Local Stability for an Inverse Coefficient Problem of a Fractional Diffusion Equation∗

2014-06-07CaixuanRENXiangXU

Chinese Annals of Mathematics,Series B 2014年3期

Caixuan RENXiang XU

1 Introduction

Nowadays time-fractional diffusion equations are of practical interest and importance,since they describe the power law decay for the diffusion in porous media perfectly.For instance,we refer to Bisquert[1],Hatano[5]and Hilfer[6]where a concrete physical experiment is designed to study the decay behavior offree-carrier density in a semiconductor with an exponential distribution of traps,and the decay ofion-recombination isothermal luminescence.In mathematics,forward problems of time-fractional diffusion equations are well studied and considerable results have been obtained both theoretically and numerically,e.g.,[2,7,9,12–13]and references cited therein.Moreover,some inverse problems which arise in fractional diffusion equations are also of great interest and attract much attention,e.g.[4,8,10–11,14,16–17].For example,Cheng et al[4]established a uniqueness result in determining the fractional order and the coefficient in the principal part simultaneously based upon the Gel’fand-Levitan theory.Xu et al[14]derived a Carleman estimate for a fractional diffusion equation with half order and obtained atype conditional stability for the Cauchy problem.Zhang et al[17]investigated an inverse source problem for a fractional diffusion equation and proved the uniqueness of the inverse problem by analytic continuation and Laplace transform.Yamamoto et al[16]applied the Carleman estimate in[14]to obtain conditional stability ofidentifying a lower order coefficient in a fractional diffusion equation from some additional data through the study of an inverse source problem.

Throughout this paper,we consider the following fractional diffusion equation with half order

where Ω=(0,l),Q=(0,l)×(0,T)anddenotes the fractional Caputo derivative in time of orderwhich is defined by

Let 0

Problem 1.1Can we estimate the leading order coefficientp(x),x∈Ω from a final observationu(x,t0)and Cauchy datahu,1(t),hu,2(t)?

The positive answer is given in Cheng et al[4]where the authors proved that Cauchy data could uniquely determinep(x)by utilizing the Gel’fand-Levitan theory.However there is no more stability indication which can be extracted from their proof.Therefore,this paper is aiming at establishing a local Hölder stability for this problem.The main idea is deriving a Carleman estimate with variable coefficients which is similar as the estimate with constant coefficients in Xu[14]and the methodology utilized in Yamamoto[15]for typical parabolic equations.

The outline of this paper is as follows.In Section 2,we reformulate the inverse coefficient problem by converting the governing equation into a fourth order equation and then present the main result.Section 3 is devoted to prove the main theorem in two steps which involves two kinds of Carleman estimates,respectively.Some concluding remarks are given in Section 4 to close the paper.

2 Formulation and the Main Theorem

In this section,we introduce some notations and present the main result,i.e.,a local Hölder conditional stability for Problem 1.1.

Throughout this paper,we denote byu(p)the solution of the initial-boundary value problem(1.1)and byv(q)the solution of a similar problem as follows:

For simplicity we restrict on some special complete function spaces.Denote byHα(Ω)the normal Sobolev space with orderα>0,i.e.,

and

Moreover,to accurately describe a local stability,we need a sub-domain which is generally characterized by the weight function de fined in(2.2)in Carleman estimate.Letandonand set

wheret0∈(0,T),β>0.Denote byQεthe local sub-domain defined as follows:

By choosing properβ>0 andd(x)>0 such that

where Φ>0 is a constant,we can easily verify thatQε⊂Q0⊂Qand Ωε⊂Ω0⊂Ω.

Now we proceed to the main result,i.e.,the local conditional Hölder stability while determining the leading order coefficientp(x).

Theorem 2.1Suppose that u(p),v(q)∈H4,3(Q)are solutions to(1.1)and(2.1),re-spectively,with positive p(x),q(x)Assume∂xv(x,t0)>0for x∈Ω,hu,1(t),hu,2(t),hv,1(t),hv,2(t)∈H4(0,T)and

Then there exists ε0>0,such that for any ε<ε0,we have θ∈(0,1)and constant C(ε,σ0)such that

Hereis considered as an a priori given bound for solutions offractional diffusion equations with the leading order coefficients under consideration.Estimation(2.4)is,in general,impossible without such a priori condition on solutions,and hence it is so-called a conditional stability estimate.

Remark 2.1To well understand the theorem,we give some following remarks.

(1)For the forward problem(1.1),the regularity of the solutionu∈H4,3(Q)can be achieved if the coefficientp(x)and Cauchy datahu,1(t),hu,2(t)are smooth enough,and satisfy sufficient compatibility conditions

(2)It should be noticed that the assumption on the final observation,i.e.,∂xv(x,t0)>0 is also reasonable.Denotingand taking derivatives with respect toxon both sides of(2.1),we have

If we add a positive boundary condition atx=l,i.e.,

for 00.Sincev(x,0)=0 which implies

the maximum principle for fractional equations(see[9])implies that the solutionz(x,t)>0 fort>0,i.e.,∂xv(x,t0)>0.

(3)Theθis complicated and the exact formulation will come out in the following section(see(3.23)).

(4)The sub-domain Ω3εin(2.4)can not be replaced by Ω due to the cut-offfunction utilized in the Carleman estimate.Hence the estimate in(2.4)is so-called a local stability.

(5)The argument can be extended to be more general,i.e.,for any sub-domainω⊂⊂Ω,there existsε>0,such thatω⊂Ω3εand hence

3 Proof of the Main Theorem

In this section,we aim at the detailed proof of Theorem 2.1.The methodology relies on a classical Carleman estimate for the parabolic equations,where its applications on various inverse coefficient problems are reviewed in[15,Section 6].

To start with,we denote

inQ.Subtracting(2.1)from(1.1)yields the following system:

whereh1(t)=hu,1(t)−hv,1(t),h2(t)=hu,2(t)−hv,2(t).

Notice thaty(x,0)=u(x,0)−v(x,0)=0 andv(x,0)=0.We obtain

It allows us to recall a technical lemma in[14].

Lemma 3.1(see[14])Let AC([a,b])be the space of absolutely continuous functions on[a,b].Assume that y∈AC([a,b])and satisfies

then the following equality holds:

where0<α,γ<1and0<α+γ≤1.

Direct calculation with Lemma 3.1 yields

with

Fix a certain timet=t0,(3.2)can be considered as a third order differential equation with respect tof(x),for instance,let

a(x)=u(x,t0)−v(x,t0)=y(x,t0),b(x)=v(x,t0),x∈Ω.

The equation(3.2)can be reduced into the following one:

For sake of simplicity,we set the following differential operators for the coefficients in(3.3)such that

Thus we firstly establish the Carleman estimate with respect tofin some appropriate subdomain of Ω for the new equation(3.3).

Notice the a priori assumption on the regularity ofp,qandu,v(consequentlya,b).We observe that the coefficientq∂xbnear the leading termfin(3.3)satisfiesq∂xb∈H3(Ω)while the rest coefficients near lower order terms are bounded.The Carleman estimate then is carried out for a general third order differential operatorLsuch that

whereL3(x)∈H3(Ω),Lj(x)∈L∞(Ω)withj=0,1,2.In addition,recall the following notation of weight functions:

We establish a technical lemma for the Carleman estimate in the form of(3.4).

Lemma 3.2Suppose that(3.4)holds true with g(x)andsuppg⊂D⊂Ω.Moreover,assume

and we can choose some s1>0and constant C>0such that

for all s≥s1.

ProofThe proof contains three steps.

Step 1We first prove the Carleman estimate for a first-order differential operatorPg=∂xg.Set

Then

Notice

and

By the assumptionand∂xψ?=0,one can chooseλsufficiently large such thatWe then obtain

and

by applying the same claims to

Step 2Now,we proceed to the principle term in(3.4)such thatwithsimilar to previous step,we letand

Standard calculation yields

and

Notice∂xϕ=λeλψ∂xψand(3.6).We thus conclude that,for sufficiently largeλ,there exists

Consequently,choosing sufficiently larges>0,we calculate(3.7)×s2+(3.8)×s+(3.9)and obtain

Step 3Finally,noticing the fact that

one can prove(3.5)easily.Sincecan be absorbed by the left-hand side of(3.5)with sufficiently large parameters.

We note that the order of the weight function with respect tosin(3.5)is different from a classic Carleman estimate which is derived directly forHowever it does not make too much difference towards the main result in Theorem 2.1.

Now we proceed to estimatef(x)in(3.3).Since a compact support is necessary to apply the Carleman estimate in Lemma 3.2,we introduce a cut-offfunction.Without loss of generality,letfunction satisfying 0≤χ≤1,and

whereQεandQ2εare defined,respectively,as in(2.3).Noticing

and substitutingχfinto the left-hand side of(3.3),we derive

In light of the definition onχ,we observe thatg3vanishes in Ω2εand only survives in ΩεΩ2ε.Moreover,the regularity ofp,q,bin Theorem 2.1 and the Sobolev embedding theorem yield

withx∈ΩεΩ2εand the constantMin Theorem 2.1.We thus apply the Carleman estimate in Lemma 3.2 toP0◦(χf)in(3.11)and obtain

for alls≥s1.

Now,a further estimate upondxin the equation(3.12)is necessary to complete the full estimate onχf.To this end,we turn to the Carleman estimate which is derived in[14].However,the local Carleman estimate there is established only with a first order time derivative term,whereas in our situation,the Carleman estimate on a tracet=t0is required which means that a high order term(see(3.13)for details)is necessarily included in the Carleman estimate.Therefore,to fit the current situation,we established another differential equation for the 2nd order time derivative

Before embarking on the estimate onwe present the second Carleman estimate for(1.1)in Lemma 3.3.The derivation is lengthy and different from Xu et al[14],for the sake of compactness of the proof structure,we give the detailed proofin Appendix A.

Denote byTthe transformed operator of(1.1)with integer order,i.e.,

Lemma 3.3(Carleman Estimate forT u)There exists λ0,such that for any λ≥λ0,we can choose s0and C such that

for all s≥s0and

Similarly,in order to use the local Carleman estimate in Lemma 3.3 without involving boundary integrals,we again imply the cut-offfunction0≤χ≤1 defined in(3.10).SinceQ0⊂⊂Qbased upon the definition ofQεwithε=0,we verifyχ(x,0)=0,and hence

withs≥1 and a fixed constantλ.The additional large parametersnear the|m2|2term makes the rest proof more consistent though one can prove the same result without it.Substituting(3.13)into(3.12)yields

In order to estimate the termdxdt,we consider the following two equations ofm1andm2which can be obtained by taking time derivative on both side of(3.2):

The equation(3.15)is a similar fourth order differential equation as[14]with respect tomi.In order to establish Carleman estimate formi,we need a further transformation as follows:

whereare chosen such thatfori=1,2 andj=0,1,2,3.

Thusand the boundary integral will vanish automatically during the integration by part while deriving the Carleman estimate in Lemma 3.3.Simple calculation gives

More precisely,based on the governing equation fory,i.e.,equation(3.1),and the assumption thatforj=0,1,2,3,further calculation gives the coefficients as follows:

Based on a priori bound assumption forhu,i(t),hv,i(t)andwe have

Replacingmibyniin equation(3.15)gives

and

fori=1,2 and(x,t)∈Q.For simplicity,we define,fori=1,2,

Noting thati=1,···,4 only survives whileε<ψ(x,t)≤2ε,by(2.3),for some fixed sufficiently largeλ>0 applying Lemma 3.3 toonQεimplies

whereM,Fare defined in Theorem 2.1.

Sinceni=mi−mi,0onQεas the definition ofχ(x,t),we have

for alls>s0.

Substituting(3.19)into(3.14),we have

for allmax{s0,s1},where the first terms on the right-hand side can be absorbed by the left-hand side.

Moreover,since Ω3ε⊂Ωεandψ(x,t0)>3εon Ω3ε,we can estimate the left-hand side of(3.20)by

Combining(3.20)and(3.21)and dividing e2sexp(3λε)on both sides,we can obtain that for allsufficiently large,

where=exp(λε)andC0=2exp(λ(Φ−3ε)).One can chooseεsmall enough such that Φ−3ε>0.

Assuming=0,by lettings→∞in(3.22),we can derivep(x)=q(x)on Ω3εwhich verifies the local uniqueness of the inverse coefficient problem.If there exists≥M,(3.22)immediately implies

by fixing a certains>0.At the same time,if

with the choice

Thus,the proof of Theorem 2.1 is completed.

4 Conclusion

Fractional diffusion equations as well as the inverse nature of these equations have attracted considerable interests in view of their potential use in physical and chemical processes and in engineering during last decades.In this paper,we investigated an inverse coefficient problem with respect to a half-order time-fractional diffusion equation.After generalizing the Carleman estimate to fractional diffusion equations with spatially varying conductivity,we implement the methodology developed in[3](see also the review paper[15])to establish a Hölder type conditional stability estimation for an inverse problem identifying the coefficient in the principal part.As for the results identifying the coefficient near the lower order term,we refer to a very recent work[16].Future works will be emphasized on deriving global Carleman estimates and Lipschitz stabilities on these inverse problems.

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Appendix A

Proof of Lemma 3.3Consider a following general fourth order differential equation with variable coefficientsT uwhere

Obviously,for Lemma 3.3,we can takeDenote the principal part ofTby

Note that to prove Lemma 3.3,it is sufficient to obtain a similar estimate for the principal part,i.e.,T0,which takes the form as follows:

where=sϕ(x)and=λμ(x).The reason is same as the second step in the proof of Lemma 3.2,i.e.,the lower order terms can be absorbed by choosing sufficiently large parameterssandλ.Here,for the sake of compactness,we omit the details.

Next throughout this section,we are aiming at proving the above estimate forT0.Letw=esϕuandSince∂xϕ=λϕμ,∂tϕ=−2β(t−t0)λϕ,we have

and

Therefore,according to the order ofwe can splitPwinto two terms,i.e.,Pw=P1w+P2w,where

Denote the inner product onQby

We can compute

In the following,we use bj(x,t),b(x,t)to represent bounded functions when(s,λ)are sufficiently large.For large λ>1,s>1,utilizing integration by parts andwe obtain

Since

we have

Let ε be any small positive constant,we have

Thus we can obtain

Similar as J11,for J12,we have

where

Combining all components of J1,we have

Continuing to estimate the remainder J2–J9,we have

where

In addition,

Combining J1–J9,we derive

Moreover,based on(A.3)and(A.7),we have

Inserting(A.14)into(A.13)yields

Noticing that in(A.15),the sign of|∂xw|2is negative,hence we have to obtain another estimate.In the following,we will estimatewdxdtto further estimatedxdt,where

By integration by parts andwe have