About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 238410, 14 pages
http://dx.doi.org/10.1155/2013/238410
Research Article

Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

1School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
2Department of Basic Curriculum, The Chinese People’s Armed Police Force Academy, Lang Fang, He Bei 065000, China
3Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China

Received 26 August 2013; Accepted 13 October 2013

Academic Editor: Sining Zheng

Copyright © 2013 Xiaowei An 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 consider the following Cauchy problem: where and are real-valued potentials and and is even, is measurable in and continuous in , and is a complex-valued function of . We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

1. Introduction

In this paper, we consider the following Cauchy problem: where and are real-valued potentials, and is even, is measurable in and continuous in , is a complex-valued function of , and is the Hilbert space: with the inner product and the norm Model (1) appears in the theory of Bose-Einstein condensation, nonlinear optics and theory of water waves (see [1, 2]).

For convenience, denote when and when . We also give some assumptions on , , and as follows.(V1) and for , .(V2), , and is bounded for all . Here is the complementary set of .(f1) is measurable in and continuous in with .Assume that, for every , there exists such that for all . Here (W1) is even and for some , .

First, we consider the local well-posedness of (1). We have a proposition as follows.

Proposition 1 (local existence result). Assume that and are true, satisfies or , and . Then there exists a unique solution of (1) on a maximal time interval such that and either or else

Definition 2. If with , we say that the solution of (1) exists globally. If with and , we say that the solution of (1) blows up in finite time.

This paper is directly motivated by [1, 35]. Since Cazevave established some results on blowup and global existence of the solutions to (1) with (V1), (f1), and (W1) in [1], we are interested in the problems such as “What are the results about the blowup and global existence of the solutions to (1) with (V2), (f1), and (W1)?” On the other hand, since Gan et al. had established some sharp thresholds for global existence and blowup of the solution to the related problems to (1) (see [35] and the references therein), it is a natural way to consider the sharp threshold for global existence and blowup of the solution to (1).

About the topic of global existence and blowup in finite time, there are many results on the special cases of (1). We will recall some results on the following Cauchy problem: In [6], Glassey established some blowup results for (8). In [7], Berestyki and Cazenave established the sharp threshold for blowup of (8) with supercritical nonlinearity by considering a constrained variational problem. In [8], Weinstein presented a relationship between the sharp criterion for the global solution of (8) and the best constant in the Gagliardo-Nirenberg inequality. In [9], Cazenave and Weisseler established the local existence and uniqueness of the solution to (8) with . Very recently, Tao et al. in [10] studied the Cauchy problem (8) with , where and are real numbers, with , and established the results on local and global well-posedness, asymptotic behavior (scattering), and finite time blowup under some assumptions. Other sharp thresholds were established by Chen et al. in [11, 12]. The following Cauchy problem is also a special case of (1), where with . In [2], Oh obtained the local well-posedness and global existence results of (9) under some conditions. In [3, 5], Gan et al. and Zhang, respectively, established the sharp thresholds for the global existence and blowup of the solutions to (9) under some conditions. In [4], Gan et al. dealt with with a singular integral operator, where with . They got the sharp threshold for global existence and blowup of the solution to (10) and the instability of the wave solutions. Very recently, Miao et al. also obtained some results on the blowup and global existence of the solution to a Hartree equation (see [1315]). Naturally, we want to establish some new sharp thresholds for global existence and blowup of the solution to (1) in this paper and generalize these results above. Although the methods of our paper are inspired by the references above, our results, which will be stated in Section 2, are new and cover theirs.

This paper is organized as follows. In Section 2, we will recall some results of [1] and state our main results; then we will prove Proposition 1 and give some other properties. In Section 3, we will prove Theorems 3 and 4. In Section 4, we establish the sharp threshold for (1) with . In Section 5, we will prove Theorem 7.

2. Our Main Results

Now we will introduce some notations. Denote

mass ( norm)

energy

In [1], Cazenave obtained some sufficient conditions on blowup and global existence of the solution to (1) with (V1), (f1), and (W1). The following two theorems can be looked at as the parallel results to Corollary and Theorem of [1], respectively.

Theorem 3 (global existence). Assume that , and are true, and for some , (and if ). Here . Suppose further that there exist constants and such that with . Then the solution of (1) exists globally. That is,

Theorem 4 (blowup in finite time). Assume that and , , , and are true. Suppose further that If (1) or (2) and , then the solution of (1) will blow up in finite time. That is, there exists such that

Denote

We will establish the first type of sharp threshold as follows.

Theorem 5 (sharp threshold I). Assume that and with . Suppose further that and there exist constants and , such that Let be a positive constant satisfying where is defined by (22). Suppose that satisfies
Then(1)if , the solution of (1) exists globally;(2)if , , and , the solution of (1) blows up in finite time.

Remark 6. Theorem 5 is only suitable for (1) with . To establish the sharp threshold for (1) with , we will construct a type of cross-constrained variational problem and establish some cross-invariant manifolds. First, we introduce some functionals as follows:

Denote the Nehari manifold and cross-manifold Define

In Section 5, we will prove that is always positive. Therefore, it is reasonable to define the following cross-manifold:

We give the second type of sharp threshold as follows.

Theorem 7 (sharp threshold II). Assume that (f1), (W1), and (23). Suppose that and there exists a positive constant such that with the same in (23). Assume further that the function satisfies and for . Here is the value of the partial derivative of with respect to at the point . If and with , then the solution of (1) blows up in finite time if and only if .

Remark 8. (1) implies that for .
(2) The blowup of solution to (1) will benefit from the role of the potential if . In some cases, the blowup of the solution to (1) can be delayed or prevented by the role of potential (see [16] and the references therein).

In the sequel, we use and to denote various finite constants; their exact values may vary from line to line.

First, we will give the proof of Proposition 1.

Proof of Proposition 1. If (V1) is true, then there exist with , , and such that Noticing that , using Hölder’s and Sobolev’s inequalities, we have for any . Consequently, we have By the results of Theorem in [1], we have the local well-posedness result of (1) in .
If (V2), (f1), and (W1) are true, similar to the proof of Theorem 3.5 in [2], we can establish the local well-posedness result of (1) in . Roughly, we only need to replace by in the proof, and we can obtain similar results under the assumptions of (V2), (f1), and (W1).

Noticing that and , following the method of [6] and the discussion in Chapter 3 of [1], one can obtain the conservation of mass and energy. We give the following proposition without proof.

Proposition 9. Assume that is a solution of (1). Then for any .

We will recall some results on blowup and global existence of the solution to (1) with (V1), (f1), and (W1).

Theorem A (Corollary 6.12 of [1]). Assume that , , and (16). Suppose that there exist and such that Then the maximal strong -solution of (1) is global and for every .

Theorem B (Theorem 6.54 of [1]). Assume that , , , and (18)–(20). If , , and , then the -solution of (1) will blow up in finite time.

Let . After some elementary computations, we obtain

We have the following proposition.

Proposition 10. Assume that is a solution of (1) with and . Then the solution to (1) will blow up in finite time if either(1)there exists a constant such that or(2) and .

Proof. Since and , we have
(1) If , integrating it from to , we get . Since , we know that there exists a such that for . On the other hand, we have which implies that there exists a satisfying Using the inequality and noticing that , we have Consequently,
(2) Similar to (46), we can get which implies that the solution will blow up in a finite time .

3. The Sufficient Conditions on Global Existence and Blowup in Finite Time

In this section, we will prove Theorems 3 and 4, which give some sufficient conditions on global existence and blowup of the solution to (1).

We would like to give some examples of , , and . It is easy to verify that they satisfy the conditions of Theorem 3.

Example 11. Consider that , , and with a real constant and .

Example 12. Consider that , , and with a real constant and .

Proof of Theorem 3. Letting , where and with , using Hölder’s and Young’s inequalities, we obtain with . Specifically, we have Using (53) and Gagliardo-Nirenberg’s inequality, we get Using Young’s inequality, from (54), we have for some . Noticing that , using Gagliardo-Nirenberg’s inequality and (55) with , we get Since , from (56), we can obtain Since means that , (57) implies that is always controlled by . That is, the solution of (1) exists globally.

We would like to give some examples of , , and . It is easy to verify that they satisfy the conditions of Theorem 4.

Example 13. Consider that , , and with and with .

Example 14. Consider that , , and with and with .

Proof of Theorem 4. Set Using (18)–(20), we have From (58) and (59), we obtain Since , whether (1) or (2), (60) will be absurd for large enough. Therefore, the solution of (1) will blow up in finite time.

4. The Sharp Threshold for Global Existence and Blowup of the Solution to (1) with and with

In this section, we will establish the sharp threshold for global existence and blowup of the solution to (1) with and with .

Before giving the proof of Theorem 5, we would like to give some examples of and . It is easy to verify that they satisfy the conditions of Theorem 5.

Example 15. Consider that , with , and , .

Example 16. Consider that , with and .

Example 17. Let be one of those in Examples 15 and 16. Let where and satisfies when and makes smooth. Obviously, .

Proof of Theorem 5. We will proceed in four steps.
Step  1. We will prove . and mean that Using Gagliardo-Nirenberg’s and Hölder’s inequalities, we can get That is, if and .
On the other hand, if , we have that is, Using (67), we can obtain from (65). Hence
Step  2. Denote We will prove that and are invariant sets of (1) with and with . That is, we need to show that for all if . Since and are conservation quantities for (1), we have for all if . We need to prove that . Otherwise, assume that there exists a satisfying by the continuity. Note that (71) implies However, the inequality above and are contradictions to the definition of . Therefore, . Consequently, (71) and imply that . That is, is an invariant set of (1) with and with . Similarly, we can prove that is also an invariant set of (1) with and with .
Step  3. Assume that and . By the results of Step 2, we have and . That is, The two inequalities imply that which means that that is, the solution exists globally.
Step  4. Assume that and . By the results of Step 2, we obtain and . Hence we get By the results of Proposition 10, the solution will blow up in finite time.

As a corollary of Theorem 5, we obtain the sharp threshold for global existence and blowup of the solution of (8) as follows.

Corollary 18. Assume that and (23). Let be a positive constant satisfying Here
Suppose that satisfies
Then(1)if , the solution of (8) exists globally;(2)if , , and , the solution of (8) blows up in finite time.

Remark 19. In Theorem 1.5 of [10], Tao et al. proved the following.
Assume that is a solution of (8) with , where , , with , , , and . Then blowup occurs.
Corollary 18 covers the result above under some conditions. In fact, if , then hence implies that . That is, our blowup condition is weaker than theirs. On the other hand, our conclusion is still true if with , , and . In other words, our result is stronger than theirs if with , , and .

5. Sharp Threshold for the Blowup and Global Existence of the Solution to (1)

Theorem 7 is inspired by [5], but it extends the results to more general case. We need subtle estimates and more sophisticated analysis in the proof.

First, we would like to give some examples of , , and . It is easy to verify that they satisfy the conditions of Theorem 7.

Example 20. Consider that , with for and with , , , and .

Example 21. Consider that , with for and with , is a real number, , and , .

Example 22. Consider that , with for and with , , and .

5.1. Some Invariant Manifolds

In this subsection, we will prove that and construct some invariant manifolds.

Proposition 23. Assume that the conditions of Theorem 7 hold. Then .

Proof. Assume that satisfying . Using Gagliardo-Nirenberg’s and Young’s inequalities, we have Using Hölder’s inequality, from (81), we can obtain Equation (82) implies that for some positive constant .
On the other hand, if , we get From (84), we obtain
Consequently,

Now, we will give some properties of , , and . We have a proposition as follows.

Proposition 24. Assume that and are defined by (22) and (28). Then one has the following.(i)There at least exists a such that (ii)There at least exists a such that

Proof. (i) Noticing the assumptions on , , and , similar to the proof of Theorem 1.7 in [17], it is easy to prove that there exists a satisfying Multiplying (89) by and integrating over by part, we can get .
Multiplying (89) by and integrating over by part, we obtain Pohozaev’s identity: From and (90), we can get .
(ii) Letting for and , we can obtain Looking at and as the functions of , setting and , we get that and . We want to prove that there exists a pair of such that and . Since , we know that the image of and the plane intersect in the space of and form a curve . Hence there exist many positive real number pairs relying on such that near with . On the other hand, under the assumptions of and , it is easy to see that for any . By the continuity, we can choose that a pair of near with satisfies both and