Haibing WANGBin WU

1 Introduction

In this paper,we discuss an inverse problem of determining two unknown coefficients,a kernel function and a source function,depending only on time in a fractional wave equation with memory effect.Let Ω be a bounded domain in RN(N=1,2,3)with smooth boundary∂Ω,QT:=Ω×(0,T)and ΣT:=∂Ω×(0,T)for a given timeT>0.We consider the following fractional integrodifferential equation:

with the Caputo time fractional derivativeof order 1<α<2,defined by

where Γ is the Gamma function,see[16]or[23]for details on the fractional derivative.Here the operatorLis uniformly elliptic on,de fined by

whereAij=Aji,1≤i,j≤N,are smooth functions,and there exist positive constantsν1andν2,such that

We supplement the above fractional wave equation with the following initial condition:

and the boundary condition

Many modern science and engineering technology areas can be described very successfully by models using fractional differential equations(see[18,27,33,35]).The direct problem was extensively studied by many authors(see,e.g.,[2,15,17,22,26]and the references therein).In practical situations,the functionkrepresents some physical property,which is very hard to be measured directly in advance.So we consider an inverse problem of determining convolution kernel functionkfrom some additional measurements onu.

In this paper,we take the following additional conditions ofintegral overdetermination:

whereare defined by

with known functionsϕi.Heregi(t)are the measurement data representing the average temperature on a small part of Ω,because the weight functionsϕi(x)are usually chosen to satisfy Supp(ϕi)b Ω in practice.

The inverse problem considered in this paper is stated as follows.

Inverse ProblemDetermineu∈C([0,T];H2(Ω))∩C1([0,T];L2(Ω)),p∈C1[0,T]andk∈C[0,T]from(1.1)–(1.3)and the additional measurements(1.4).

As for inverse convolution kernel problems for the integer order integrodifferential equation withα=1 or 2,Colombo and Guidetti gave an efficient strategy to prove global in time existence and uniqueness results based by using analytic semigroup theory(see[7]).Colombo and Guidetti showed that a semilinear integrodifferential parabolic inverse problem has a unique solution global in time under suitable growth conditions for the nonlinearity involved in the evolution equation(see[7]).Lorenzi and Rocca[21]proved the local existence and the global uniqueness of an identification problem,which focuses on recovering two unknown convolution kernels in a phase-field system coupling two hyperbolic integro-differential equations.For inverse problems related to other models,we refer to[4–6,8–10,13,20].

Recently,the subject ofinverse problem for the fractional differential equations received much attention.Cheng,Nakagawa and Yamamoto[3]obtained the uniqueness in determiningαand a diffusion coefficient varying spatial variable on the basis of Gel’fand-Levitan theory.Sakamoto and Yamamoto[25]proved the well-posedness in the Hadamard sense for an inverse problem of determining a spatially varying function of the source by final over-determined data.As for other kinds ofinverse problems related to the fractional differential equations,we refer to[14,19,29–30,32,34].However,all these papers focused on the cases without integral term,i.e.,k≡0.To the authors’knowledge,there are no works published concerning the identification of the kernel function and the source function simultaneously for the fractional integrodifferential equation,even the single source function identification problem.

In this paper,we investigate the global existence and uniqueness of our inverse problem.First,we prove a local existence of(u,p,k)in a suitable Sobolev space by using a fixed point argument.Then,we give the proof of global uniqueness result.Finally,with the aid of splitting process of convolution term,which was successfully used to prove the global existence for the strongly damped wave equation in[5],we prove the global existence of(u,p,k).

Next we give some notations which will be repeatedly used in the sequent sections.

For any integerm,we denote byHm(Ω)the usual Sobolev spaces defined for spatial variable(see,e.g.,[1]).For a given Banach spaceVon Ω,we use the notationCm([0,T];V)to denote the following space:

We endowCm([0,T];V)with the following norm making it be a Banach space:

Forr∈L1(0,T)andq:(0,T)→V,we define the convolution

whenever the integral has a meaning.We next define Banach spaceXTby

XT:=C1([0,T];L2(Ω))∩C([0,T];H2(Ω))

with the norm

Furthermore,we set

YT=XT×C1[0,T]×C[0,T]

endowed with the norm

In order to discuss the uniformly elliptic operator−Lconveniently,we denote the domain of−LbyIt is well-known that the operator−Lhas only real and simple eigenvaluesλn,and with suitable numbering,we haveByφn,we denote the eigenfunction corresponding toλn,which satisfies1,where(·,·)denotes the inner product in Hilbert spaceL2(Ω).Then forγ∈R,we define the functionD((−L)γ)by

and thatD((−L)γ)is a Hilbert space with norm

Moreover,we introduce the Mittag-Leffler function in[23]

withα>0 andβ∈R.It is known thatEα,β(z)is an entire function inz∈C.

We now give the definition of weak solution to(1.1)–(1.3),which is introduced by Sakamoto and Yamamoto in[24].

Definition 1.1We call u a weak solution to(1.1)–(1.3)if(1.1)holds in L2(Ω)and u(·,t)∈for almost all t∈(0,T),u,∂tu∈C([0,T];D((−L)γ))and

with some

We make the following assumptions:

Remark 1.1In(H1),implieswhich will be used in Lemma 3.4 below.Indeed,by(see[23,(2.114)]),we have

whereis the Riemann-Liouville fractional derivative,defined by

for 0<2−α<1.Obviously,due to 0<2−α<1.Therefore,from(1.6)andwe conclude thatgi∈W2,1(0,T),→H1(0,T).

Remark 1.2In order to guaranteewe could give a usual regularity conditiongi∈C3[0,T],such thatIn fact,by integration by parts andwe have

This gives

because ofα−1∈(0,1).

Remark 1.3(H2)–(H3)are the consistency conditions for our problem(1.1)–(1.4)when dealing with smooth solutions.

Remark 1.4In engineering,ϕi(x)can be thought of as an internal(tiny)sensor(see[12,28])measuring the mean temperature in measurement area.Supp(ϕ)is always chosen small enough to make the measurement area very small.So hypothese(H5)is reasonable.

Our main result in this paper is the following global existence and uniqueness for our inverse problem.

Theorem 1.1Under hypotheses(H1)–(H5),there exists a solution(u,p,k)∈XT×C1[0,T]×C[0,T]to the inverse problem(1.1)–(1.4)for any T.

In order to prove Theorem 1.1,we need the following two lemmas.

Lemma 1.1Under hypotheses(H1)–(H5),there exists a sufficiently small τ>0,such that the inverse problem has a unique solution(u,p,k)∈Xτ×C1[0,τ]×C[0,τ].

Lemma 1.2Under hypotheses(H1)–(H5),for given measurement data gi(t)for i=1,2in(1.4),if the inverse problem(1.1)–(1.4)has two solutions(uj,pj,kj)∈XT×C1[0,T]×C[0,T](j=1,2)for any time T,then(u1,p1,k1)=(u2,p2,k2)in[0,T].

The proofs of these two lemmas will be given in Sections 3–4.

2 Preliminary Results

In this section,we present some preliminary results,including well-posedness for a fractional differential equation,an equivalent lemma for our inverse problem and a technique result,which will be used in the proofs of our main results.

We first consider the following initial and boundary problem:

Based on the results of[24],we will prove the following well-posedness of(2.1).Different from[24],we need a better regularity to construct the fixed pointed operator in next section.Moreover,the constantCbelow in(2.2)and(2.4)should be taken apart fromt,which is necessary to apply contraction mapping.

Lemma 2.1Let aThen thereexists a unique weak solution u∈XTto(2.1),such that

for all t∈[0,T],where the constant C is dependent on α,Ωand the coefficients of L,but independent of T.Furthermore,we have

For any t∈(0,T],we also have the following estimate:

Remark 2.1The estimate(2.2)will be used to construct the fixed pointed operator in the next section,which is the key ingredient to prove the local in time existence.In the proof of the global in time existence,we need(2.4)to extend repeatedly the local solution to a larger time interval.

To prove Lemma 2.1,we first give a property of the Mittag-Leffler function,i.e.,the following Lemma 2.2.

Lemma 2.2Let β∈Rbe arbitrary andµsatisfy<µ

The proof of this lemma can be found in[23],and we shall omit here.

Now we give the proof of Lemma 2.1.

Proof of Lemma 2.1We first split(2.1)into the following two initial and boundary value problems:

Foraby Theorems 2.2–2.3 proved by Sakamoto and Yamamoto[24],there exist uniqueC1([0,T];L2(Ω))andwsatisfying(2.6)and(2.7),respectively.And we have

Therefore,(2.1)has a unique solutionu=v+wgiven by(2.3).

Next we prove(2.2).By[24,Corollary 2.7,Theorem 2.3],we have

and

On the other hand,using[23,(1.83)],we have

from which it follows that

So,combining the above result with Lemma 2.2 andwe have

Similarly,we have

Here in the last inequality of(2.15),we have used

From(2.10)–(2.11)and(2.14)–(2.15),we get the desired estimate(2.2).

Finally,we prove(3.4).Differentiating(2.8)with respect totleads to

By the estimate forλnin[11],

we can chooseC∗sufficiently large,such that

which implies

Then noticing thatand applying(2.17)and Lemma 2.2,we obtain

On the other hand,using(2.13)and

we have

Summing up(2.18)–(2.19)yields(2.4).This completes the proof of Lemma 2.1.

The next lemma aims to transfer the original inverse problem(1.1)–(1.4)to a new form including the explicit expression ofp(t)andk(t).

Lemma 2.3Let

If the inverse problem(1.1)–(1.4)is solvable,then so is the following system:

with

where c0is the same one in(H4)and Ni(i=1,2)are defined by(2.25)below.On the other hand,if(2.20)–(2.22)has a solution and the compatibility conditions(H2),(H3)and the technical condition(H4)hold,then there exists a solution to the inverse problem(1.1)–(1.4).

Remark 2.2From Lemma 2.3,we know that(2.20)–(2.22)is an equivalent form of the original inverse problem(1.1)–(1.4).So,in the following several sections,we turn to discuss(2.20)–(2.22),other than the original one.

Proof of Lemma 2.3We split the proofinto two steps.

Step 1Assume that(1.1)–(1.4)has a solution(u,p,k)∈YT.Applyingto both sides of(1.1)yields

We note thatlThen by integration by parts,we get the following equality:

With the help of(2.24),we can rewrite(2.23)as

Due to(H4),we can solve this system to get(2.21)and

Furthermore,by differentiating(2.26)with respect tot,we get(2.22).

Step 2Now we assume that(u,p,k)satisfies(2.20)–(2.22).In order to prove that(u,p,k)is the solution to the inverse problem(1.1)–(1.4),it suffices to show that(u,p,k)satisfies(1.4).ApplyingHito the equation in(2.20),we have

On the other hand,from(H2),we easily see that

We get(2.26)by integrating(2.22)over[0,t].From(2.21)and(2.26),we conclude that

Then substituting(2.28)into(2.27),and using(H3),we have that(i=1,2)satisfy

By means of Laplace transform of the Caputo derivative(see[23]),we have that

whereandare the Laplace transforms ofHiandk,respectively.This leads to(s)=0,which implies thatHi(t)=0.SoThis completes the proof of Lemma 2.3.

At the end of this section,we give a technical lemma which will be used to estimatep,kin suitable Sobolev space in subsequent sections.

Lemma 2.4Let(H1)and(H5)hold.Then for all u∈YTand all l∈C1([0,T]),there exists a constant C>0depending on gi,ϕi,but independent of T,such that

where i=1,2and Niare the same as those in(2.25).

ProofBy the Hölder’s inequality,we see that

On the other hand,a direct calculation yields

Here we note thatl(0)=0.By integration by parts,it follows from(H5)that

which gives

Hence,we have

Combination(2.32)with(2.36)yields the desired estimate(2.31).This completes the proof of Lemma 2.4.

3 Proof of Lemma 1.1

We are now in a position to prove local in time existence,i.e.,Lemma 1.1,which proceeds by the fixed point arguments.First,we define the function set

HereMis a large constant depending on the initial and boundary dataa,band measurement datagi.For givenwe consider

and

to generate(u,p,k),whereis de fined byare the same as those in(2.25).

Remark 3.1Usually,we useon the right-hand sides of(3.2)–(3.3)to generatepandk.But,if we do so,we can not choose suitableTandRto proveandbecause there is a lack ofTin the terms includinguin(2.31).In this situation,the fixed point argument can not be applied to our problem.So we take the solutionuto(3.1)to generatepandk.This process in principle is similar to the Gauss-Seidel iteration.

By(2.17)and the Hölder’s inequality,we have

which implies

Using this result together withwe have

Therefore,Lemma 2.1 ensures that there exists a unique solutionu∈XTto(3.1).Then(3.2)–(3.3)define the functionsp(t)andk(t)in terms ofu.Furthermore,by Lemma 3.4,we have

Note thatWe obtain

Substituting(3.6)into(3.5)yields

This implies thatp∈C1[0,T]andk∈C[0,T].

Thus the mapping

given by(3.1)–(3.3)is well-defined.

Now we show thatSmapsZT,Minto itselffor sufficiently smallT>0.More precisely,we have the following result.

Lemma 3.1Fordefine

Then for properly small τ>0,we have

and

for all T∈(0,τ].

Throughout the following proof,we useCto denote a constant which depends on Ω,α,the initial dataa,b,the known functionsf,ϕiand measurement datagi,but independent ofMandT.

Proof of Lemma 3.1From Lemma 2.1 and(3.4),it follows that

On the other hand,by(3.2)–(3.3),together with Lemma 2.4 and(3.6),we can show thatandare bounded with

Then,adding up(3.10)–(3.11)leads to

where the functionω1(T)is of the form

and therefore satisfiesNow we takeM,such thatM=2Cwith the constantCin(3.12).Then there exists a sufficiently smallτ>0,such that

for allT∈(0,τ].That is,SmapsZT,Minto itselffor each fixedT∈(0,τ].

Next we estimate the increment of operatorS.To this end,we deduce the differences(u−U,p−P,k−K)from(3.1)–(3.3)to yield

and

whereandsatisfyandrespectively.

Using Lemma 2.1 and(3.4),we obtain

where

satisfiesMoreover,from(2.25)andwe can easily see that

Therefore,it follows that

Then,from(3.15)–(3.16),together with(3.19),we derive

From(3.17)and(3.20),we deduce that

Because ofwe can obtain(3.9),if we chooseτsufficiently small,such thatThe proofis complete.

Now we are in position to prove Lemma 1.1.

Proof of Lemma 1.1Lemma 3.1 shows that there exists a sufficiently smallτ>0,such thatSis a contraction mapping fromZτ,MtoZτ,M.Therefore the Banach fixed point theorem concludes that for sufficiently smallτ,there exists a unique solution(u,p,k)∈Xτ×C1[0,τ]×C[0,τ]to the problem constituted by(3.1)–(3.3).As a consequence,the problem constituted by(1.1)and(1.4)also admits a solution(u,p,k)in[0,τ]by Lemma 2.3.The proofis complete.

4 Proof of Lemma 1.2

In this section,we give the proof of the global in time uniqueness of solutions to our inverse problem,i.e.,Lemma 1.2.

Proof of Lemma 1.2By Lemma 2.3,we know that(2.20)–(2.22)is equivalent to our considered inverse problem.So,in the following we turn to prove the global uniqueness of(2.20)–(2.22).

Given any timeT,we assume that(ui,pi,ki)(i=1,2)are two solutions to the problem(2.20)–(2.22)in[0,T]with the regularity(ui,pi,ki)∈XT×C1[0,T]×C[0,T].This implies

whereC∗is depending onα,Ω,T,the initial dataaandb,the known functionsf,ϕiand the measurement datagi.

Let

Thensatisfies

and

Here,in a way similar to l,the functionssatisfyWe need to prove

Define

T0=inf{t∈(0,T]:

If(4.5)is not true,then it is obvious that T0is well-defined and satisfies 00,such that

From the definition of T0,we see that

Applying Lemma 3.1 to(4.2)in[0,tn+ε]and using(3.4),(4.1)and(4.6),we obtain

Due towe have

Substituting(4.8)into(4.7)yields

Additionally,from a direct calculation,it follows for i=1,2 that

and

by which and theinequality,we have

Here we have used that

Noticing thatand applying(4.3)–(4.4),(4.12),we have the following estimate forpandk:

From(4.9)and(4.13),we deduce that

with

So,forε>0 small enough,such thatC(T,ϕi,gi,C∗)ω3(ε)<1,we are led to

By takingn→∞,we obtain

which contradicts with the definition ofT0.Therefore(4.5)is proved.Now we can conclude that

for any timeT.The prooffor Lemma 1.2 is complete.

5 Proof of Theorem 1.1

Now we prove the global solvability Theorem 1.1 for our inverse problem.More precisely,for every given timeT>0,we will prove the existence of solutions to the problem constituted by(2.20)–(2.22),which is an equivalent form of our inverse problem.

In order to prove Theorem 1.1,we first show that local solution can be extended to a larger time interval in Subsection 5.1.Then,we give a preliminary estimate for the extension solution in Subsection 5.2.Finally,we prove Theorem 1.1 in Subsection 5.3.

5.1 Extension of the solution

Lemma 1.1 ensures that there exists a unique solution(2.20)–(2.22)for sufficiently smallτ>0.In this subsection,we show that the unique solutionin[0,τ]can be extended to a larger time interval[0,τ′],where 0<τ<τ′≤min{2τ,T}.We state the result as follows.

Lemma 5.1Letbe the unique solution to(2.20)–(2.22)in[0,τ].Then thereexists a τ′∈(τ,min{2τ,T}),such thatcould be uniquely extended to a solution(u,p,k)in[0,τ′],belonging toYτ′.

ProofWe consider

and

It suffices to show that(5.1)–(5.3)has a solution(U(x,τ+t),P(x,τ+t),K(τ+t))∈Yτ′−τ.Then(u,p,k)defined by

is an extension ofin[0,τ′],which has the regularity(u,p,k)∈Yτ′.Furthermore,from the global uniqueness result in Lemma 1.2,it follows that(u,p,k)given by(5.4)is the unique solution to the problem constituted by(2.20)–(2.22)int∈(0,τ′).This shows that the extension is unique.

Let

In the sequel,,lτand other functions with superscriptτare defined analogously.To prove the solvability of(5.1)–(5.3),we use the following splitting of convolution introduced by Colombo[11]:

which was successfully used to prove the global uniqueness of an inverse problem concerning with the strongly damped wave equation with memory.In a way similar to(5.5),we have

Then we find that

Based on(5.5)and(5.7),we rewrite the problem(5.1)–(5.3)as

and

whereh1andh2are known functions in terms ofandgi,defined by

For givenwe haveC1[0,τ′−τ],h2∈C[0,τ′−τ].Then repeating the arguments of the proof of Lemma 1.1,we could show that there exists a>0,such that(5.8)–(5.10)has a unique solution(U(x,t+Then(u,p,k)defined by(5.4)is the unique extension offrom[0,τ]to[0,τ′],if we chooseThis completes the proof of Lemma 5.1.

5.2 A preliminary estimate for the extended solution

We give here an a priori estimate on(u,p,k)defined by(5.4),which is the unique extension ofin[0,τ′].It will be used to guarantee that the process of extension can be repeated.

Lemma 5.2Assume that the hypotheses(H1)–(H5)hold.Then for(u,p,k)∈Yτ′defined by(5.4),we have the following estimate:

where C depends on α,Ω,τ,τ′,the initial data a and b,the known functions f,ϕiand the measurement data gi.

In order to prove Lemma 5.2,we need the following two results.

Lemma 5.3Assume that the hypotheses(H1)–(H5)hold.Then for the solution(uτ,kτ,pτ)to(5.8)–(5.10),we have the following estimate:

for t∈[0,τ′−τ],where C depends on α,Ω,τ,τ′,the initial data a and b,the known functions f,ϕiand the measurement data gi.

ProofBy Lemma 1.1,we see that there exists a positive constantC∗,depending onα,Ω,τ,the initial dataaandb,the known functionsf,ϕiand the measurement datagi,such that

Applying Lemma 2.1 to(5.8),we obtain for allt∈[0,τ′−τ]that

From(2.4)in Lemma 2.1 and(3.4),we deduce that

Note that 0≤t≤τ′−τ≤τ.Then by(4.4),we have

Moreover,by the Hölder’s inequality,we obtain for allt∈[0,τ′−τ]that

and

Finally,substituting(5.13),(5.15)–(5.18)into(5.14),and noticing thatt≤τ,we get(5.12).This completes the proof.

Lemma 5.4Assume that the hypotheses(H1)–(H5)hold.Then for the solution(uτ,kτ,pτ)to(5.8)–(5.10),we have the following estimate:

for t∈[0,τ′−τ],where C depends on α,Ω,τ,τ′,the initial data a and b,the known functions f,ϕiand the measurement data gi.

ProofFirst by(5.9)–(5.10),we have

Next we estimate the two terms on the right-hand side of(5.20).We easily see that

from which it follows that

Furthermore,noticing that

(see Remark 1.1)and

we have

Similarly,we have

Then from the de finitions ofand(2.34),together with(5.22)–(5.23),we deduce that

Moreover,a simple calculation gives

Finally,substituting the estimates(5.25)–(5.26)into(5.20)yields the desired estimate(5.19).The proof of Lemma 5.3 is complete.

Now we give the proof of Lemma 5.2.

Proof of Lemma 5.2Noticing

it suffices to show that

Adding up(5.12)in Lemma 5.2 and(5.19)in Lemma 5.3 yields that,∀t∈[0,τ′−τ],

Observe that,∀t∈(0,τ′−τ],

Inserting(5.29)into(5.28),we obtain

Hence(5.27)can be obtained by applying the Gronwall’s inequality to(5.30).

5.3 Proof of Theorem 1.1

Proof of Theorem 1.1Set

T:={τ∈(0,T]:the problem constituted by(2.20)–(2.22)has

at least a solution(u,p,k)∈Yτin[0,τ]}.

Obviously,we haveT=∅from Lemma 1.1.DefineT1:=sup(T).By a similar argument to the proof of Theorem 1.3 in[31],together with Lemmas 5.1–5.2,we could proveT1=T.This completes the proof of Theorem 1.1.

AcknowledgementWe would like to thank Professor Jijun Liu for his fruitful discussion.

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