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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 947642, 15 pages
http://dx.doi.org/10.1155/2014/947642
Research Article

Klein-Gordon Equations on Modulation Spaces

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
3Department of Mathematics, Xiamen University, Xiamen 361005, China

Received 16 January 2014; Accepted 26 April 2014; Published 20 May 2014

Academic Editor: Simeon Reich

Copyright © 2014 Guoping Zhao 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 Cauchy problem for a family of Klein-Gordon equations with initial data in modulation spaces . We develop the well-posedness, blowup criterion, stability of regularity, scattering theory, and stability theory.

1. Introduction

In this paper, we consider the Cauchy problem for the following nonlinear Klein-Gordon equation in the space : where is a complex-valued function in for some time interval containing , the initial data lies in the product of modulation spaces (, ), and the nonlinear term is any -time product of and , . To understand this research problem and its historical developments, the reader may see Ruzhansky et al. [1] for a brief survey of nonlinear evolution equations on the modulation spaces. Concerning the well-posedness of solution to the Schrödinger equation in the modulation space, readers can refer to [2, 3].

We give some remarks about our results. The known study of the Klein-Gordan equations (or other dispersive equations) on modulation spaces must be based on the assumption that the nonlinear term is a polynomial. This assumption is also necessary in this paper; in fact, this is an open problem that if holds for any positive real constant .

We recall that is the critical index for (1). Up to now, we cannot solve (1) in for the case that (the sup-critical case). On the other hand, we notice that the modulation space has low regularity property. More precisely, for sufficiently large , we have the following embedding:

In other words, the modulation space has lower regularity than for large . So, for large (high dimension for instance), one can solve (1) in which contains sup-critical initial dates in for .

The local well-posedness of (1) in (, ) is a result of Be ́nyi and Okoudjou [4]; see also Wang [2] for a global result with small initial data. These results say that, for , there exists a positive such that (1) has a unique solution . Moreover, the lifetime of the solution can be proved to be bounded below by a decreasing positive function depending on ; that is, . It also asserts that if a strong solution keeps its norm bounded in a bounded interval, it can be extended beyond the endpoint. Hence, the following blowup criterion holds:

In this paper, we will develop a stronger blowup criterion which says that a blowup solution cannot blow up too slowly (see Corollary 8 and Remark 9). We also study the regularity of solutions and show that the regularity is stable along the lifetime. As an application, the global existence of low regularity ensures the global existence of high regularity.

Compared with (used in [4]), the space seems more suitable for applying continuity argument, which is the key point for obtaining the perturbation theorem, especially the long-time version. So we choose as our work space and establish the nonlinear estimate associated with this work space in Section 2.

In Section 3, we will establish the local theory. We first use the fixed point theorem to construct a local-in-time solution to (1). Then, we verify that such solution is a strong solution in the sense that and is unique in the category of strong solution. Finally, we study the regularity of solutions and deduce a stronger blowup criterion which implies the high rate of blowup. We will develop the scattering results in Section 4. In Section 5, we establish a stability theory for (1) and obtain the continuous dependence as a corollary.

Denote the operator , and for any function . Using this notation, we define and the Klein-Gordon semigroup:

We now state our main results. Unless otherwise specified, we assume that the letters are integers such that and is -admissible (Definition 25). First, we have the following local theorem.

Theorem 1 (local well-posedness). Let be a compact time interval that contains . Let satisfy for some , where is a small constant (depending only on , ). Then, there exists a unique solution to (1). Moreover, u is a strong solution to (1) in the sense that , and one also has

From Lemma 19, we can verify the condition (8) by choosing sufficiently small. So this theorem already gives local existence for large data. On the other hand, by inequality (63), we have the following global result as an application.

Corollary 2 (global well-posedness for small fine data). Let . Assume that satisfies for some small constant . Then, there exists a unique global solution . Similarly, u is also a strong solution; that is, . One also has the bound

More precisely, we have the following global well-posedness result which gives the decay rate of solutions.

Corollary 3 (another form of global well-posedness for small fine data). Assume that satisfies for some small constant . Then, there exists a unique global solution :

In the proof of Theorem 1, uniqueness is an immediate conclusion by the fixed point theorem. But, in fact, we have the following stronger result.

Theorem 4 (unconditional uniqueness in ). Let be a time interval containing , , , and let be two strong solutions to (1) in the sense of (45) with the same initial data , ; then in for all .

By combining the above uniqueness result with the local theorem, one can define the maximal interval of the strong solution; thus, we have the following standard blowup criterion.

Theorem 5 (blowup criterion). Let , , and let be the maximal interval. If , then one has

The above blowup criterion will be improved soon as an application of Lemmas 31 and 32. For completeness, we also give a proof for this weak version. Then, we give a regularity result.

Theorem 6 (persistence of regularity). Let ,, and , and let be a strong solution to (1) with its maximal existence interval . If and for some , then is also a strong solution with the same maximal interval.

Remark 7. Combining the above theorem with global result in in [2], one can easily get the global well-posedness in with small initial data for and .

Thus, it is not possible to develop a singularity which causes the norm to blow up while the norm remains bounded. We also see that the regularity is stable, because if a solution lies in and is not in at some initial time , it never belongs to at any later (or earlier) time. As an application (see also Lemmas 31 and 32), we have the following stronger version of blowup criterion.

Corollary 8 (stronger blowup criterion). Let , , and a strong solution of Cauchy problem (1) blows up in a finite time if and only if

Remark 9. From another point of view, the above blowup criterion implies that cannot blow up too slowly when tends to a finite blowup time ; that is, for every .

We also obtain a scattering theorem for these equations provided a bounded norm.

Theorem 10 ( bounds imply scattering). Let , and let be a global strong solution to (1) such that for some constant . Then, there exist such that are solutions to the free Klein-Gordon equation , and as .

Finally, we will discuss the stability theory. The stability theory for (1) means that given an approximate solution to (1), with and , small in a suitable space, is it possible to show that the genuine solution to (1) stays very close to in some sense (for instance, in the )? Note that the question of continuous dependence of the data corresponds to the case and the uniqueness theory to the case , . We have the following short-time perturbations and long-time perturbations.

Theorem 11 (short-time perturbations). Let be a compact time interval, and let be an approximate solution to (1) in the sense of (19). Assume that has a uniform bound: for some constant . Let and let , be close to , , respectively, in the sense that for some . Moreover, assume the following smallness conditions: for some , where is a small constant.
Then, there exists a solution to (1) with initial values at time satisfying

Theorem 12 (long-time perturbations). Let be a compact time interval, and let be an approximate solution to (1) in the sense of (19) for some function . Assume that for some constants , and . Let and let , be close to , , respectively, in the sense that for some . Moreover, assume the following smallness conditions: for some , where is a small constant. Then, there exists a solution to (1) with initial values , at time satisfying

As applications of the above stability theorems, we have the following corollaries.

Corollary 13 (continuous dependence). Assume that , and is a strong solution to (1) with initial data , . If in , then For every compact interval , let be the solution to (1) on with initial , , and then we have with ; that is, in .

Also, one can deduce continuous dependence for directly without using perturbation theorem, and the proof is not difficult, so we omit the details.

Corollary 14. Assume that is a -admissible pair. Denote by the subset of , such that, for every , the Cauchy problem (1) with initial data has a global strong solution on and . Then, the set is open in .

2. Preliminaries

If and are two quantities (typically nonnegative), we will often use the notation to denote the statement that for some absolute constant , where can depend on , but it might be different from line to line. Given , we write , , and . Let denote the Banach space of functions whose norm

The norm is defined with the usual modification. We also abbreviate for , or , when there is no confusion. We use to denote the space-time norm: with the usual modifications when , or is infinite. For the operator , the operator and the Klein-Gordon semigroup have been defined in Section 1. Thus, we may recall Duhamel’s formula:

Also, we recall the integral form of Gronwall’s inequality.

Lemma 15 (Gronwall inequality and integral form [5]). Let be continuous and nonnegative, and suppose that obeys the integral inequality for all , where and is continuous and nonnegative. Then, we have for all .

Let be the Schwartz space and the tempered distribution space. We introduce the definition of modulation space, which was introduced by Feichtinger [6] in 1983 by short-time Fourier transform. We will also display some basic properties of this function space.

Applying the frequency-uniform localization techniques, one can get an equivalent definition of modulation spaces (see [7] for details) as follows. Let be the unit cube with the center at , so constitutes a decomposition of . First, we construct a smooth cut-off function. Let and let be a smooth function satisfying for and for . Let be a translation of ,

We see that in , so for all . Denote Then, satisfies the following properties:

In fact, constitutes a smooth decomposition of and , in which

The frequency-uniform decomposition operators can be exactly defined by for .

Definition 16 (modulation space). Let , and one defines the modulation space

Below, we list some basic properties for the space .

Lemma 17 (embedding). Let and . (1)If , , and , then .(2)If and , then .

Proposition 18 (isomorphism). Let , . Then, is an isomorphic mapping.

The proof of Proposition 18 can be found in [6] for the cases and [7, 8] for the cases . We denote .

Lemma 19 (for uniform boundedness of in , see [9]). Let , , and . One has where the constant depends only on , , , and .

One can also find these estimates in [10] and a more general estimate on with , in Chen and Fan [11]. Chen and Fan also showed that the exponent is the best possible in the factor if equals 1 [11].

Lemma 20 (for truncated decay estimate of , see Proposition  4.2 in [2]). Let , , and , ;
One has where the constant depends only on , , , and .

Lemma 21 (algebra property [4]). Let . Assume that , with for . One has where is independent of .

Lemma 22 (Leibniz rule for modulation space [3]). Let and , and for . One has where C is independent of , and if and vanishes otherwise. Particularly, if we choose to be or , and for . We have

Lemma 23 (for Bernstein multiplier theorem, see Proposition 1.11 in [10]). Let , . Then, is a multiplier on , . Moreover, there exists a constant such that

Lemma 24. Let , , and let be a compact subset of . Then, and supp is dense in .

Definition 25 (-admissible pair). One calls the exponent -admissible if there exists another exponent such that

Remark 26. From Definition 25, if is -admissible, we can easily verify that
Moreover, we have the following inequality:

Remark 27. If , there exist -admissible pairs. The condition can ensure that .

Definition 28. One defines the strong solution to (1) as follows: the distribution is the solution to (1) in the sense of (45) with the initial data .

We establish the following nonlinear estimate.

Proposition 29 (nonlinear estimate). Let , . For any -admissible pair , one has

Proof. Observe that . For any -admissible pair , using the general Minkowski inequality, Proposition 18, and Lemma 20, we have for any .
If , then and there exists such that
With this , we have . Observing Definition 25 and Remark 27, we have and . So in this case we choose satisfying (66) and exploit Lemma 17 and the Hardy-Littlewood-Sobolev inequality to have
If , then there exists such that
With this , we have and . Taking advantage of Young’s inequality and Hölder’s inequality, we obtain In general, we get the first result. Now, by Lemma 21 and Hölder’s inequality, we have
By the fact and the embedding theorem (Lemma 17), we obtain

From Proposition 29, we have an immediate corollary as follows.

Corollary 30. Let , , and then one has

3. Local Well-Posedness

In this section, we establish the local theory for the Cauchy problem (1). In the rest of this paper, we assume that satisfy , so there exist -admissible exponents by Remark 27.

3.1. Proof of Existence Part of Theorem 1

We use the fixed point argument to construct a local solution. Let , and define a map on :

We want to choose suitable and so that is a contraction. By corollary of Proposition 29, we have

Let satisfy and choose , and then we have . Indeed,

We also have if .

So we can shrink to so that and find a possible smaller constant . Then, when , we have and it is nothing but a contraction map. We now obtain which is the fixed point of that solves Cauchy problem (1) in the sense of integral form. Of course, .

3.2. Proof of Strong Solution Part of Theorem 1

In this subsection, we will verify that the local solution is a strong solution in the sense that .

3.2.1. To Prove

It is equivalent to prove that as . We may assume without loss of generality that ,

Recall that . For , by density Lemma 24, Lemma 19, triangle inequality, and the definition of , we only need to prove that for .

Using Hausdorff-Young inequality, we have

Since , so as , by Lebesgue’s dominated convergence theorem. Since , there exists only the finite number of such that , so we have

Thus,

For ,

First, by the Minkowski inequality, we have that

For and , we have , so . Since , therefore, .

Secondly, as in the proof of Proposition 29,we get that

Since , . We have

Because , it leads to

Accordingly, (78) holds; that is, .

3.2.2. Exists and Is Continuous in Sense

First of all, we consider the existence of derivative in sense.

Recall that

For , , and the definition of , we should only deal with the derivatives of for and .

By the density Lemma 24, for every , there exists , such that . For the derivative of at for , we have