Research Article | Open Access
Self-Similar Solutions for Nonlinear Schrödinger Equations
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.
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  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 ([4–6]) 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
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  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 ([8–11]). Cazenave and Weissler [12, 13] (also Ribaud and Youssfi ) 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
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 ). 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 .
4. The Proof of Main Results
To prove the main results, we need the following lemmas.
Lemma 4.1 (see ). Let and with . Suppose that If , then there exists a constant such that for all .
Lemma 4.2. Let , where for ; for , then
Lemma 4.3 (see ). Suppose that ; , then one has for .
Lemma 4.4 (see ). Let and , then where.
4.1. The Proof of Theorem 3.3
We first prove (i). Defining the following map by (3.1),
Let and choose
for all .
This implies that is a contraction map from into . Thus, there exists a unique solution of (1.1) with .
for , where . As a consequence, we get by Lemma 4.3 that
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
In the case of , the proof above can see that of (iii) below.
For a proof of (ii) see .
We now prove (iii). Note that and under the assumption in (iii).
Because , So we derive from (4.9) and the Sobolev embedding theorem 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
The continuity with respect to of is obvious; so .
The proof of Theorem 3.5 is thus completed.
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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