Space-Time Estimates on Damped Fractional Wave Equation

Abstract

We obtain space-time estimates on the solution u ( t , x ) 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 t [arrow right] [superscript] 0 + [/superscript] is also studied, with the initial data satisfying certain regularity conditions.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Jiecheng Chen 1 and Dashan Fan 2 and Chunjie Zhang 3

Academic Editor:Nasser-eddine Tatar

1, Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2, Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
3, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

Received 26 July 2013; Accepted 27 January 2014; 4 May 2014

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.

1. Introduction

Let ( t , x ) ∈ [ 0 , ∞ ) × ^{... n} , α > 0 , a , b ∈ ... , and let Δ be the Laplace operator. We consider the following Cauchy problem: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF] Here, as usual, the fractional Laplacian ( - Δ ^{) α} is defined through the Fourier transform: [figure omitted; refer to PDF] for all test functions f . The partial differential equation in (1) is significantly interesting in mathematics, physics, biology, and many scientific fields. It is the wave equation when a = 1 , b = 0 , and α = 1 and it is the half wave equation when a = 0 , 2 b = i , and α = 1 / 2 . 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 a = 0 , 2 b = i , and α = 1 . 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 a = 0 , b = 1 / 2 , and α = 1 .

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 2 b _{u t} ( t , x ) in the wave equation, one obtains the damped wave equation [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] where α , b > 0 are fixed constants. According to our best knowledge, the fractional damped wave equation was not studied in the literature, except the wave case α = 1 . So our plan is to first study the linear equation (5) and to prove some ^{L p} [arrow right] ^{L q} 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 [figure omitted; refer to PDF] where L is the Fourier multiplier with symbol ^{b 2} - | ξ ^{| 2 α} (see Appendix). Thus our interest will focus on the operators [figure omitted; refer to PDF] Using dilation, we will restrict ourselves to the case b = 1 so the theorems are all stated for u ( f , g ) = _{u 1} ( f , g ) (see Remark 6). We now denote [figure omitted; refer to PDF] These two operators are both convolution. We denote their kernels by _{Ω α} ( t ) and _{K α} ( t ) . Thus, we may write [figure omitted; refer to PDF] To state our main results, we need the following definition of admissible triplet.

Definition 1.

A triplet ( p , q , r ) is called σ -admissible if [figure omitted; refer to PDF] where 0 < r ...4; p ...4; + ∞ , r < q < ∞ , and σ > 0 .

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

Theorem 2.

Let α > 0 and let ( p , q , r ) be n / 2 α -admissible and 1 ...4; p ...4; + ∞ . Then for any β > n α | 1 / p - 1 / 2 | , one has [figure omitted; refer to PDF] Here, ^{L γ p} ( ^{... n} ) denotes the homogeneous Sobolev ^{L p} space with order γ , and ^{H r} denotes the real Hardy space.

Theorem 3.

Let α = 1 , ( p , q , r ) be n / 2 -admissible and 1 ...4; p ...4; + ∞ . Then the damped wave operators satisfy [figure omitted; refer to PDF] for any β > ( n - 1 ) | 1 / p - 1 / 2 | .

By the above theorems, we easily obtain the following space-time estimates on the solution u ( t , x ) .

Theorem 4.

Let α > 0 and let ( p , q , r ) be n / 2 α -admissible and 1 ...4; p ...4; + ∞ . For the solution u ( t , x ) of (5), one has [figure omitted; refer to PDF]

Theorem 5.

Let α = 1 , ( p , q , r ) be n / 2 -admissible and 1 ...4; p ...4; + ∞ . The solution u ( t , x ) of the damped wave equation satisfies [figure omitted; refer to PDF] for any β > ( n - 1 ) | 1 / p - 1 / 2 | .

Remark 6.

For (5) with general b > 0 , it is not hard to see that [figure omitted; refer to PDF] where s = b t and _{f b} ( x ) = f ( ^{b - 1 / α} x ) . Therefore, [figure omitted; refer to PDF] and by applying Theorem 4, we have [figure omitted; refer to PDF] For α = 1 , we have a similar result using Theorem 5.

In the statement of these theorems, the notation A ... B means that there is a constant C > 0 independent of all essential variables such that A ...4; C B . Also, throughout this paper, we use the notation A ≈ B to mean that there exist positive constants C and c , independent of all essential variables such that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF]

Using the linearization [figure omitted; refer to PDF] for small | ξ | , we have [figure omitted; refer to PDF] Thus for small | ξ | , [figure omitted; refer to PDF] This indicates that, for | ξ | near zero, _{T α} behaves like the fractional heat operator (see [11, 29, 30, 36, 37]).

For large | ξ | , we similarly have [figure omitted; refer to PDF] This indicates that as | ξ | near ∞ , ^{e t} _{T α} behaves like the wave operator if α = 1 and like the Schrödinger operator if α = 2 ; see [12, 16, 38, 39].

In the same manner, the operator _{S α} ( t ) behaves the same as the operator _{T α} . 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 u ( t , x ) as t [arrow right] ^{0 +} . The similar convergence theorem for Schrödinger operator ^{e i t Δ} f ( x ) has been widely studied; see [3, 40-44].

2. Estimates on Kernels

As we mentioned in the first section, we will estimate the kernels _{K α} ( t ) and _{Ω α} ( t ) based on their different frequencies. So we will divide this section into several subsections.

2.1. Estimate for | ξ | near Zero

Let _{[varphi] 1} be a ^{C ∞} radial function with support in { ξ ∈ ^{... n} : | ξ ^{| 2 α} ...4; 1 / 2 } and satisfy _{[varphi] 1} ...1; 1 whenever | ξ ^{| 2 α} ...4; 1 / 3 . In this section we are going to obtain the decay estimates on the kernels [figure omitted; refer to PDF] With those decay estimates, we then are able to obtain two ^{H p} bounds for the convolutions with the above two kernels. Without loss of generality, we assume 0 < 2 α < 1 . This assumption is not essential by tracking the following proofs.

Proposition 7.

Let _{K α , 0} and _{Ω α , 0} be defined as above. For all t > 0 , one has [figure omitted; refer to PDF]

Proof.

The estimates of two inequalities are the same, so we will prove the first one only.

(i) If ( 1 + t ^{) - 1 / 2 α} | x | ...4; 1 and 0 < t ...4; 1 , then it is obvious to see [figure omitted; refer to PDF]

(ii) If ( 1 + t ^{) - 1 / 2 α} | x | ...4; 1 and t > 1 , then by scaling [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] we have [figure omitted; refer to PDF]

(iii) If ( 1 + t ^{) - 1 / 2 α} | x | > 1 and t > 1 , then by (ii) we know [figure omitted; refer to PDF] Using the Leibniz rule, we have [figure omitted; refer to PDF] Observe that [figure omitted; refer to PDF] For k ...5; 1 , using an induction argument we have [figure omitted; refer to PDF] where _{... k} ...5; 0 , ^{ψ j k} ( ξ ) ... | ξ ^{| 2 j α - k} , and [figure omitted; refer to PDF] For each fixed x ∈ ^{... n} , there exists at least one variable _{x i} such that | _{x i} | ...5; | x | / n . By integration by parts n times on the variable _{ξ i} , we obtain [figure omitted; refer to PDF] The main terms needed to be estimated are [figure omitted; refer to PDF] with j = 1,2 , ... , n . The other terms can be treated easily by further taking integration by parts.

We let Φ be a ^{C ∞} radial function satisfying Φ ( ξ ) = 1 if | ξ | ...4; 1 and Φ ( ξ ) = 0 if | ξ | > 2 . Let Ψ ( ξ ) = 1 - Φ ( ξ ) . By the partition of unity we write [figure omitted; refer to PDF] We note that t > 1 , and the support of _{[varphi] 1} ( ^{t - 1 / 2 α} ξ ) together with (28) implies [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] By integration by parts, [figure omitted; refer to PDF] Here, an easy computation gives [figure omitted; refer to PDF] For _{J 2} , noting that [figure omitted; refer to PDF] and ^{Ψ [variant prime]} ( s ) is supported in [ 1 / 2,2 ] , we have [figure omitted; refer to PDF]

(iv) If ( 1 + t ^{) - 1 / 2 α} | x | > 1 and 0 < t ...4; 1 , then a similar argument, without scaling, shows that [figure omitted; refer to PDF] The proposition now follows from (i)-(iv).

Proposition 8.

Let f ∈ ^{H r} ( ^{... n} ) . Then for any t > 0 and 0 < r ...4; p < + ∞ , [figure omitted; refer to PDF] Particularly, we have [figure omitted; refer to PDF]

Proof.

We prove the proposition for the kernel _{K α , 0} only, since the proof for the other one is exactly the same. Let us first consider the case p = + ∞ and 0 < r < 1 . Invoking an interpolation argument [45, 46], we may assume that n ( 1 / r - 1 ) is a positive integer. Thus the dual space of ^{H r} is the homogeneous Lipschitz space _{Λ n ( 1 / r - 1 )} ( ^{... n} ) (one can see the definition in [46]), which is exactly the homogeneous Hölder space ^{C n ( 1 / r - 1 )} ( ^{... n} ) . By duality we have [figure omitted; refer to PDF] If t ...5; 1 , it is easy to check that [figure omitted; refer to PDF] where P ( ξ ) is a homogeneous polynomial of degree n ( 1 / r - 1 ) . Thus, using the same argument as before we obtain [figure omitted; refer to PDF] If 0 < t ...4; 1 , [figure omitted; refer to PDF] This shows that, for all 0 < r < 1 , [figure omitted; refer to PDF] On the other hand, if we write [figure omitted; refer to PDF] then by checking the proof of Proposition 7, we find [figure omitted; refer to PDF] for all multi-indices k . So by the Calderón-Torchinsky multiplier theorem [47], we also have, for all 0 < r < 1 , [figure omitted; refer to PDF] Now interpolating between (51) and (54), we finish the proof for 0 < r < 1 .

For the case 1 ...4; r ...4; + ∞ , we use Young's inequality to get [figure omitted; refer to PDF] where 1 / r + 1 / q = 1 / p + 1 . By Proposition 7, [figure omitted; refer to PDF]

2.2. Estimate for | ξ | Lying in the Mid-Interval

Let _{[varphi] 2} be a ^{C ∞} radial function with support in { ξ ∈ ^{... n} : 1 / 4 ...4; | ξ ^{| 2 α} ...4; 200 } and satisfy _{[varphi] 1} ...1; 1 whenever 1 / 3 ...4; | ξ ^{| 2 α} ...4; 100 . We first will obtain the decay estimate on the kernels [figure omitted; refer to PDF] and then prove the mapping properties of the convolution operators with the above kernels. As in Section 2.1, we assume 0 < 2 α < 1 without loss of generality.

Proposition 9.

For all t > 0 and N > 0 , we have [figure omitted; refer to PDF]

Proof.

If ( 1 + t ^{) - 1 / 2 α} | x | ...4; 1 , then the proof is the same as (i) and (ii) in the proof of Proposition 7. So we assume ( 1 + t ^{) - 1 / 2 α} | x | > 1 and t > 1 . In the case of t ...4; 1 , we use the same proof as the following argument for t > 1 , without taking the scaling kernel.

For t > 1 , consider the scaling kernel [figure omitted; refer to PDF] By the Leibniz rule, [figure omitted; refer to PDF] Next we prove the following estimate: [figure omitted; refer to PDF] In fact, using Taylor's expansion, we have [figure omitted; refer to PDF] Then by an easy computation, [figure omitted; refer to PDF] Thus, by the induction, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] So (61) is proved. Note by the compact support of _{[varphi] 2} ( ^{t - 1 / 2 α} ξ ) , we have [figure omitted; refer to PDF] and we will prove, for all such ξ , [figure omitted; refer to PDF] If t / 4 ...4; | ξ ^{| 2 α} ...4; t , (69) then is a consequence of (61) and (28). If t < | ξ ^{| 2 α} ...4; 5 t / 4 , then [figure omitted; refer to PDF] When 5 t / 4 < | ξ ^{| 2 α} ...4; 200 t , similar to (33), we get [figure omitted; refer to PDF] which is further bounded (note also t > 1 ) by [figure omitted; refer to PDF] Thus we have proved (69). Fix an x ∈ ^{... n} and let _{x i} be the variable such that _{x i} > | x | / n . Using integration by parts ( n + 1 ) times on _{ξ i} , we obtain [figure omitted; refer to PDF] By (61), (69), and the compact support of _{[varphi] 2} , we have [figure omitted; refer to PDF] The second term _{K 2} 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 f ∈ ^{H r} ( ^{... n} ) . Then for any t , N > 0 and r ...4; p < + ∞ , [figure omitted; refer to PDF] Particularly, we have [figure omitted; refer to PDF]

2.3. Estimates for | ξ | near the Infinity

Let _{[varphi] 3} be a ^{C ∞} radial function with support in { ξ ∈ ^{... n} : | ξ ^{| 2 α} ...5; ^{2 6} } and satisfy _{[varphi] 3} ...1; 1 whenever | ξ ^{| 2 α} ...5; 100 . Defining [figure omitted; refer to PDF] we have the following proposition.

Proposition 11.

Let 1 ...4; p ...4; + ∞ and α > 0 . Then there exists a _{δ p} > 0 such that for any β > n α | 1 / 2 - 1 / p | and t > 0 , we have [figure omitted; refer to PDF]

Proof.

We will show the case n ...5; 2 and leave the easy case n = 1 to the reader. Again, we will only show the inequality of _{K α , ∞} ( t ) * f since the proof of the other one is similar.

Define an analytic family of operators [figure omitted; refer to PDF] By the Plancherel formula, we have [figure omitted; refer to PDF] If we can show [figure omitted; refer to PDF] for Re z > n α / 2 and some λ > 0 , the proposition easily follows by a complex interpolation on these two inequalities for 1 ...4; p ...4; 2 . Then we can use a trivial dual argument to achieve the proposition for the whole range of p . Also, without loss of generality, we prove (81) with z = β > n α / 2 . Let Φ be a standard cutoff function with support in { ξ : 1 / 2 ...4; | ξ | ...4; 2 } satisfying [figure omitted; refer to PDF] Defining [figure omitted; refer to PDF] then (81) will follow if we prove [figure omitted; refer to PDF] In fact, (84) implies [figure omitted; refer to PDF] Noting that ^{2 - j} ≈ | ξ | in the support of Φ ( ^{2 - j} ξ ) , we get (81) from the above inequality.

Next we prove (84). Let _{[real] α , j} be the kernel of ^{W j α} ( t ) . By Young's inequality, it suffices to show [figure omitted; refer to PDF] for some λ > 0 . By the definition, without loss of generality, we may write [figure omitted; refer to PDF] Using the Taylor expansion with integral remainder, for r ∈ supp ... ( Φ ) , we write [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This gives [figure omitted; refer to PDF] for ^{2 α j} ...5; 100 and 1 / 2 ...4; r ...4; 2 . By the definition of g it is easy to see that for any integer m ...5; 0 [figure omitted; refer to PDF] uniformly for ^{2 α j} ...5; 100 and 1 / 2 ...4; r ...4; 2 .

Now we write [figure omitted; refer to PDF] where the phase function [Weierstrass p] is defined as [figure omitted; refer to PDF] Let sets _{E 1} , _{E 2} , and _{E 3} be defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and m = 1 / M . Hence, [figure omitted; refer to PDF] where _{χ E} denotes the characteristic function of a set E .

Furthermore, we let [figure omitted; refer to PDF] for m = 1,2 , ... , n . Then [figure omitted; refer to PDF]

For each _{χ E 1 , m [real] α , j} , using integration by parts on the _{ξ m} variable, it is easy to obtain that, for m = 1,2 , ... , n , [figure omitted; refer to PDF] for any positive number N .

By the polar decomposition, [figure omitted; refer to PDF] where the phase function P is defined by [figure omitted; refer to PDF] Using integration by parts on the inner integral, we obtain [figure omitted; refer to PDF] for any positive number N .

By the Proposition in [48, page 344], [figure omitted; refer to PDF] Thus, if t ^{2 j α} ...5; 1 [figure omitted; refer to PDF] If t ^{2 j α} ...4; 1 , [figure omitted; refer to PDF] For _{χ E 2 [real] α , j} , if t ^{2 j α} ...4; 1 , [figure omitted; refer to PDF] If t ^{2 j α} > 1 and then we choose N = n , [figure omitted; refer to PDF] Finally we estimate _{χ E 1 [real] α , j} . For each m = 1,2 , ... , n , [figure omitted; refer to PDF] If the set { x : t ^{2 j ( α - 1 )} ... | x | ... ^{2 - j} } is not empty, we write [figure omitted; refer to PDF] Clearly [figure omitted; refer to PDF] Also, choose a sufficiently large N , and then [figure omitted; refer to PDF]

If the set { ^{2 - j} ...5; | x | ... t ^{2 j ( α - 1 )} } is empty, then we also have [figure omitted; refer to PDF] The proposition is proved.

3. Proof of Theorems 2 and 3

Proof of Theorem 2.

Recalling the definition of [figure omitted; refer to PDF] in Section 2, we have [figure omitted; refer to PDF] By the triangle inequality and Propositions 8, 10, and 11, we only have to verify that, for any n / 2 α -admissible triplet ( p , q , r ) , [figure omitted; refer to PDF] These two inequalities are obviously true if [figure omitted; refer to PDF] For 1 / q = ( n / 2 α ) ( 1 / r - 1 / p ) , denote [figure omitted; refer to PDF] By Proposition 8, we have [figure omitted; refer to PDF] This indicates that, for any λ > 0 , there exists a positive constant C independent of λ and f such that [figure omitted; refer to PDF] This shows that _{K α , 0} ( t ) is a bounded mapping from ^{H r} ( ^{... n} ) to the mixed norm space ^{L q , ∞} ( [ 0 , ∞ ] , ^{L p} ( ^{... n} ) ) for any admissible triplet ( p , q , r ) . Now we choose admissible triplets ( p , _{q 1} , _{r 1} ) and ( p , _{q 2} , _{r 2} ) satisfying [figure omitted; refer to PDF] Then by the Marcinkiewicz interpolation, we easily obtain [figure omitted; refer to PDF] Similarly we can show that, for any n / 2 α -admissible triplet ( p , q , r ) , [figure omitted; refer to PDF]

Proof of Theorem 3.

By checking the above proof, we only need to show the following proposition.

Proposition 12.

There is a _{δ p} > 0 for which if β > ( n - 1 ) | 1 / p - 1 / 2 | , then [figure omitted; refer to PDF] hold for all 1 ...4; p ...4; ∞ .

Proof.

Let [figure omitted; refer to PDF] where _{[varphi] 3} is defined in Section 2.3 (corresponding to α = 1 ). We will prove, for any β > ( n - 1 ) / 2 , that [figure omitted; refer to PDF] with some λ > 0 . 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 _{W β} ( t ) by [figure omitted; refer to PDF] By Young's inequality, it suffices to show that if β > ( n - 1 ) / 2 , then [figure omitted; refer to PDF] Let Φ be the cutoff function defined in Section 2.3. Then we have [figure omitted; refer to PDF] where, by [49, Ch. 4], [figure omitted; refer to PDF] In the last integral, [figure omitted; refer to PDF] and _{J ν} ( s ) is the Bessel function of order ν .

So, by the Minkowski inequality, [figure omitted; refer to PDF]

First, we assume t ...5; 1 . Changing variables, we have [figure omitted; refer to PDF]

Using the Taylor expansion with integral remainder, for r ∈ supp ... ( _{[varphi] 3} ) , we write [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This gives [figure omitted; refer to PDF] for k ...5; 6 and 1 / 2 ...4; r ...4; 2 . By the definition of g it is easy to see that if we denote h ( r ) = g ( 1 / ^{2 k} r ) , then [figure omitted; refer to PDF] Also, for any integer m ...5; 0 , [figure omitted; refer to PDF] uniformly for k ...5; 10 and 1 / 2 ...4; r ...4; 2 .

When [figure omitted; refer to PDF] using the known estimate [figure omitted; refer to PDF] it is easy to see [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] When [figure omitted; refer to PDF] we use the asymptotic expansion of _{J ( n - 2 ) / 2} ( r ) : for any integer N ...5; 0 , [figure omitted; refer to PDF] where _{c 1} , _{c 2} , ... , _{c N} are constants.

In this case, [figure omitted; refer to PDF] where, without loss of generality, we denote [figure omitted; refer to PDF] It is easy to see that, for a suitable integer N , [figure omitted; refer to PDF] Thus it remains to show that, for each j , [figure omitted; refer to PDF] Since the estimates of all _{Y k , j} are similar, we will only show [figure omitted; refer to PDF] Using integration by parts and noting _{[varphi] 3} ( ^{2 k} r ) ...1; 1 if k > ^{2 8} and r ∈ supp ... ( Φ ) , it is easy to check that one has [figure omitted; refer to PDF] if ^{2 - k} t > 1 , for any positive integer m , and [figure omitted; refer to PDF] if ^{2 - k} t ...4; 1 , for any positive integer μ . Thus, we have the following lemma.

Lemma 13.

Let ^{2 k} | x | ...5; 10 . For any m ...5; 0 , one has [figure omitted; refer to PDF] if ^{2 - k} t ...4; 1 .

Also, for any μ ...5; 0 , [figure omitted; refer to PDF] if ^{2 - k} t > 1 .

Now we continue the proof of the proposition. Write [figure omitted; refer to PDF] In _{A 1} , noting β - ( n - 1 ) / 2 > 0 , we use the lemma with μ = 1 / 2 and m = 1 : [figure omitted; refer to PDF]

Similarly, in Lemma 13 we let μ = m = n : [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Using Lemma 13, we write [figure omitted; refer to PDF] Here, the last term [figure omitted; refer to PDF] Use the polar coordinate and Lemma 13 for μ = 1 / 2 : [figure omitted; refer to PDF] Similarly, we can show [figure omitted; refer to PDF]

When 0 < t ...4; 1 , the proof is the same with only minor modifications.

4. Almost Everywhere Convergence

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

Theorem 14.

Let s > 1 / 2 . If f belongs to the inhomogeneous Sobolev space ^{L s 2} ( ^{... n} ) and g ∈ ^{L s - α 2} ( ^{... n} ) , then the solution u ( t , x ) of (5) converges to f ( x ) a.e. x ∈ ^{... n} as t [arrow right] ^{0 +} .

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

Lemma 15 (see [50]).

Let n ...5; 2 and 1 < d < n . Then [figure omitted; refer to PDF]

Proposition 16.

Let n ...5; 2 and let m ( t , | ξ | ) be defined on ^{... +} × ^{... n} and satisfy [figure omitted; refer to PDF] Denote the maximal function [figure omitted; refer to PDF] Then if γ > 0 , we have [figure omitted; refer to PDF] If γ ...4; 0 , then [figure omitted; refer to PDF]

Proof.

Making t into a function of x , we only have to bound [figure omitted; refer to PDF] where t ( x ) : ^{... n} [arrow right] ^{... +} is any measurable function.

By the polar decomposition, [figure omitted; refer to PDF] By Minkowski's inequality, change of variables, and Lemma 15, we have [figure omitted; refer to PDF]

When γ > 0 , we have [figure omitted; refer to PDF] Here we have to let [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF] Obviously we have to let [figure omitted; refer to PDF] which, together with (170), implies [figure omitted; refer to PDF] If γ ...4; 0 , then [figure omitted; refer to PDF] Proposition 16 is proved.

Proof of Theorem 14.

Denote [figure omitted; refer to PDF] It is not hard to verify that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] Theorem 14 will be proved if we can show, as t [arrow right] ^{0 +} , [figure omitted; refer to PDF] for g ∈ ^{L s - α 2} ( ^{... n} ) and [figure omitted; refer to PDF] for f ∈ ^{L s 2} ( ^{... n} ) . 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 f . So a further boundedness on the maximal function [figure omitted; refer to PDF] that [figure omitted; refer to PDF] is enough to imply Theorem 14.

Next we will prove (181). By (176) and Proposition 16, [figure omitted; refer to PDF] Fix s > 1 / 2 . Taking 1 < d < s + 1 / 2 and l close to d - 1 / 2 , we have l < s and thus [figure omitted; refer to PDF] Applying Proposition 16 with γ = α and 1 < d < 1 + 2 α , we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] we proved (181) when n ...5; 2 (note that Proposition 16 was proved only when n ...5; 2 ).

For n = 1 , instead of (181), we will show [figure omitted; refer to PDF] which is also enough to obtain the pointwise convergence. Taking h ( x ) ∈ ^{L 2} ( [ - N , N ] ) , we have [figure omitted; refer to PDF] Noting that [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] and by duality [figure omitted; refer to PDF] from which (186) follows.

Acknowledgment

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

Conflict of Interests

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

References

[1] H. Bellout, A. Friedman, "Blow-up estimates for a nonlinear hyperbolic heat equation," SIAM Journal on Mathematical Analysis , vol. 20, no. 2, pp. 354-366, 1989.

[2] M. D. Blair, H. F. Smith, C. D. Sogge, "On Strichartz estimates for Schrödinger operators in compact manifolds with boundary," Proceedings of the American Mathematical Society , vol. 136, no. 1, pp. 247-256, 2008.

[3] T. Cazenave Semilinear Schrödinger Equations , vol. 10, of Courant Lecture Notes in Mathematics, pp. xiv+323, New York University Courant Institute of Mathematical Sciences, New York, NY, USA, 2003.

[4] T. Cazenave, F. B. Weissler, "Rapidly decaying solutions of the nonlinear Schrödinger equation," Communications in Mathematical Physics , vol. 147, no. 1, pp. 75-100, 1992.

[5] J. Chen, D. Fan, L. Sun, "Hardy space estimates for the wave equation on compact Lie groups," Journal of Functional Analysis , vol. 259, no. 12, pp. 3230-3264, 2010.

[6] Y. Giga, "Solutions for semilinear parabolic equations in ^{L p} and regularity of weak solutions of the Navier-Stokes system," Journal of Differential Equations , vol. 62, no. 2, pp. 186-212, 1986.

[7] M. Hieber, J. Prüss, "Heat kernels and maximal ^{L p} - ^{L q} estimates for parabolic evolution equations," Communications in Partial Differential Equations , vol. 22, no. 9-10, pp. 1647-1669, 1997.

[8] M. Keel, T. Tao, "Endpoint Strichartz estimates," American Journal of Mathematics , vol. 120, no. 5, pp. 955-980, 1998.

[9] H. Lindblad, C. D. Sogge, "On existence and scattering with minimal regularity for semilinear wave equations," Journal of Functional Analysis , vol. 130, no. 2, pp. 357-426, 1995.

[10] B. Marshall, W. Strauss, S. Wainger, " ^{L p} - ^{L q} estimates for the Klein-Gordon equation," Journal de Mathématiques Pures et Appliquées , vol. 59, no. 4, pp. 417-440, 1980.

[11] C. Miao, B. Yuan, B. Zhang, "Strong solutions to the nonlinear heat equation in homogeneous Besov spaces," Nonlinear Analysis: Theory, Methods & Applications , vol. 67, no. 5, pp. 1329-1343, 2007.

[12] A. Miyachi, "On some Fourier multipliers for ^{H p} ( ^{... n} ) ," Journal of the Faculty of Science. University of Tokyo IA , vol. 27, no. 1, pp. 157-179, 1980.

[13] G. Mockenhaupt, A. Seeger, C. D. Sogge, "Local smoothing of Fourier integral operators and Carleson-Sjölin estimates," Journal of the American Mathematical Society , vol. 6, no. 1, pp. 65-130, 1993.

[14] D. Müller, E. M. Stein, " ^{L p} -estimates for the wave equation on the Heisenberg group," Revista Matemática Iberoamericana , vol. 15, no. 2, pp. 297-334, 1999.

[15] M. Nakao, K. Ono, "Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations," Mathematische Zeitschrift , vol. 214, no. 2, pp. 325-342, 1993.

[16] J. C. Peral, " ^{L p} estimates for the wave equation," Journal of Functional Analysis , vol. 36, no. 1, pp. 114-145, 1980.

[17] A. Seeger, C. D. Sogge, E. M. Stein, "Regularity properties of Fourier integral operators," Annals of Mathematics , vol. 134, no. 2, pp. 231-251, 1991.

[18] G. Staffilani, D. Tataru, "Strichartz estimates for a Schrödinger operator with nonsmooth coefficients," Communications in Partial Differential Equations , vol. 27, no. 7-8, pp. 1337-1372, 2002.

[19] A. Stefanov, "Strichartz estimates for the Schrödinger equation with radial data," Proceedings of the American Mathematical Society , vol. 129, no. 5, pp. 1395-1401, 2001.

[20] E. Terraneo, "Non-uniqueness for a critical non-linear heat equation," Communications in Partial Differential Equations , vol. 27, no. 1-2, pp. 185-218, 2002.

[21] G. Todorova, B. Yordanov, "Critical exponent for a nonlinear wave equation with damping," Journal of Differential Equations , vol. 174, no. 2, pp. 464-489, 2001.

[22] M. C. Vilela, "Inhomogeneous Strichartz estimates for the Schrödinger equation," Transactions of the American Mathematical Society , vol. 359, no. 5, pp. 2123-2136, 2007.

[23] F. B. Weissler, "Existence and nonexistence of global solutions for a semilinear heat equation," Israel Journal of Mathematics , vol. 38, no. 1-2, pp. 29-40, 1981.

[24] N. Hayashi, E. I. Kaikina, P. I. Naumkin, "Damped wave equation with a critical nonlinearity," Transactions of the American Mathematical Society , vol. 358, no. 3, pp. 1165-1185, 2006.

[25] R. Ikehata, K. Tanizawa, "Global existence of solutions for semilinear damped wave equations in ^{... N} with noncompactly supported initial data," Nonlinear Analysis: Theory, Methods & Applications , vol. 61, no. 7, pp. 1189-1208, 2005.

[26] R. Ikehata, Y. Miyaoka, T. Nakatake, "Decay estimates of solutions for dissipative wave equations in ^{... N} with lower power nonlinearities," Journal of the Mathematical Society of Japan , vol. 56, no. 2, pp. 365-373, 2004.

[27] P. Marcati, K. Nishihara, "The ^{L p} - ^{L q} estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media," Journal of Differential Equations , vol. 191, no. 2, pp. 445-469, 2003.

[28] A. Matsumura, "On the asymptotic behavior of solutions of semi-linear wave equations," Publications of the Research Institute for Mathematical Sciences , vol. 12, no. 1, pp. 169-189, 1976.

[29] C. Miao, B. Yuan, B. Zhang, "Well-posedness of the Cauchy problem for the fractional power dissipative equations," Nonlinear Analysis: Theory, Methods & Applications , vol. 68, no. 3, pp. 461-484, 2008.

[30] C. Miao, B. Zhang, "The Cauchy problem for semilinear parabolic equations in Besov spaces," Houston Journal of Mathematics , vol. 30, no. 3, pp. 829-878, 2004.

[31] T. Narazaki, " ^{L p} - ^{L q} estimates for damped wave equations and their applications to semi-linear problem," Journal of the Mathematical Society of Japan , vol. 56, no. 2, pp. 585-626, 2004.

[32] T. Narazaki, " ^{L p} - ^{L q} estimates for damped wave equations with odd initial data," Electronic Journal of Differential Equations , vol. 2005, pp. 1-17, 2005.

[33] K. Nishihara, " ^{L p} - ^{L q} estimates of solutions to the damped wave equation in 3-dimensional space and their application," Mathematische Zeitschrift , vol. 244, no. 3, pp. 631-649, 2003.

[34] H. Volkmer, "Asymptotic expansion of ^{L 2} -norms of solutions to the heat and dissipative wave equations," Asymptotic Analysis , vol. 67, no. 1-2, pp. 85-100, 2010.

[35] Q. S. Zhang, "A blow-up result for a nonlinear wave equation with damping: the critical case," Comptes Rendus de l'Académie des Sciences I , vol. 333, no. 2, pp. 109-114, 2001.

[36] J. Chen, Q. Deng, Y. Ding, D. Fan, "Estimates on fractional power dissipative equations in function spaces," Nonlinear Analysis: Theory, Methods & Applications , vol. 75, no. 5, pp. 2959-2974, 2012.

[37] Z. Zhai, "Strichartz type estimates for fractional heat equations," Journal of Mathematical Analysis and Applications , vol. 356, no. 2, pp. 642-658, 2009.

[38] A. Miyachi, "On some estimates for the wave equation in ^{L p} and ^{H p} ," Journal of the Faculty of Science. University of Tokyo IA , vol. 27, no. 2, pp. 331-354, 1980.

[39] A. Miyachi, "On some singular Fourier multipliers," Journal of the Faculty of Science. University of Tokyo IA , vol. 28, no. 2, pp. 267-315, 1981.

[40] K. M. Rogers, "A local smoothing estimate for the Schrödinger equation," Advances in Mathematics , vol. 219, no. 6, pp. 2105-2122, 2008.

[41] L. Carleson, "Some analytic problems related to statistical mechanics," Euclidean Harmonic Analysis , vol. 779, of Lecture Notes in Mathematics, pp. 5-45, Springer, Berlin, Germany, 1980.

[42] S. Lee, "On pointwise convergence of the solutions to Schrödinger equations in ^{... 2} ," International Mathematics Research Notices , vol. 2006, 2006.

[43] P. Sjölin, "Regularity of solutions to the Schrödinger equation," Duke Mathematical Journal , vol. 55, no. 3, pp. 699-715, 1987.

[44] L. Vega, "Schrödinger equations: pointwise convergence to the initial data," Proceedings of the American Mathematical Society , vol. 102, no. 4, pp. 874-878, 1988.

[45] J. Bergh, J. Löfström Interpolation Spaces. An Introduction , of Grundlehren der Mathematischen Wissenschaften, no. 223, pp. x+207, Springer, Berlin, Germany, 1976.

[46] H. Triebel Theory of Function Spaces , vol. 78, of Monographs in Mathematics, pp. 284, Birkhäuser, Basel, Switzerland, 1983.

[47] A.-P. Calderón, A. Torchinsky, "Parabolic maximal functions associated with a distribution. II," Advances in Mathematics , vol. 24, no. 2, pp. 101-171, 1977.

[48] E. M. Stein Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , vol. 43, of Princeton Mathematical Series, pp. xiv+695, Princeton University Press, Princeton, NJ, USA, 1993.

[49] E. M. Stein, G. Weiss Introduction to Fourier Analysis on Euclidean Spaces , of Princeton Mathematical Series, no. 32, pp. x+297, Princeton University Press, Princeton, NJ, USA, 1971.

[50] S. L. Wang, "On the weighted estimate of the solution associated with the Schrödinger equation," Proceedings of the American Mathematical Society , vol. 113, no. 1, pp. 87-92, 1991.

Appendix

Appendix

We study the Cauchy problem [figure omitted; refer to PDF] We claim that the solution, in the Fourier transform side, is given by [figure omitted; refer to PDF] To verify this fact, we write the solution as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Take derivative, [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] This shows the claim.