Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 428909, 17 pages http://dx.doi.org/10.1155/2014/428909

Research Article

Space-Time Estimates on Damped Fractional Wave Equation

Jiecheng Chen,1 Dashan Fan,2 and Chunjie Zhang3

1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

2 Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI53201, USA

3 School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

Correspondence should be addressed to Chunjie Zhang; purezhang@hdu.edu.cn Received 26 July 2013; Accepted 27 January 2014; Published 4 May 2014 Academic Editor: Nasser-eddine Tatar

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.

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 ^ 0+ is also studied, with the initial data satisfying certain regularity conditions.

1. Introduction

Let (t,x) e [0,œ) x r", a > 0, a,b e c, and let A be the Laplace operator. We consider the following Cauchyproblem:

autt (t) (x) + 2but (t, x) + (-A)au (t, x) = 0, (1)

with initial conditions

u(0,x) = f(x), ut (0,x) = g(x). (2)

Here, as usual, the fractional Laplacian (-A)a is defined through the Fourier transform:

(=ATf(t) = \t\2af(t) (3)

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 a = 1 and it is the half wave equation when a = 0,2b = i, and a = 1/2. As known, the wave equation is one of the most fundamental equations in physics. Another fundamental equation in physics is the Schrodinger equation which can be deduced from (1) by letting a = 0, 2b = i, and a = 1. The Schrodinger 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 a = 1.

As we all know, wave equation, Schroodinger 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 2but(t,x) in the wave equation, one obtains the damped wave equation

utt (t, x) + 2but (t, x) - Au (t, x) = 0, b > 0. (4)

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:

utt (t, x) + 2but (t, x) + (-A)au (t, x) = 0,

u (0, x) = f (x), ut (0, x) = g (x),

where a,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 a = 1. So our plan is to first study the linear equation (5) and to prove some Lp ^ Lq 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

ub (f, g) (t, x)

e-ht cosh (t^L)f + e-ht (bf + g)

where L is the Fourier multiplier with symbol b2 - \Z,\2a (see Appendix). Thus our interest will focus on the operators

T^b (t) := e-bt cosh (t^L), -btsinh (t^L)

Sa,b (t) := e

Using dilation, we will restrict ourselves to the case b = 1 so the theorems are all stated for u(f,g) = u1(f,g) (see Remark 6). We now denote

Sa (t) = Satl (t) = e

tsinh (t^1 - (-A)01) - (-A)a '

Ta (t) = Tai (t) = e-t cosh (t^1-(-Af).

These two operators are both convolution. We denote their kernels by Qa(t) and Ka(t). Thus, we may write

Ta (t) f = Ka (t) * f, Sa (t) f = na (t) * f. (9)

To state our main results, we need the following definition of admissible triplet.

Definition 1. A triplet (p, q, r) is called a-admissible if

- < a(---

q \r p

where 0 < r < p < +œ, r < q < >x>, and a > 0.

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

Theorem 2. Let a > 0 and let (p, q, r) be n/2a-admissible and 1 < p < Thenfor any ¡3 > na\1/p - 1/2\, one has

\K (t)*f\\U.K¥t)

If \K (t

\\Hr(R")

\\Hr(R")

IIK(R")'

Here, Lpy(Rn) denotes the homogeneous Sobolev Lp space with order y, and H denotes the real Hardy space.

Theorem 3. Let a = 1, (p,q,r) be n/2-admissible and 1 < p < Then the damped wave operators satisfy

LP(R")dtj - \\f Hh^R") + \\f lltp„(R")'

I \\^1 (t)*f\\

o \\Q1 (t)*f\\lp( W)dt

^ \\/\\Hr(R») + \\J\\lp (R»)>

for any ß> (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 a > 0 and let (p, q, r) be n/2a-admissible and 1 < p < +ot. For the solution u(t, x) of (5), one has

1/q dt) <

\\Hr(R") '

\\lp„(r») + \\Jm_

\\Hr(R")

») + Ma

Theorem 5. Let a = 1, (p,q,r) be n/2-admissible and 1 < p < The solution u(t, x) of the damped wave equation satisfies

U\\LP( R»)dt) <

\\Hr(R")

») + \\/\\lp.(R") +

+ \\g\\Hr(R") + \\g\\L^1(R")'

for any ß> (n - 1)\1/p- 1/2\.

Remark 6. For (5) with general b > 0, it is not hard to see that Ta,b (t)f(x) = Ta l (s)fb (b1/ax),

f 1 \ (!5)

Sa,b (t)f(x) =Sa,i (s)(lfh)(b1/ax),

where s = bt and fb(x) = f(b-1/ax). Therefore,

ub (f, g) (t, x) = u (fb, ±gb) (bt, b1/ax) (16)

and by applying Theorem 4, we have

j^p dt

-b^r-m-uq (mHr + b-1Mw) (17)

+ b-p/a-1/« (Ml? +b\\f\\LU + \\g\\LU )■

For a = 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 < 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.

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

Ka (t) (x) = e— \ cosh (iVHC)

Jr" v /

5,nh (WHiT),. (19)

Q, (!) (x) = e- I -V, >-e{'l)dl

Using the linearization

(1 + S)1/2 -1 + -, 2

for small j£j,we have

cosh —

- cosh (i(l -

et(1-|?|2"/2) + e-t(1-|Ç|2"/2)

Thus for small j£j,

cosh ( £ A/1 —

t t(1-|?|272K 0-^-t(1-|?|2"/2)

-t|?|2"/2

This indicates that, for near zero, Ta behaves like the fractional heat operator (see [11, 29, 30, 36, 37]). For large |£|,we similarly have

cosh (fVHÎr) = cosh (if^" V(l-I^r2"))

git|?r - g-it|?r

This indicates that as |£| near >x>, e*Ta behaves like the wave operator if a = 1 and like the Schrodinger operator if a = 2; see [12,16, 38, 39].

In the same manner, the operator Sa (t) behaves the same as the operator Ta. 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 ^ 0+. The similar convergence theorem for Schrodinger operator e'tAf(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 Ka(t) and Qa(t) based on their different frequencies. So we will divide this section into several subsections.

2.1. Estimate for near Zero. Let 01 be a Cm radial function with support in e r" : |£|2" < 1/2} and satisfy 01 = 1 whenever |£|2a < 1/3. In this section we are going to obtain the decay estimates on the kernels

Kafi (t) (x) = e-t £» 01 (Ç) cosh (i^l-I^I2")

sinh MHKf),£)

Qa>o (i) (x) = e-t I 01 (Ç)-\

With those decay estimates, we then are able to obtain two H^ bounds for the convolutions with the above two kernels. Without loss of generality, we assume 0 < 2a < 1. This assumption is not essential by tracking the following proofs.

Proposition 7. Let and Q 0 be defined as above. For all t > 0, one has

Ko (i) (*)| * (1 + i)-"/2"(l + (1 + i)-1/2a |%|)" Ko (i) (*) | * (1 + i)-"/2a(1 + (1 + i)-1/2a |x|)

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

(i) If (1 + i)-1/2,|x| <1 and 0 < t < 1, then it is obvious to see

Ko (i)(*)|*\ |0i (26)

(ii) If (1 + i)-1/2,|x| <1 and t > 1, then by scaling

= i"/2V' \R" 01 (?) cosh (f VHSf) e'^^ (27) = e- | 01 (i-1/2acosh ( Since

e- cosh (^i2 -i|?|2")

* exp I

1 + V1-I?r/i

|?|2"/2

we have

Itn/2aKa0 (t) (t1/2ax)\ < j k (i—1/2^)\ e—|?|"/2^ < 1.

(iii) If (1 + i)—1/2a|x| >1 and t > 1, then by (ii) we know

tn/2aKa0 (t) (t1/2ax)

|Rn 01 (t-1'2^) cosh )

Using the Leibniz rule, we have

dl (<¿2 (i"1'2-?) cosh (V^MT))

= 1 (*2 (^'^M, (cosh (^

Observe that

( cosh ( Vi2 - i

sinh (Vi2 -ffl*)

i2 -i\ç\

m2a-%.

For k> 1, using an induction argument we have

3k (cosh (V^^f )) = *('.*)!

k ¿-v; ^

(Vi2 -W

where ek > 0, < |£|2;a—k, and

F(i,^) = cosh (V^2 -ii^i2-) or sinh (V^2 -ii^i2-)-

For each fixed x e Rn, there exists at least one variable xt such that > |x|/m. By integration by parts n times on the variable we obtain

tn'2aKafi (i) (i1/2-x)

= ¡■1 X ck i Ck (01 (r^K

k=0 JK

x (cosh (Vi2

1 n— 1 k

= ^ Hcne—t k=1j=1

jR„ Ck (01 (^—1/2a *))*('• a i-vk (?)

> -^\2 n

■i j 01 (r1/2a:^(a)

® - ^

\ 2a \

i2 ) x cosh (Vi2 -i^ry«'*^.

The main terms needed to be estimated are

i-Vn (?)

-1e—t j 01 (r1/2^)F(a) ^n Jr» v 7

V^2 -i|^|2a)

with j = 1,2,..., n. The other terms can be treated easily by further taking integration by parts.

We let O be a Cm radial function satisfying 0(£) = 1 if i^i < 1 and O(0 = 0 if i£i > 2. Let ¥(£) = 1 - O(£). By the partition of unity we write

1 e—t j 01 (r1/2^)F(a) ^n Jr» v '

tW (?)

-1e—t j 01 (f—1/2aÇ)®(|*|Ç)F(a)

Jr» v 7

+ —e

1e—t j 01 (i—1/2aÇ)¥(|x|Ç)F(i,Ç)

en JR» v ;

V'2 - 'ii\î

= A + *2-

We note that t > 1, and the support of ^(t 1/2a£) together with (28) implies

. - Vf2-»!?!2" ^ I I * e e ' * exp ( --

Therefore

i e-!?|2"/2 |*1 (t-1/J*^(WW^

|x| Jr» 1 1

* i f itf^

By integration by parts,

* —77e

(i-1/2a^)F(i,^)

tj< (?)

t2 -t|rï+

= /1 + /2-

Here, an easy computation gives

M J|f|>1/|x|

|x| J!?!>1/M

|*r+1' |x|"+2a 1 '

For /2, noting that

and T'(s) is supported in [1/2,2], we have

J1/2!x!<!?!<2/!*!

(iv) If (1 + t) 1/2>| >1 and 0 < t < 1, then a similar argument, without scaling, shows that

Ko (t)(*)|*

,i«+2a *

(39) The proposition now follows from (i)-(iv).

Proposition 8. Let f e Hr(r"). Thenforanyt > 0 and 0 < r < p < +œ,

|K*(t)* f||№(R») *(l+t)-("/2«)(1/r-1/^)|

11 ^^^ ( R»),

llP-o(t) * /IIhpcr-, ^(1 + t)-("/2")(1/r-1/^)||/||Hr(R„).

Particularly, we have

K0(t)*/||L~(R») ^(1+t)-"/2"r||/||нr(R„), Ko(t)*/||L~(R") ^(1+t)-"/2"r||/||Hr(R„).

Proof. We prove the proposition for the kernel Ka>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 Hr is the homogeneous Lipschitz space An(1/r_1)(r") (one can see the definition in [46]), which is exactly the homogeneous Holder space C"(1/r_1)(r"). By duality we have

Ko(t)*/||L~ * supKo(t)(*--)||

xeR" v '

If t > 1, it is easy to check that sup||Ka>o(t)(x - -)||

= sup ||^a,0(t)(t1/2"x - ■)||p»(i/r-i)

xeR»"

*t-"/2ar sup If 01 (t-1/2aÇ)P(Ç).

xeR» I JR»

x cosh (Vt2 -t|?|2a)ei<?'">d?

where P(£) is a homogeneous polynomial of degree n(1/r-1). Thus, using the same argument as before we obtain

Ko(t)*f||L~ *t-"/2"r||

If 0 < t < 1,

sup||K«>Q (i) (% - -)|| ¿"(l/r-l)

sup I f (Ç) P (Ç) e-t cosh (iVl-|?|2")

xeR" IJ R" V '

This shows that, for all 0 < r < 1, On the other hand, if we write

llHr •

m (i, Ç) = ^(i) (Ç) = (Ç) cosh (iV1 -then by checking the proof of Proposition 7, we find

for all multi-indices fc. So by the Calderon-Torchinsky multiplier theorem [47], we also have, for all 0 < r < 1,

Kott * /IIht *

llHr •

Now interpolating between (51) and (54), we finish the proof for 0 < r < 1.

For the case 1 < r < +œ, we use Young's inequality to get

Ko *(f)/L = Ko(i)||L,||/L> (55)

where 1/r + 1/^ = 1/_p + 1.By Proposition 7,

«,o (t)lli«

< (f |(1 + i)-/2«(l + (1 + i)-!/2« < (1 + i)-(«/2«)(1/r-1/p)

(56) □

2.2. Estimate for Lying in tfoe Mid-Interval. Let 02 be a Cm radial function with support in e r" : 1/4 < |£|2a < 200} and satisfy 0: = 1 whenever 1/3 < |£|2" < 100.We first will obtain the decay estimate on the kernels

^ (i) (*) = e-t 02 (?) cosh (i^1-|?|2")

Q«>m (i)(x) = e-t f 02 (?)-V

sinh (i^!-!?!2«)

and then prove the mapping properties of the convolution operators with the above kernels. As in Section 2.1, we assume 0 < 2a < 1 without loss of generality.

Proposition 9. For all i > 0 and N > 0,we have

Km (i) Ml * (1 + i)-/2-N(1 + (1 + i)-l/2« Mp2a,

Km (i) (*)| * (1 + i)-/2-N(1 + (1 + i)-1/2" jxj)-"-2".

Proof. If (1 + /-)-1/2ajxj < 1, then the proof is the same as (i) and (ii) in the proof of Proposition 7. So we assume (1 + i)-1/2>| > 1 and t > 1.In the case of t < 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

i^m W(i1/2a*)

f 02 (?) cosh (iV1-|?|2a)e,<?^>d^ (59)

(53) _ e-t fR» 02 (i-1/n) cosh ( V^2 -i|?r)

By the Leibniz rule,

ag (02 (r1/2<^) cosh (V^MT))

_fC^ (02 (i-1/2«?))9^ (cosh (^

Next we prove the following estimate: (cosh (V^2 -i|?|2"))

In fact, using Taylor's expansion, we have cosh (V^ ) = f

Z(i2 -ik|2«)'-1

Then by an easy computation,

a, (cosh (v^r)Xf (2Z)! 92 (cosh (Vi2 -i|?r))

= f ^D^ ^ I * I T

■ |2«\'-1

+f ^12^(|?r4?12

Abstract and Applied Analysis Thus, by the induction, we have (cosh (V^2 -^l2"))

Z(Z-1)---(Z-j+1)(f2-^|2»)

£ £ (2W

we obtain

vf (?) * |?|2ja-fc-

/(/-1)---(/-;+l) < 1

(2 (/-;))!'

Z(Z-1)...(Z-j+1)(f2 -^T

M (2 (/-;))!

So (61) is proved. Note by the compact support of 02(i 1/2"£), we have

- < |?|2" < 200i, 4 lsl '

and we will prove, for all such

cosh ( Vi2 -i|?|2") * & > 0.

If i/4 < |?|2" < i, (69) then is a consequence of (61) and (28). If t < |?|2a < 5i/4, then

tgVtii^ = eVt|?i2"-t2-t < g-f/2 ^ e-2|?|2"/5

When 5i/4 < |?|2" < 200i, similar to (33), we get e-ta| (cosh (¿Vi|?|2a -i2))

which is further bounded (note also t > 1)by

j=i j=i

Thus we have proved (69). Fix an x e r" and let x{ be the variable such that x{ > |x|/n. Using integration by parts (n+1) times on , we obtain

i"/2"(i)(i1/2"x)

k=0 JR

x ^ (cosh ( Vi2 -i|?|2")) e<?'*>d?

Ick+i j 9"-k+1 (02 (m))*"*

k=1 JR

x (cosh ( Vi2 -i|?|2a))ei<?'*>^

JL L dj*1 (O).-

x cosh (Vi2 -i|?|2") e,<?'*>d?

= *1 +^2

By (61), (69), and the compact support of 02, we have

-N/2«

H^U IC*' * (r№{)l

-1 K+1 f r

x i(-K+fc-1)/2a+j+N/2ae-fc|Ç|2'

x |^|2ja-fc#

1 K+1 f r -,

H^L- lir^v*- «

-N/2«

-N/2« 1

The second term K2 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 e Hr(r"). Thenforany t,N > 0 and r < p < +œ,

llHr(R")'

llHr(R")'

KmW * /||№(R")

IKmW * /||№(R") *(1+i)~ Particularly, we have

IIWWIUr») * a+ ^11/11^ (R" ).

* /llL~(R») * (1 + î)-nII/IIH^(R»)•

2.3. Estimates for |£| near the Infinity. Let 03 be a Cm radial function with support in e r" : |£| a > 2 } and satisfy 03 s 1 whenever |£|2a > 100. Defining

(i) (x) = e-t |rb 03 (Ç) cosh - 1 )

f sinh (ii

(x) = e-t I 03 (Ç)-\

Jr" ¿v

and -1

we have the following proposition.

Proposition 11. Let 1 < p < +œ> and a > 0. Then there exists a >0 such that for any p > na|1/2 - 1/p| and t > 0, we have

IK™ W*/|U") ^a + f)*' ||/|U

Proof. We will show the case n > 2 and leave the easy case n = 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

Tz (/)(*)

= e-t i 03 c.

By the Plancherel formula, we have

||TZ(/)||L2(R")

IIl2(r")

||TZ(/)||LI(R") < (1 + t) e

for Re z > «a/2 and some A > 0, the proposition easily follows by a complex interpolation on these two inequalities for 1 < p < 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 = ^ > na/2.

Let O be a standard cutoff function with support in : 1/2 < |£| <2} satisfying

V£ =0.

Defining Wf (f)/(*)

= e-t I ¿'V^o (2"'Ç) 03 (Ç) / (Ç) e^'0^,

then (81) will follow if we prove

||Wf(i)/(x)||il * e-t(1 + i)A2J"a/2||/||Li• In fact, (84) implies

£||2-("a/2+E)j w? (i)/(x)||Ll

< ^2-jE(1 + i)

Ve > 0.

Noting that 2 ' - |£| in the support of O(2 we get (81) from the above inequality.

Next we prove (84). Let Ra>j be the kernel of W"(f). By Young's inequality, it suffices to show

||R«y ||LI ^Ü+i)A2j"a/2,

for some A > 0. By the definition, without loss of generality, we may write

(*,f) = e-< f ®(2-'"

= e-t2J" f O^e^'I^V2'«^.

Using the Taylor expansion with integral remainder, for r e supp(O), we write

Vr2a - 1 = r" + «(-),

r« fi

^(r) = -T f0 (l -Sf2a)

for Re z = 0. (80) This gives

eitV22i"r2"-1 = eit2iar" eit3(1/r2ia)

for 2ay >100 and 1/2 < r < 2. By the definition of g it is easy to see that for any integer m > 0

,»t3(1/2t"r)

uniformly for 2aj > 100 and 1/2 < r < 2.

Abstract and Applied Analysis Now we write

R„ , (x, i) = e"t2J" f e'^'^O tf) e^(1/2J"|?|)d?, (92)

where the phase function p is defined as

p(Uj) = i2^r + 2'"&*>.

Let sets £1, E2, and £3 be defined as £1 = (x£ r" : |x| >Mi2j(a-1)j, £2 = (x£ r" : |x| <rni2j(a-1)j, £3 = (x£ r" : mi2j(a-1) < |x| < Mi2j(a-1)j , where

M = n2a+100 max -¡a,1

i a J and m = 1/M. Hence,

I I R«J I I L1 = I I ^ I I L1 + I I XE2I I L1 + I I I I L1 > (96)

where denotes the characteristic function of a set £. Furthermore, we let

£1,rn = e £1 : km| > N for i = 1, 2,..., (97) for m = 1,2,...,«. Then

IIXe, R«,j||Li < Z R«>.

■jIIL1'

For each ^ Raj, using integration by parts on the variable, it is easy to obtain that, for m = 1,2,..., n,

im (*) (x, i)| * e-t2-%,„ (*) min {l, (2j j*j)~

for any positive number N. By the polar decomposition,

Xe2 (*) R«,j i)

= e-t2j" | 1 (J™ O0 (r) e'^dr) da (?')

where the phase function P is defined by

P(r,i,j) = i2ja r" + 2jr(^',x),

O0 (r) = 0(r)eWr). Using integration by parts on the inner integral, we obtain

|*e2 (*) R«j (*, i)| * (x) min {l, (i2^")-N} ,

for any positive number N.

By the Proposition in [48, page 344], K (*) Rj (*, i)| * (x) min {l, (i^)

Thus, if i2ja > 1

lh rJL 1 2j" j (i2ja)-"/2dx

-i ^n/22j«w/2

(94) If í2ja < l,

Xk,*e-t2;" Í

* e-^)" < e-t.

For ^ R«.-, if í2;" <1,

IIxe rJL 1 *e-t2j" Í dx

- e-^)" - e-t. If i2J" > 1 and then we choose N = n,

IkRJL - e-t2J"(i2ja)-N(i2j(a-1))"

= e-t(i2ja)-Ni"2j"a - e-t. Finally we estimate ^ R •. For each m = 1,2,..., n,

IIxei„R JL1 - f ( min {1, ^ |x|)-N} dx.

If the set jx :

- |x| - 2 is not empty, we write

|kmR«,j||Li *e-t2j" Í i> . dx 11 1,m Jt2J("-1>i|x|<2-J

+ e-t2j" Í min {l,(i2ja)-N}dx J2"J'<|x| 1 J

J2-J<|x| = /1 +/2.

Clearly

/1 < e-t.

Also, choose a sufficiently large N, and then

72 *e-t2j" Í (2j jxj) Ndx

* e-t.

If the set {2 J > jxj > í2j(a 1)} is empty, then we also have

mR«,;||ii * *2'n Í . (2^ jxj)

11 1,m J|x|>2-J

The proposition is proved.

dx * e (112) □

3. Proof of Theorems 2 and 3

Proof of Theorem 2. Recalling the definition of

(i), Ka>m (i), (i), Qa>o (i), Qa>m (i), Qa>m (i)

in Section 2, we have

K (f)*/|

-K,0 (0*/| + Km (0*/| + K>m (i)*/|,

K (i)*/|

- K,0 W*/| + K,m W*/| + KTO (i)*/|-

By the triangle inequality and Propositions 8, 10, and 11, we only have to verify that, for any n/2a-admissible triplet (p, <?> r),

llHr(R")'

(Jo KoCWllV)*) (R»)-

These two inequalities are obviously true if

1 n (1 1 ^ 2a V r p

For 1/^ = (n/2a)(1/r - 1/p), denote

F(i)/ = ||X«,o(i) */||iP(R-)-By Proposition 8, we have

F(i)/^(1+i)-1/?||/||Hr.

This indicates that, for any A > 0, there exists a positive constant C independent of A and / such that

|{i:|F(i)/|>A}|

< |{i : Cf

-i/?||

This shows that -Ka>0(i) is a bounded mapping from Hr(r") to the mixed norm space L?'œ([0, ot], L^(r")) for any admissible triplet (_p, r). Now we choose admissible triplets (p,<?i,ri) and (p, satisfying

^ < r < r2 < OT, ^ < ^ < ^2 < OT,

1-1. 1—^ <? <?1 <?2 '

1=1 I"0

r r, r9

Then by the Marcinkiewicz interpolation, we easily obtain

Ko w*/llV)di

llHr(R")'

Similarly we can show that, for any n/2a-admissible triplet (p, <?>r),

' 1-TO \ 1/?

Jo Ko(')*/||V)diJ -||/||№). (122)

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

Proposition 12. There is a <5^ > 0 for which if p > (n-1)|1/p-1/2|, then

*/ILr») *(1 + f)s'e-

Nl^r»)'

*/IUr») ^(1+i)Spe-tll/lli^R»)

hold for all 1 < p < ot. Proof. Let

tVii2-

W^ (i)/(x)-e-t f

^ /(O^dfc (124)

where 03 is defined in Section 2.3 (corresponding to a = 1). We will prove, for any ^ > (n - 1)/2, that

||Wj3(i)/(x)||Ll ^e-t(1+i)A||/||Ll, (125)

with some A > 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^f) by

©^ (x,i)-e-t f

03 (?)

e'^dÇ.

By Young's inequality, it suffices to show that if ^ > (n - 1)/2, then

||©^'Î)||lI < (1 + i)("+1)/2e-t.

Let O be the cutoff function defined in Section 2.3. Then we have

;(X'i)-e-t X I fc=6J»

œ r e»iV!ir-i

tVi?!1

0(2-fc |?|)03 (|?|)e'<s'^

where, by [49, Ch. 4],

^ViiF-i

Î'1 V is! -i

Îœ pitVr2-1

o —— 03 (r) O (2-fcr) V(„-2)/2 (r M)r"-1dr.

In the last integral,

V„ (s) =

/] (s)

and /v(s) is the Bessel function of order v. So, by the Minkowski inequality,

IF ML XMI* • (131)

First, we assume t > 1. Changing variables, we have

Yfc (*)

|..|(2-n)/2 /-TO I-;-

W [ e"iV2^O(r)03 (2fcr}/(„-2)/2 (132)

x(2fcrM)r"/2-^1/2dr.

Using the Taylor expansion with integral remainder, for r e supp(03), we write

This gives

Vr2 - 1 = r + g (r), f

gitV22tr2-1 = eit2lreit3(1/2lr)

for fc > 6 and 1/2 < r < 2. By the definition of g it is easy to see that ifwedenote fe(r) = g(1/2fcr),then

|k(m) (r)| * 2-fc. Also, for any integer m > 0,

>f0(2*r)

,>t0(2*r)

* 1 if t2 K < 1,

* (i2 fc)m if t2 fc >1

uniformly for fc > 10 and 1/2 < r < 2. When

2fc M < 24,

using the known estimate

Z(„-2)/2 (r) = O (r("-2)/2) , as r —> 0, (139) it is easy to see

№ Ml*

.|(2-«)/29-fc03-«/2-1)(2fc |x|)("-2)/2 = 9-fc(^-«)

fc=6 fc=6

Y2-fc(^-K) f

dx * e

2fc |x| > 24,

we use the asymptotic expansion of 7(„_2)/2(r): for any integer N > 0,

(r)="±" ^ + O (r-(N+1)-1/2) , (143)

r7+1/2

\i=or /

where q, c2,..., cN are constants. In this case,

Yfc (x)

N |x|(1-n)/2-i

'J 2fc(^-«/2-1/2+j)

x fTO eöVttMy^Mo (r) (2fcr) r"/2-^'dr

+ O (|X|(-"+1)/2-N-12-fc(^-"/2+1/2+N))

= ICjyfc>j (x) + O (|x|(-"+1)/2-N-12-fc(^-"/2+1/2+N)) , j=0

where, without loss of generality, we denote

Yfcj (x)

|(1-«)/2-j

2fc(^-«/2-1/2+j)

x fTO ¿^MV*^ (r) 03 (2fcr) r"/2-^'dr. It is easy to see that, for a suitable integer N,

TO /•

l2-fc(^-"/2+1/2+N) f M(-"+1)/2-N-1dx * e-. ^ Jixi>2-t+4

Thus it remains to show that, for each j,

HMM>2-(||lI - i("+1)/2e-t. (147) fc=6

Since the estimates of all Yfc- are similar, we will only show

e- Zll^fc.oXli. fc=6

M>2-*+4}||Li * i("+1)/2e-t. (148)

Using integration by parts and noting 03(2fcr) = 1 if k > 2 and r e supp(O), it is easy to check that one has

J J™ e,2*,.(t-W)e,ti?(2M0 (f) w (2fcr)

* min|l)2-mfc|i-|x||-m(2-fci)m} if 2-fci > 1, for any positive integer m, and

J J™ ^^wy^M 0 (f) ^ (2fcr) r"/2-^di

* min jl,2-^fc|i-

if 2 t < 1, for any positive integer Thus, we have the following lemma. □

Lemma 13. Let 2fc|x| > 10. For any m > 0, one has

|7fc>0 (x)| < W(1-")/22-fc(£-"/2-1/2)2-mfc|i - |x||-m (151)

if 2-fci < 1.

Also, for any ^ > 0,

(*)| * |x|(1-")/22-fc(^-"/2-1/2)2-^fc|i- |x||-^(2-fci)"

if 2-fci > 1.

Now we continue the proof of the proposition. Write

XI|Yfc>0^li^i>2-l+4}||Li(R»)

œœ ç

œ /•

■Il №,0 (*)|d*

fc=6 J|x|>100t

fc=6 J10/2t<|*|<t/2

+ Z| |yfc,0 (*)

^=6 Jt/2<|x|<100t

— 1 + ^^ 2 + ^^ 3*

In A 1,noting^-(n-1)/2 > 0, we use the lemma with 1/2 and m — 1 :

A1 * Z2-fc(^-"/2+1/2) fc=6

'10/2fc<|x|<t/2

|*|(-"+1)/2 d*

+ T1 Z 2-

fc=log t+1

-fc(^-"/2+1/2)

J10/21 <|x|<t/2

|*|(-"+1)/2d*

("+1)/2

Similarly, in Lemma 13 we let ^ — m — n:

x|(1-")/2-"2-2"fcdx

J100t<|x|

2 S J"

+ Z 2-fctf-"/2-1/2) | 2-"fc|x|(1-")/2-"dx

,__. 1 J100t<|x|

fc=log t+1 ("+1)/2

E — {xe r" : 2 < m < 100i} ,

£fc —jxe r" : |f- |%|| < 2-fc}. Using Lemma 13, we write

A3 * Z2

-fc(^-"/2-1/2)

x J |x|(1-")/22-^fc|i-|x||-^(2-fci)^dx

JE\Et ™

+ Z 2-fc(^-"/2-1/2)

fc=log t+1

xJ |x|(1-")/22-mfc|i-|x||-mdx ™ f

+ Z2-fc(^-"/2-1/2) J |x|(1-")/2d* fc=6

— + ^2 + ^3-

(153) Here, the last term

™ r 1

£3 — Z2-fc(^-"/2-1/2) J |%| —d* fc=6 je* 2

™ rt+2-fc

* y2-t(^-»/2-1/2) J r(1-")/2+"-1 * i("+1)/2

" è6 Jt-2-fc

Use the polar coordinate and Lemma 13 for ^ = 1/2:

^ — Z2-fc(^-"/2+1/2)i1/2 J |x|(1-")/2|i - |x||-1/2dx fc=6 ^

log t 100 t

* Z 2-fc(^-"/2+1/2)i1/2 ( J r("-1)/2(r - ¿)-1/2dr fc=6 VJt+2-t

ft-2-1 \ + J r("-1)/2(r-i)-1/2drj

log t / f100t

*tn/2 Y 2-k(ß-n/2+1/2) ( \ (r-t)-1/2dr te

+ [ (r-t)-1/2dr) Jt/2 )

(n+1)/2

Similarly, we can show

f T«P-nl2-H2) ( ixi(l-n)l22~mk\t-\x\\->»dx

k=log t+1

(n+1)/2

When 0 < t < 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 L2S(r") and g e L2sa(r"), then the solution u(t, x) of (5) converges to f(x) a.e. x e r" as t ^ 0+.

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

Lemma 15 (see [50]). Let n > 2 and 1 < d < n. Then

ig(u)ex'"da(u) [ \g (u)\2da (u).

R"|J S"-1 \x\d Js»-1

Proposition 16. Let n > 2 and let m(t, \?\) be defined on r+ x r" and satisfy

Denote the maximal function

m* f (x) = sup \m (t, D) f (x)

Then if y > 0, we have

llm7M|L?(W-^) * Ml?>

If y < 0, then

llm7M|L?(W-^) * Ml]'

'■-2y d

-- <l< -, d> 1.

l> d-y--> d>1. (165)

Proof. Making t into a function of x, we only have to bound

m(t(x),D)f(x)={ e,{xt>m(t(x),\Z\)f(Z)dZ, (166)

where t(x) : r" ^ r+ is any measurable function. By the polar decomposition,

\m(t(x) ,D) f (x)\

C -- /

rn-lm (t (x), r) J f (r?) e,(rx't }da (?) dr

j™ rn-1(1 + r)-y | £ 1 f (r?) e,(rx'^')da (?)

By Minkowski's inequality, change of variables, and Lemma 15, we have

¡¡m(t(x),D)f(x)\lmxrädx)

*\ rn-1 (1 + r)-y 0

HL /w^wo

= \ r"/2+d/2-1 (1 + r)-Y 0

L?(dx/\x\d)

L?(dX/\Xf)

t f(rï)e'^da(ï)\

< rn/2+d/2-1(1 + r)-y(js-1 ^(r^do^y'dr

Î2 rœ\

+ j )rn/2+d/2-1 (1 + r)-v

(js„-i mrïpa^fdr

= L1 + L 2.

When y > 0,we have

1 *j0 rn/2+d/2-1(js-i |f(rï)|2d<ï)) dr

d-1-2l , \

I r dr)

(Îo r2lr"-1 L- (169)

f0rd-1-2ldr)1/2(j„ w2«)1*

L?(R"Y

Here we have to let

On the other hand,

! - 1 -2Z < -1.

L 2 * I r'

if |/K)|2^')) dr

/ rrn \ 1/2

(i rd-1-2^ dr)

x(ifr2" L i/k)i2m?>)1/2 (171)

(i2°° ^ ^(L

- 11/lit2 (R")'

Obviously we have to let

d- 1 -2/-2y > -1, which, together with (170), implies d - 2a

< I < -. 22

If y < 0, then

- -(JO *) X L

Îœ \ 1/2

x(if ^d-1-2r^ ^ |/"(r^')|2da(^')dr)

Proposition 16 is proved. Proof of Theorem 14. Denote

mi (i;|^|) = e-t cosh (iVl-|?|2a),

sinh (iVH?r) m2 (^|) = ^-)-2- ' .

It is not hard to verify that

V(f,£) e r+ x r".

(174) □

u (i, x) = mx (i, D) / (%) + m2 (f, D) / (%) + m2 (i, D) g (%) = w (i, D) / (%) + m2 (i, D) # (%),

Theorem 14 will be proved if we can show, as i ^ 0+,

m2 (i, D) # (%) —> 0, a.e. xc r" (178)

for g c L2s-a(r") and

w (i, D) / (x) —> f (x), a.e.xc r" (179)

for / c L2s(R"). 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

w*/ (x) = sup (i, D) / (x)

||w*/(x)

L^dx/M^) - ||/||L2(R")' 5 > 2'

is enough to imply Theorem 14.

Next we will prove (181). By (176) and Proposition 16,

K/M^wi^) - ¡/L(RT

V/>d-1. (182) 2

Fix s > 1/2. Takng 1 <d<s+1/2 and I close to d - 1/2, we have I < s and thus

L2(dx/|x|d) - ||/||L2(R").

Applying Proposition 16 with y = a and 1 < d < 1 + 2a, we have

IK7MIL2 wi^) * II/IIl2„(r») ^ II/ILÎ(R-). (184)

w*/ (%) < m*/ (x) + m*/ (x),

we proved (181) when n > 2 (note that Proposition 16 was proved only when n > 2).

For n = 1, instead of (181), we will show

|K/(x)||T2(r _T]) - N

||L2(r-N,N])

' 5 > 1, (186)

which is also enough to obtain the pointwise convergence. Taking h(x) e L2([-N, N]), we have

w (t (x), D) f (x) h (x) dx

= \N f w(t(x),^)f(^)ei{x'(-)d^h(x)dx J-N JR

= f f(Z) fN w(t(x),Z)ei{x'0h(x)dxd$ Jr J-N (187)

f \f(S)\2{l + \S\2)Sdt)m

JR \ \ '

\ N \2 \ 1/2

f \lNw(t(x),Oei{x*)h(x)dx\

Noting that

w (t (x) ,$) = m1 (t (x) ,^) + m2 (t (x) V (x, £),

we have

Therefore

w(t(x),t)ei<x'°h(x)dx

* NllH^Ni). (189)

w (t (x), D) f (x) h (x) dx * Nm\\

and by duality

\\w(t(x),D)f(x from which (186) follows.

m-NN]) * N WJ Wl]

, s>-, (191)

Appendix

We study the Cauchy problem

utt (t, x) + 2but (t, x) + (-A)au (t, x) = 0,

(t,x)e[0,rn)x r", (A.1)

u (0, x) = f (x), ut (0, x) = g (x).

We claim that the solution, in the Fourier transform side, is given by

i(t,0 = e-ht cosh (t^b2 -\tfa)f(V

sinh (A-2) -V- 7 (bf(V + g(V).

To verify this fact, we write the solution as

u (t, x) = A (t, L) f (x) + T (t, L) g (x), (A.3)

-u sinh {t^L)

r (t, L) = e-

A (t, L) = be-

Take derivative,

sinh (tVL)

■e-ht cosh (tVL),

L = b2 - (-A)a.

2-tsinh (tVL)-be-ht cosh (tVL)

ut = - ( b e

xf- be-ht cosh (tVL) f + e-ht VL sinh (t VL) f

t sinh (t VL)

g + e ht cosh (tVL) g

= -b2e-htsinh (tfL) f + e-htb2 -(~AT sinh (tVL) f

+ e ht cosh (t VL) g -be ht ——V——) g.

t ±*r

sinh (tVL) f + eht cosh (tVL) g

sinh (tVL)

(-A)a .

utt = be ht V— sinh (tVL) f - be ht cosh (tVL) g

2 - h t

sinh (tVL)

g - e-ht(-A)a cosh (tVL) f

+ e-htVLsinh (tVL) g- be-ht cosh (tVL) g.

Therefore,

2behtut = 2b sinh (tVL) f + 2b cosh (tVL) g

sinh (tVL)

e utt = b

(-A)a . vl

sinh (tVL) f-2b cosh (tVÏ) g

.sinh (t^Z) . _

+ b2-V^0- ^A)a cosh (V) f

+ VIsinh (tVZ) g.

bt bt e utt + 2be ut

-(-AT Vl

b sinh(tVZ)f-b

,sinh (tVL)

7.2 _ /_A

- (-A)a cosh (tVL)f+-sinh (tVZ) g

-(-A)a vl

b sinh (tVZ)f- (-A)a cosh (tVZ) f

(-Ar...

= -(-A)a

sinh (tW) g

, _ sinh (t^L) cosh (tVZ)f+-^^ (bf + g)

utt + 2but = -(-Af (u)

ibsinh (tVL)

= -(-A)a

cosh (tVZ))f

sinh (tV~L)

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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[12 [13

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