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
For , by the Hausdorff-Young inequality and the Lebesgue dominated convergence theorem, we have
As , so there is only the finite number of such that . Thus, we get as ; that is, in , for .
For , by the Bernstein multiplier theorem, we have
Using the almost orthogonality of modulation space, we obtain .
For , by Lemma 19, we have .
Accordingly, for ,
For the nonlinear part, If , then we have . In fact, taking advantage of (92) and the Lebesgue dominated convergence theorem, we can get
Recall that and apply (92) and the Lebesgue dominated convergence theorem to the first term of (93); we have
Consequently,
Next, the proof of time continuity of is similar to . It only needs to take care of the difference of smoothness and the action of the Bessel potential. Finally, we obtain ; that is, it is the strong solution.
3.3. Global Well-Posedness for Small Fine Data
Let us construct a time decaying norm as
This idea for the NLS can be traced back to the work of Strauss [12] and Wang and Hudzik [2]. We consider the following mapping: in the metric space with . For any , in view of Lemma 20, we get that and (recall the notation that )
Owing to , it follows that and that
Combining the above four inequalities, we have
Hence, if we choose and small enough, then using the standard contraction mapping argument the Cauchy problem (1) has a unique solution satisfying (13).
3.4. Uniqueness
In order to prove the uniqueness in a category of strong solution, we only need to verify it locally; that is, to prove that the set is open. If for , we can choose an interval sufficient small so that and
So on , which concludes that is an interior point. So is open.
3.5. Blowup Criterion
We assume that , write the integral equation on , and use nonlinear estimate to deduce that
Then, let close to sufficiently enough so that , where is the one we chose in Theorem 1. Then, , such that , so we can extend on by local existence and uniqueness, a contradiction.
3.6. Persistence of Regularity
We want the norm which can hold the continuation of solution to be as low regularity as possible. It is interesting to make an assertion that the boundedness of low regularity norm suffices to ensure the boundedness of high regularity norm and thus ensure the continuation of strong solution. By the improved version of product lemma (Lemma 22) and the embedding theorems of modulation spaces, we can establish Lemmas 31 and 32 to achieve our goal.
Firstly, we “reduce” the regularity index from to .
Lemma 31. Let be a bounded time interval containing , let , and let be a strong solution to (1). If the quantity is finite, then one has where is some positive constant associated with the length of .
Proof. Without loss of generality, we may assume and for some . Writing the integral equation, using Lemma 19, and embedding, we have for . Let and , and both and are continuous and nonnegative on . Using Gronwall inequality, we have for and obtain the conclusion immediately.
Secondly, we reduce another regularity index from to 0. By the Leibniz rules for modulation space Lemma 22, we can deduce the following lemma. The proof of this lemma is very similar to the proof of Lemma 31, so we omit the details.
Lemma 32 32. Let be a bounded time interval containing , , and let be a strong solution to (1). If the quantity is finite, then one has where is some positive constant associated with the length of .
So, for any strong solution in a bounded interval , if it keeps norm bounded, it will keep norm bounded in the same interval. Conversely, if is bounded in a bounded time interval , by embedding , will be in . Thus, one can extend a strong solution in if and only if the quantity remains bounded. This implies that the blowup phenomenon is independent of exact and , so if an initial data lies in both and (, ), they have the same maximal existence interval.
4. Scattering Theorems
The goal of this section is to derive scattering results.
4.1. Proof of Scattering Theorem
Without loss of generality, we assume , and let For ,
By the fact we used in Section 3.2.1 that is from Definition 25 for -admissible pair , there exists such that
Take advantage of the nonlinear estimate and Hölder inequality:
Since , we have as . This implies that is Cauchy in as . Denote to be the limit:
In a similar way, we obtain
Recall that ; so taking advantage of the nonlinear estimates, we get as . So is , respectively. In fact, in our proof, we also have .
One may ask whether there exists a global solution to Cauchy problem (1) which is scattering associated with . The following remark gives us a partial answer.
Remark 33. Let and , and then there exists a solution of (1) on for some such that as .
To construct , we only need to solve the Cauchy problem from to some finite time with initial data at , that is, to solve the following equation:
on . In fact, we have . If we choose large enough, where is as in Theorem 1, then we can solve this equation by the same technique used in the proof of Theorem 1 and deduce that there exists a solution to (1) on and . So, if is large enough, we have as . A similar argument applies in the situation .
5. Stability Theory
In this section, we will derive the stability theory including short-time result and long-time result and deduce some corollaries.
5.1. Proof of Short-Time Perturbations
By time symmetry, we may assume that . Let . Then, satisfies the following initial value problem:
The integral equation is
For , define
By nonlinear estimate and the smallness condition, for every ,
By Product Lemma 21, we deduce
Noticing that , we have for every .
A standard continuity argument then shows that if we shrink sufficiently small, then for any , which implies (23). Using (124) and (127), one can easily obtain (24). By Lemma 19, we have
By Lemma 20 and Hölder inequality, we have that for every Similarly, we have and then we use the integral equation to conclude (25) and (26). Finally, (27) and (28) are easy conclusions from (25), (26), and (20).
After proof of short-time perturbations, we will derive long-time perturbations by an iterative procedure. In long-time perturbations, the smallness condition will be replaced by a bound condition . In our proof, we need to divide interval into several subintervals to gain a small size of . The -admissible condition which ensures allows us to do so.
5.2. Proof of Long-Time Perturbations
For convenience, we also assume that . We subdivide into subintervals such that where is as in Theorem 11. Choose in Theorem 12 and apply short-time perturbations on to obtain
Using the integral equation on , we have
So we can shrink such that
We also have So we can use the short-time perturbations to obtain
For , repeat the process above to shrink to continue the inductive argument. Accordingly, after a finite number of “shrinks” of , we have, , and then, for some and , we have where only depends on . So we have
By smallness assumption, we also have
Using integral equation and Hölder inequality, we conclude that
In the same way, we have
Finally, we can use (29) to deduce (37) and (38).
5.3. Proof of Continuous Dependence
Since , is a -admissible pair. For every compact interval , noticing that
we apply long-time perturbations with and to conclude that there exists a small constant such that if then (1) has a strong solution on with the initial data and So is .
The proof of Corollary 14 is similar, and we omit it here.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the NSF of China (Grant no. 11271330) and PSF of Zhejiang Province (BSH 1302046).