Space-Time Estimates on Damped Fractional Wave Equation

and Applied Analysis 3 all essential variables such that A ≤ CB. 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 cB ≤ A ≤ CB. (18) 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 K α (t) (x) = e −t ∫ R cosh (t√1 − 󵄨󵄨󵄨 󵄨 ξ 󵄨 󵄨 󵄨 󵄨 2α ) e i⟨x,ξ⟩ dξ, Ω α (t) (x) = e −t ∫ R sinh(t√1 − 󵄨󵄨󵄨 󵄨 ξ 󵄨 󵄨 󵄨 󵄨 2α ) √1 − 󵄨 󵄨 󵄨 󵄨 ξ 󵄨 󵄨 󵄨 󵄨 2α e i⟨x,ξ⟩ dξ. (19) Using the linearization

Here, as usual, the fractional Laplacian (−Δ)  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  = 1,  = 0, and  = 1 and it is the half wave equation when  = 0, 2 = , and  = 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  = 0, 2 = , and  = 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  = 0,  = 1/2, and  = 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 .
With an extra damping term 2  (, ) in the wave equation, one obtains the damped wave equation   (, ) + 2  (, ) − Δ (, ) = 0,  > 0. ( 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][25][26][27][28][29][30][31][32][33][34][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:   (, ) + 2  (, ) + (−Δ)   (, ) = 0,  (0, ) =  () ,   (0, ) =  () , where ,  > 0 are fixed constants.According to our best knowledge, the fractional damped wave equation was not studied in the literature, except the wave case  = 1.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   (, ) (, ) = { − cosh ( √ )  +  − sinh ( √ ) √  ( + )} , where  is the Fourier multiplier with symbol  2 − || 2 (see Appendix).Thus our interest will focus on the operators  , () :=  − cosh ( √ ) ,  , () :=  − sinh ( √ ) √  .
The following theorems are part of the main results in the paper.
In the statement of these theorems, the notation  ⪯  means that there is a constant  > 0 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 cosh Thus for small ||, This indicates that, for || near zero,   behaves like the fractional heat operator (see [11,29,30,36,37]).
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  → 0 + .The similar convergence theorem for Schrödinger operator  Δ () has been widely studied; see [3,[40][41][42][43][44].

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

Estimate for |𝜉| near Zero.
Let  1 be a  ∞ radial function with support in { ∈ R  : || 2 ≤ 1/2} and satisfy  1 ≡ 1 whenever || 2 ≤ 1/3.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 0 < 2 < 1.This assumption is not essential by tracking the following proofs.
Proposition 7. Let  ,0 and Ω ,0 be defined as above.For all  > 0, one has Proof.The estimates of two inequalities are the same, so we will prove the first one only.
(i) If (1 + ) −1/2 || ≤ 1 and 0 <  ≤ 1, then it is obvious to see (ii) If (1 + ) −1/2 || ≤ 1 and  > 1, then by scaling Since we have (iii) If (1 + ) −1/2 || > 1 and  > 1, then by (ii) we know Using the Leibniz rule, we have Observe that For  ≥ 1, using an induction argument we have where   ≥ 0,    () ⪯ || 2− , and For each fixed  ∈ R  , 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  = 1, 2, . . ., .The other terms can be treated easily by further taking integration by parts.We let Φ be a  ∞ radial function satisfying . By the partition of unity we write We note that  > 1, and the support of  1 ( −1/2 ) together with (28) implies Therefore By integration by parts, Here, an easy computation gives For  2 , noting that and || > 1 and 0 <  ≤ 1, then a similar argument, without scaling, shows that The proposition now follows from (i)-(iv).
Proposition 8. Let  ∈   (R  ).Then for any  > 0 and 0 <  ≤  < +∞, Particularly, we have Proof.We prove the proposition for the kernel  ,0 only, since the proof for the other one is exactly the same.Let us first consider the case  = +∞ and 0 <  < 1. Invoking an interpolation argument [45,46], we may assume that (1/ − 1) is a positive integer.Thus the dual space of   is the homogeneous Lipschitz space Λ (1/−1) (R  ) (one can see the definition in [46]), which is exactly the homogeneous Hölder space Ċ (1/−1) (R  ).By duality we have If  ≥ 1, it is easy to check that sup where () is a homogeneous polynomial of degree (1/−1).Thus, using the same argument as before we obtain ⪯ sup This shows that, for all 0 <  < 1, On the other hand, if we write then by checking the proof of Proposition and then prove the mapping properties of the convolution operators with the above kernels.As in Section 2.1, we assume 0 < 2 < 1 without loss of generality.
Proposition 9.For all  > 0 and  > 0, we have Proof.If (1 + ) −1/2 || ≤ 1, then the proof is the same as (i) and (ii) in the proof of Proposition 7.So we assume (1 + ) −1/2 || > 1 and  > 1.In the case of  ≤ 1, we use the same proof as the following argument for  > 1, without taking the scaling kernel.
For  > 1, consider the scaling kernel By the Leibniz rule, (61) In fact, using Taylor's expansion, we have cosh Then by an easy computation, Abstract and Applied Analysis 7 Thus, by the induction, we have where Since If /4 ≤ || 2 ≤ , (69) then is a consequence of ( 61) and (28).
When 5/4 < || 2 ≤ 200, similar to (33), we get which is further bounded (note also  > 1) by Thus we have proved (69).Fix an  ∈ R  and let   be the variable such that   > ||/.Using integration by parts (+1) times on   , we obtain By ( 61), (69), and the compact support of  2 , we have The second term  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.
Proof.We will show the case  ≥ 2 and leave the easy case  = 1 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  > /2 and some  > 0, the proposition easily follows by a complex interpolation on these two inequalities for 1 ≤  ≤ 2. 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  =  > /2.
Let Φ be a standard cutoff function with support in { : Defining for some  > 0. By the definition, without loss of generality, we may write Using the Taylor expansion with integral remainder, for  ∈ supp(Φ), we write where This gives Let sets  1 ,  2 , and  3 be defined as where For each   1, R , , using integration by parts on the   variable, it is easy to obtain that, for  = 1, 2, . . ., , 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 .

Proof of Theorems 2 and 3
These two inequalities are obviously true if By Proposition 8, we have This indicates that, for any  > 0, there exists a positive constant  independent of  and  such that This shows that  ,0 () is a bounded mapping from   (R  ) to the mixed norm space  ,∞ ([0, ∞],   (R  )) for any admissible triplet (, , ).Now we choose admissible triplets (,  1 ,  1 ) and (,  2 ,  2 ) satisfying Then by the Marcinkiewicz interpolation, we easily obtain Similarly we can show that, for any /2-admissible triplet (, , ), Proof of Theorem 3. By checking the above proof, we only need to show the following proposition.
Proposition 12.There is a hold for all 1 ≤  ≤ ∞.
where  3 is defined in Section 2.3 (corresponding to  = 1).We will prove, for any  > ( − 1)/2, that with some  > 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   () by By Young's inequality, it suffices to show that if  > ( − 1)/2, then      F  (, ) Let Φ 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  ≥ 1. Changing variables, we have Using the Taylor expansion with integral remainder, for  ∈ supp( 3 ), we write where This gives for  ≥ 6 and 1/2 ≤  ≤ 2. By the definition of  it is easy to see that if we denote ℎ() = (1/2  ), then Also, for any integer  ≥ 0, uniformly for  ≥ 10 and 1/2 ≤  ≤ 2. When using the known estimate it is easy to see Thus, When we use the asymptotic expansion of  (−2)/2 (): for any integer where  1 ,  2 , . . .,   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 In  1 , noting −(−1)/2 > 0, we use the lemma with  = 1/2 and  = 1 : Similarly, in Lemma 13 we let  =  = : Using Lemma 13, we write Here, the last term Use the polar coordinate and Lemma 13 for  = 1/2: Abstract and Applied Analysis 13 Similarly, we can show When 0 <  ≤ 1, the proof is the same with only minor modifications.

Almost Everywhere Convergence
Next we will study the pointwise convergence of the solution (, ) of ( 5) to the initial data.We will prove the following.
When  > 0, we have Abstract and Applied Analysis Here we have to let On the other hand,