Space-Time Estimates on Damped Fractional Wave Equation

Chen, Jiecheng; Fan, Dashan; Zhang, Chunjie

doi:https://doi.org/10.1155/2014/428909

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 428909 | https://doi.org/10.1155/2014/428909

Space-Time Estimates on Damped Fractional Wave Equation

Jiecheng Chen,¹Dashan Fan,²and Chunjie Zhang³

Academic Editor: Nasser-eddine Tatar

Received26 Jul 2013

Accepted27 Jan 2014

Published04 May 2014

Abstract

We obtain space-time estimates on the solution to the Cauchy problem of damped fractional wave equation. We mainly focus on the linear equation. The almost everywhere convergence of the solution to linear equations as is also studied, with the initial data satisfying certain regularity conditions.

1. Introduction

Let , , , and let be the Laplace operator. We consider the following Cauchy problem: with initial conditions Here, as usual, the fractional Laplacian is defined through the Fourier transform: for all test functions . The partial differential equation in (1) is significantly interesting in mathematics, physics, biology, and many scientific fields. It is the wave equation when , , and and it is the half wave equation when , , and . As known, the wave equation is one of the most fundamental equations in physics. Another fundamental equation in physics is the Schrödinger equation which can be deduced from (1) by letting , , and . The Schrödinger equation plays a remarkable role in the study of quantum mechanics and many other fields in physics. Also, (1) is the heat equation when , , and .

As we all know, wave equation, Schrödinger equation, heat equation, and Laplace equations are most important and fundamental types of partial differential equations. The researches on these equations and their related topics are well-mature and very rich and they are still quite active and robust research fields in modern mathematics. The reader is readily to find hundreds and thousands of interesting papers by searching the Google Scholar or checking the MathSciNet in AMS. Here we list only a few of them that are related to this research paper [1–23].

With an extra damping term in the wave equation, one obtains the damped wave equation We observe that there are also a lot of research articles in the literature addressing the above damped wave equation. Among numerous research papers we refer to [24–35] and the references therein. From the reference papers, we find that the damped wave equation (4) is well studied in many interesting topics such as the local and global well-posedness of some linear, semilinear, and nonlinear Cauchy problems and asymptotic and regularity estimates of the solution. We observe that the space frames of these studies focus on the Lebesgue spaces and the Lebesgue Sobolev spaces.

These observations motivate us to consider the Cauchy problem of a more general fractional damped wave equation: where are fixed constants. According to our best knowledge, the fractional damped wave equation was not studied in the literature, except the wave case . So our plan is to first study the linear equation (5) and to prove some estimates. In our later works, we will use those estimates to study the well-posedness of certain nonlinear equations. We can easily check that the solution of (5) is formally given by where is the Fourier multiplier with symbol (see Appendix). Thus our interest will focus on the operators Using dilation, we will restrict ourselves to the case so the theorems are all stated for (see Remark 6). We now denote These two operators are both convolution. We denote their kernels by and . Thus, we may write To state our main results, we need the following definition of admissible triplet.

Definition 1. A triplet is called -admissible if where , , and .

The following theorems are part of the main results in the paper.

Theorem 2. Let and let be -admissible and . Then for any , one has Here, denotes the homogeneous Sobolev space with order , and denotes the real Hardy space.

Theorem 3. Let , be -admissible and . Then the damped wave operators satisfy for any .

By the above theorems, we easily obtain the following space-time estimates on the solution .

Theorem 4. Let and let be -admissible and . For the solution of (5), one has

Theorem 5. Let , be -admissible and . The solution of the damped wave equation satisfies for any .

Remark 6. For (5) with general , it is not hard to see that where and . Therefore, and by applying Theorem 4, we have For , we have a similar result using Theorem 5.

In the statement of these theorems, the notation means that there is a constant independent of all essential variables such that . Also, throughout this paper, we use the notation to mean that there exist positive constants and , independent of all essential variables such that It is easy to see that, by the linearity, we only need to prove Theorems 2 and 3. To this end, we will carefully study the kernels

Using the linearization for small , we have Thus for small , This indicates that, for near zero, behaves like the fractional heat operator (see [11, 29, 30, 36, 37]).

For large , we similarly have This indicates that as near , behaves like the wave operator if and like the Schrödinger operator if ; see [12, 16, 38, 39].

In the same manner, the operator behaves the same as the operator . Based on these facts, we will estimate the kernels in their low frequencies, median frequencies, and high frequencies, separately, by using different methods. We will estimate the kernels in Section 2 and complete the proofs of main theorems in Section 3. Finally, in Section 4, we will study the almost everywhere convergence for the solution as . The similar convergence theorem for Schrödinger operator has been widely studied; see [3, 40–44].

2. Estimates on Kernels

As we mentioned in the first section, we will estimate the kernels and based on their different frequencies. So we will divide this section into several subsections.

2.1. Estimate for near Zero

Let be a radial function with support in and satisfy whenever . In this section we are going to obtain the decay estimates on the kernels With those decay estimates, we then are able to obtain two bounds for the convolutions with the above two kernels. Without loss of generality, we assume . This assumption is not essential by tracking the following proofs.

Proposition 7. Let and be defined as above. For all , one has

Proof. The estimates of two inequalities are the same, so we will prove the first one only.
(i) If and , then it is obvious to see
(ii) If and , then by scaling Since we have
(iii) If and , then by (ii) we know Using the Leibniz rule, we have Observe that For , using an induction argument we have where , , and For each fixed , there exists at least one variable such that . By integration by parts times on the variable , we obtain The main terms needed to be estimated are with . The other terms can be treated easily by further taking integration by parts.
We let be a radial function satisfying if and if . Let . By the partition of unity we write We note that , and the support of together with (28) implies Therefore By integration by parts, Here, an easy computation gives For , noting that and is supported in , we have
(iv) If and , then a similar argument, without scaling, shows that The proposition now follows from (i)–(iv).

Proposition 8. Let . Then for any and , Particularly, we have

Proof. We prove the proposition for the kernel only, since the proof for the other one is exactly the same. Let us first consider the case and . Invoking an interpolation argument [45, 46], we may assume that is a positive integer. Thus the dual space of is the homogeneous Lipschitz space (one can see the definition in [46]), which is exactly the homogeneous Hölder space . By duality we have If , it is easy to check that where is a homogeneous polynomial of degree . Thus, using the same argument as before we obtain If , This shows that, for all , On the other hand, if we write then by checking the proof of Proposition 7, we find for all multi-indices . So by the Calderón-Torchinsky multiplier theorem [47], we also have, for all , Now interpolating between (51) and (54), we finish the proof for .
For the case , we use Young's inequality to get where . By Proposition 7,

2.2. Estimate for Lying in the Mid-Interval

Let be a radial function with support in and satisfy whenever . We first will obtain the decay estimate on the kernels and then prove the mapping properties of the convolution operators with the above kernels. As in Section 2.1, we assume without loss of generality.

Proposition 9. For all and , we have

Proof. If , then the proof is the same as (i) and (ii) in the proof of Proposition 7. So we assume and . In the case of , we use the same proof as the following argument for , without taking the scaling kernel.
For , consider the scaling kernel By the Leibniz rule, Next we prove the following estimate: In fact, using Taylor's expansion, we have Then by an easy computation, Thus, by the induction, we have where Since we obtain So (61) is proved. Note by the compact support of , we have and we will prove, for all such , If , (69) then is a consequence of (61) and (28). If , then When , similar to (33), we get which is further bounded (note also ) by Thus we have proved (69). Fix an and let be the variable such that . Using integration by parts times on , we obtain By (61), (69), and the compact support of , we have The second term can be calculated directly to finish the whole proof.

By Proposition 9 and the same argument in proving Proposition 8, we have the following boundedness.

Proposition 10. Let . Then for any and , Particularly, we have

2.3. Estimates for near the Infinity

Let be a radial function with support in and satisfy whenever . Defining we have the following proposition.

Proposition 11. Let and . Then there exists a such that for any and , we have

Proof. We will show the case and leave the easy case to the reader. Again, we will only show the inequality of since the proof of the other one is similar.
Define an analytic family of operators By the Plancherel formula, we have If we can show for Re and some , the proposition easily follows by a complex interpolation on these two inequalities for . Then we can use a trivial dual argument to achieve the proposition for the whole range of . Also, without loss of generality, we prove (81) with . Let be a standard cutoff function with support in satisfying Defining then (81) will follow if we prove In fact, (84) implies Noting that in the support of , we get (81) from the above inequality.
Next we prove (84). Let be the kernel of . By Young's inequality, it suffices to show for some . By the definition, without loss of generality, we may write Using the Taylor expansion with integral remainder, for , we write where This gives for and . By the definition of it is easy to see that for any integer uniformly for and .
Now we write where the phase function is defined as Let sets , and be defined as where and . Hence, where denotes the characteristic function of a set .
Furthermore, we let for . Then
For each , using integration by parts on the variable, it is easy to obtain that, for , for any positive number .
By the polar decomposition, where the phase function is defined by Using integration by parts on the inner integral, we obtain for any positive number .
By the Proposition in [48, page 344], Thus, if If , For , if , If and then we choose , Finally we estimate . For each , If the set is not empty, we write Clearly Also, choose a sufficiently large , and then
If the set is empty, then we also have The proposition is proved.

3. Proof of Theorems 2 and 3

Proof of Theorem 2. Recalling the definition of in Section 2, we have By the triangle inequality and Propositions 8, 10, and 11, we only have to verify that, for any -admissible triplet , These two inequalities are obviously true if For , denote By Proposition 8, we have This indicates that, for any , there exists a positive constant independent of and such that This shows that is a bounded mapping from to the mixed norm space for any admissible triplet . Now we choose admissible triplets and satisfying Then by the Marcinkiewicz interpolation, we easily obtain Similarly we can show that, for any -admissible triplet ,

Proof of Theorem 3. By checking the above proof, we only need to show the following proposition.

Proposition 12. There is a for which if , then hold for all .

Proof. Let where is defined in Section 2.3 (corresponding to ). We will prove, for any , that with some . Then by repeating the complex interpolation argument in the proof of Proposition 11, with (81) replaced by (125), we finish the proof of the proposition.
Next we turn to the proof of (125). Denote the kernel of by By Young's inequality, it suffices to show that if , then Let be the cutoff function defined in Section 2.3. Then we have where, by [49, Ch. 4], In the last integral, and is the Bessel function of order .
So, by the Minkowski inequality,
First, we assume . Changing variables, we have
Using the Taylor expansion with integral remainder, for , we write where This gives for and . By the definition of it is easy to see that if we denote , then Also, for any integer , uniformly for and .
When using the known estimate it is easy to see Thus, When we use the asymptotic expansion of : for any integer , where are constants.
In this case, where, without loss of generality, we denote It is easy to see that, for a suitable integer , Thus it remains to show that, for each , Since the estimates of all are similar, we will only show Using integration by parts and noting if and , it is easy to check that one has if , for any positive integer , and if , for any positive integer . Thus, we have the following lemma.

Lemma 13. Let . For any , one has if .
Also, for any , if .

Now we continue the proof of the proposition. Write In , noting , we use the lemma with and

Similarly, in Lemma 13 we let : Let Using Lemma 13, we write Here, the last term Use the polar coordinate and Lemma 13 for : Similarly, we can show

When , the proof is the same with only minor modifications.

4. Almost Everywhere Convergence

Next we will study the pointwise convergence of the solution of (5) to the initial data. We will prove the following.

Theorem 14. Let . If belongs to the inhomogeneous Sobolev space and , then the solution of (5) converges to a.e. as .

To prove this theorem, we need Lemma 15 and Proposition 16.

Lemma 15 (see [50]). Let and . Then

Proposition 16. Let and let be defined on and satisfy Denote the maximal function Then if , we have If , then

Proof. Making into a function of , we only have to bound where is any measurable function.
By the polar decomposition, By Minkowski's inequality, change of variables, and Lemma 15, we have
When , we have Here we have to let On the other hand, Obviously we have to let which, together with (170), implies If , then Proposition 16 is proved.

Proof of Theorem 14. Denote It is not hard to verify that Since Theorem 14 will be proved if we can show, as , for and for . The proof of the two limits is similar and we will only show the second convergence. Note that the above convergence always holds for Schwarz function . So a further boundedness on the maximal function that is enough to imply Theorem 14.
Next we will prove (181). By (176) and Proposition 16, Fix . Taking and close to , we have and thus Applying Proposition 16 with and , we have Since we proved (181) when (note that Proposition 16 was proved only when ).
For , instead of (181), we will show which is also enough to obtain the pointwise convergence. Taking , we have Noting that we have Therefore and by duality from which (186) follows.

Appendix

We study the Cauchy problem We claim that the solution, in the Fourier transform side, is given by To verify this fact, we write the solution as where Take derivative, Thus Therefore, Thus, Thus, This shows the claim.

Conflict of Interests

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

Acknowledgment

This work is partially supported by the NSF of China (Grant nos. 11271330 and 11201103).

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Copyright © 2014 Jiecheng Chen 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.

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