Abstract and Applied Analysis

Volume 2014 (2014), Article ID 181254, 14 pages

http://dx.doi.org/10.1155/2014/181254

Research Article

The Property of the Solution about Cauchy Problem for Fourth-Order Schrödinger Equation with Critical Time-Oscillating Nonlinearity

Cuihua Guo,¹ Hongping Ren,² and Shulin Sun³

¹School of Mathematical Science, Shanxi University, Taiyuan, Shanxi 030006, China

²School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China

³School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 15 April 2014; Accepted 9 June 2014; Published 21 July 2014

Academic Editor: Yushun Wang

Copyright © 2014 Cuihua Guo et al. 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 property of the solution in Sobolev spaces for the Cauchy problem of the following fourth-order Schrödinger equation with critical time-oscillating nonlinearity , where , , and is a periodic function. We obtain the asymptotic property of the solution for the above equation as under some conditions.

1. Introduction

In this paper we study the Cauchy problem of the following fourth-order Schrödinger equation with a time oscillating critical nonlinearity

where is a -periodic function and is a real constant. We define the solution of (1) to be .

In this paper we plan to study the behavior of as . We define to be the average value of on . Naturally, we think of the following equation:

We define the solution of (2) to be . We will study the relation of and as .

Definition 1. For two integers and , we say that is an admissible pair if the following condition is satisfied:

To begin with, we give the following local well-posed results by similar techniques in [1].

Proposition 2. Let . Then there exists a unique -solution on a maximal time interval for (1). Moreover, for any admissible pair and , . If (or), then or.

Energy-critical fourth-order Schrödinger equations are very important equations which arise in many physical application fields [2, 3]. They have been discussed not only by physical researchers but also from mathematical viewpoint by many authors [4–8]. In particular, if the coefficient of nonlinear term is a periodic function in time, condensation and blowup phenomenon may appear from physical experiments. Naturally, from mathematical point, we will analyze why does it happen? What is the condition when condensation or blowup phenomenon appears? What is the property if condensation appears? For the Schrödinger equation, in [9] Fang and Han studied the Schrödinger equation with time-oscillating critical nonlinearities. They gave the asymptotic property of local solution and global solution for large frequencies (as ). As far as we know, there are few results about the asymptotic property of the solutions for the energy-critical fourth-order Schrödinger equations with periodic coefficient in time. So we will utilize the ideals and techniques of [9] to study the asymptotic behavior of the solution for (1) as . The difficulty of our work is how to seek power index. The index is dependent to space dimension. And the order of equation is four. This will make it more difficult to look for proper index for us. The aim of this paper is to study the property of the solution for the Cauchy problem of fourth-order Schrödinger equation with time-oscillating critical nonlinearities.

The main results of this work are the following theorems.

Theorem 3. Suppose that . For arbitrary initial data , we define and , respectively, as the maximal solution of (1) and (2) on maximal time . Then we have

where and is arbitrary admissible pair.

Theorem 4. Suppose that . Let ,. Assume that is the global solution of (2), and . Then the solution of (1) is global if is sufficiently large. Moreover, we have

for any admissible pair .

The succeeding section is devoted to establishing the dispersive estimates for the linear equation related to (1) and (2), and we will present the nonlinear estimates of nonlinearity. In Section 3 we present the proofs of Theorems 3 and 4.

2. Notations and Preliminaries

Given and a function space on , we denote by and , respectively, the following norm and the corresponding function space on .

For ,

and for ,

Later we will particularly take . For simplicity of the notations, we, respectively, abbreviate and as, respectively, and . In particular, for the case , we abbreviate as . We also abbreviate . In the following, we will introduce our four working spaces.

For any time interval , we denote

The fundamental solution of the linear equation related to (1) and (2) is given by the following oscillatory integral:

We denote by the fundamental solution operator

So (1) and (2) have the following integral forms, respectively:

Lemma 5 ((Strichartz estimates) (see [10])). Assume , , and is a solution on of the following initial value problem:

then for all admissible pairs and , we have

Lemma 6. Let be a compact time interval containing . Then we have

Proof. By the definition of , we obtain

Using Hardy-Littlewood-Sobolev inequality, we have

which completes the proof.

Lemma 7. For any compact time interval , we have

Proof. Using Sobolev embedding (see [11]) , we have

Similarly, using Sobolev embedding and noting that is an admissible pair, we have

which completes the proof of the first inequality.

Next we prove the second inequality.

Using interpolation inequality (see [12]), we obtain

Using Sobolev embedding, we obtain

So we have

which completes the proof of the second inequality.

Assume that the nonlinear function . Then we have the following two lemmas.

Lemma 8. Let be a compact time interval. Then, we have

Proof. Using Lemma of [13] and Lemma 7, we have

For the second inequality, we have

For , using Lemma of [13] and Lemma 7, we obtain

For , using Lemma of [13] (, , ) and Sobolev inequality, we have

From (28), using interpolation inequality and Lemma 7, we have

From (26), (27), and (29), we have

Similarly, we can prove

which completes the proof.

Using Lemma 8, we immediately obtain the following lemma.

Lemma 9. Let . Then we have

Using the proof techniques in [14], similarly we can obtain the following lemma.

Lemma 10. For any admissible pair , , , we have

Lemma 11. Let ,. For any initial value , , assume that

then we have

where is admissible pair.

Proof. First we prove that .

By (34), there must be constants and such that

From (11) and (12), we have

where and .

By Lemma 10, we have

Using Hölder inequality and Sobolev embedding inequality, we obtain

where .

Using (38) and (39), we have

In the following we will prove that .

Since , we can divide the time interval into subintervals ,, where ,, such that in each part .

On , since , we have

For the case , we have

For the case , we have

On , we have

Similarly for the case , we have

For the case , we have

By induction, we have

for .

So we have

Using the above estimate and (40), we have

In the following we discuss the estimate .

By (11) and (12), we have

where and .

Using (11)-(12) and Lemma 5, we obtain

Noting that

so .

Using Lemma 5 and Lemma 10, we have

Using Hölder inequality, Sobolev embedding, and Lemma 7, we obtain

Substituting (53)-(54) into (51), we obtain

Taking such that , , , using Young’s inequality and Lemma 7, we obtain

Evidently, in order to get the estimate , we have to estimate the norm .

By (11)-(12), Lemmas 6 and 9, we obtain

Let . Noting that

so .

Using Lemma 10, we obtain

From (57)–(59), we get

Let such that , thus we have

where .

If and are small such that

we obtain

Taking (63) into (56), we can get

If , and are small such that

then we can obtain

So we have

Let . By continuous extension method and contradiction method, we can prove that .

At last, we discuss the estimate .

As in Lemmas 8 and 9, we define .

By (11) and (12), we have

where , , , and , , and are as follows:

are all matrixes.

Using (11)-(12) and Lemma 5, we obtain

For the case , using Hölder inequality and Sobolev embedding, we have

Since , we have ; thus by Lemma 10,

In the following, we analyze the norm for .

Using Hölder inequality and Sobolev embedding, we obtain

where .

Similarly, we can get

Combing (74) and (75), we have

At last, we analyze the norm . We divide it into two cases.

Case I (). Using Hölder inequality, Sobolev embedding, and (73), we can get

Similarly, we can get

Combing (77) and (78), we can obtain

From (70), (73), (76), and (79), we have

Taking , we can get

Furthermore, if and are small enough such that

then we have

Case II (). Noting that

where , , , , and ;

thus we have

Combing (78) and (85), we can get

From (70), (73), (76), and (86), we have

Taking , , and , we can get

Combing (63) and (88), we can obtain

Taking , , small such that   + , then we can get

From (83) and (90), we have

Similarly, let . By continuous extension method and contradiction method, we can prove that .

From (53), (67) and (91), the desired result holds.

Lemma 12. Suppose that . is as in Lemma 11. If for any given , for the case , there exists such that , then we have

where is arbitrary admissible pair and and are dependent on .

Proof. Using (11), Strichartz estimates, and Hölder inequality, noting that (to be sure that ), we can get

taking , we can get

Similarly, we can get

which completes the proof.

3. Proofs of Theorems

Proof of Theorem 3. For any given , suppose that ; obviously, we have .

For any given (which will be decided later), we divide into , such that

On , we can get from (12)

Let , we have

On , we can get from (12)

Noting that , we have

Similarly, on , we can get from (12)

Again on , we can get from (11)

So we have

Let ; we have

So we know that is uniformly bounded. By Lemma 11, we obtain .

On , we have

Noting that , we have

Similarly, we have

So we have

thus by Lemma 11 and Strichartz estimates, Theorem 3 can be obtained.

Proof of Theorem 4. For any , we can obtain from Theorem 3

For any , if is sufficiently large, we immediately get

So using Strichartz estimates and the above inequality, we have

If is sufficiently large, we can get

Using Lemma 12, we have

for sufficiently large .

Similarly, we can get

Using Proposition 2 we know is global.

From Theorem 3, we have

For the case , we have

changing the variable to , we have

Using Strichartz estimates and Hölder inequality, we can get

So we have

Let ; we can get

which completes the proof of Theorem 4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by Natural Science of Shanxi province (no. 2013011003-2, no. 2013011002-2, and no. 2010011001-1), Natural Science Foundation of China (no. 11071149), and Research Project Supported by Shanxi Scholarship Council of China (no. 2011-011).

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