By (2.4) in Proposition 2.2, we have We get from (3.3) in Proposition 3.2 Therefore, we obtain from Proposition 2.3 and (2.5)

Lemma 4.3 (see [21]). Suppose that ; , then one has for .

Lemma 4.4 (see [22]). Let and , then where.

4.1. The Proof of Theorem 3.3

We first prove (i). Defining the following map by (3.1),

For , we have from Lemma 4.2 and (3.7) in Proposition 3.2,

Let and choose

then we get by (4.11) and (4.12)

for all .

This implies that is a contraction map from into . Thus, there exists a unique solution of (1.1) with .

Let , where . Then we derive from (3.4) and (3.6)

for , where . As a consequence, we get by Lemma 4.3 that

It follows that from (4.13), . So, (4.16) implies that

Taking the norm in both sides of (3.1), we obtain from the definition of generalized Besov spaces, Lemma 4.3 and (3.4) and (3.5)

which implies . Therefore, in the case of , we have

For , let , and , then we obtain from the assumption in (i) . In the case of , according to the equivalent norm of Besov spaces and Hölder inequality it follows that

Using the Sobolev embedding theorem , we get that

Consequently, from Remark 2.7, (3.4), (3.5), and (4.21), it follows that

Similarly

By using (3.4), (3.5), and (4.21), and arguing similarly as in deriving (4.18) one obtain that

From (4.11) and (4.12), it follows that

Thus, by (4.22)–(4.27) we have

Letting , and choosing , then (4.28) and (4.29) imply that is a contraction map from into . By the Banach contraction mapping principle we conclude that there is a unique solution such that

In the case of , the proof above can see that of (iii) below.

For a proof of (ii) see [18].

We now prove (iii). Note that and under the assumption in (iii).

Let , then by using (4.8) in Lemma 4.4 and arguing similarly as in deriving (4.24) we have

On the other hand, since , where , it follows from Proposition 3.2, Lemma 4.1, and (4.9) in Lemma 4.4 that

where .

Because , So we derive from (4.9) and the Sobolev embedding theorem that

By arguing similarly as in deriving (4.24) and (4.25) we get

it follows from (4.31)–(4.35) that

Let with and choose , then (4.36) and (4.37) imply that is a contraction map from into . By the Banach contraction mapping principle we obtain that there is a unique solution such that

This complete the proof of Theorem 3.3.

4.2. The Proof of Theorem 3.5

Without loss of generality we only consider the case . From Theorem 3.3 it follows that the Cauchy problem (1.1) has a unique solution provided that is suitably small. If, in addition, , then we have by letting that

From Theorem 3.3 it follows that the problem (1.1)–(1.4) has a unique solution such that provided that with enough small . Then we have that from (4.39)

The continuity with respect to of is obvious; so .

The proof of Theorem 3.5 is thus completed.

Acknowledgment

This research was supported by the Natural Science Foundation of Henan Province Education Commission (no. 200711013), The Research Foundation of Zhejiang University of Science and Technology (no. 200806), and the Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2010). The Science and Research Project of Zhejiang Province Education Commission (no. Y200803804).

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