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Research article

Some results for the family of univalent functions related with Limaçon domain

  • Received: 07 October 2020 Accepted: 30 November 2020 Published: 19 January 2021
  • MSC : 30C45, 30C50, 30C55

  • The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which ςf(ς)f(ς) and (ςf(ς))f(ς) map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg ¨o inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

    Citation: Afis Saliu, Khalida Inayat Noor, Saqib Hussain, Maslina Darus. Some results for the family of univalent functions related with Limaçon domain[J]. AIMS Mathematics, 2021, 6(4): 3410-3431. doi: 10.3934/math.2021204

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  • The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which ςf(ς)f(ς) and (ςf(ς))f(ς) map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg ¨o inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.


    Fractional diffusion involves phenomena that have spatial and temporal correlations [1,2]. Anomalous diffusion through fractional equations is associated with super-statistics and can be linked to a generalized random walk [3]. The phenomenon of anomalous diffusion has received widespread attention in the fields of natural sciences, engineering, technology, and mathematics [4,5,6]. The fractional diffusion equations which serve as models for describing this phenomenon are of utmost importance [7,8,9,10,11]. Numerous publications have been dedicated to this field so far (e.g., Sakamoto and Yamamoto [12]). In contrast to classical parabolic equations, the time fractional diffusion equations replace the traditional local partial derivative t with the nonlocal fractional derivative αt. The fractional equations are highly regarded in mathematical physics and present distinct properties that challenge conventional differential equations. Nevertheless, some properties, such as the maximum principle, remain valuable in our research. This paper plans to describe the behavior of time fractional diffusion equations.

    In this paper we consider the following initial boundary value problem (IBVP) of a time fractional diffusion equation with a period structure

    {αtuϵ(x,t)div(Bε(x)uε(x,t))=fε(x,t),xΩ,t(0,T),uε(x,t)=0,xΩ,t(0,T),uε(x,0)=uε0(x),xΩ, (1.1)

    where 0<α<1, T>0, ΩRd is bounded domain with C2class boundary Ω, ε>0 is a scale parameter, Bε(x)=B(xε) is a diffusion matrix which satisfies some appropriate conditions, B(y) is periodic, and fε(x,t) and uε0(x) are the source function and the initial function, respectively.

    The existence and uniqueness of solutions to the initial boundary value problem (1.1) have been investigated widely. Sakamoto and Yamamoto [12] derived a kind of solution in terms of the Fourier series; Kubicam, Ryszewska and Yamamoto [13] gave the variational formulation; Hu and Li [14] gave the formally homogenized equation by the multiple scale expansion as ε0+ and Kawamoto, Machida and Yamamoto [15] gave the homogenized equation

    {αtu0(x,t)div(B0u0(x,t))=f0(x,t),xΩ,t(0,T),u0(x,t)=0,xΩ,t(0,T),u0(x,0)=u0(x),xΩ, (1.2)

    where B0 is the homogenized coefficient matrix, and then proved the precise homogenization theorem; they also discussed the inverse problem between different structures in the one dimensional case and in the layered material case where Bε(x) is a diagonal matrix with an unknown element when f=0. The aim of this paper is to generalize this result from the case with only one unknown element to the case with multiple unknown elements.

    The rest of this paper is organized as follows. In Section 2, we introduce some necessary tools, including the well-posedness and homogenization of fractional diffusion equations with oscillating diffusion matrix, the eigenvalue problem, and the Mittag-Leffler function. In Section 3 and Section 4 we state main results and prove them. In Section 5, we draw concluding remarks.

    In this section, we state some basic tools to investigate the inverse problems of the initial boundary value problem (1.1) and its homogenized equation (1.2), including the well-posedness, homogenization theory, the eigenvalue problem of the corresponding elliptic operator, and the Mittag-Leffler function, see [13,16].

    We recall the Riemann-Liouville fractional integral operator

    (Jαu)(t)=1Γ(α)t0(tτ)α1u(τ)dτ,uL2(0,T),0<α<1, (2.1)

    then the domain D(Jα)=L2(0,T) and the range R(Jα)=Hα(0,T) with

    Hα(0,T):={Hα(0,T),0α<12,{uH12(0,T)|T0|u(t)|2tdt<},α=12,{uHα(0,T)|u(0)=0},12<α1, (2.2)

    where Hα(0,T) is the Sobolev space. Moreover, Jα:L2(0,T)Hα(0,T) is a homeomorphism with

    uHα(0,T)={uHα(0,T),0α1,α12,(u2H12(0,T)+T0|u(t)|2tdt)12,α=12. (2.3)

    Therefore, the general fractional derivative of the Caputo type is defined by

    αt=(Jα)1:Hα(0,T)L2(0,T). (2.4)

    Obviously, αt is also a homeomorphism.

    We now consider the initial boundary value problem

    {αtu(x,t)div(B(x)u(x,t))=f(x,t),xΩ,t(0,T),u(x,t)=0,xΩ,t(0,T),u(x,0)=u0(x),xΩ, (2.5)

    where u0(x)L2(Ω), f(x,t)L2(0,T;H1(Ω)), and the matrix B(x)=(bij(x))d×d satisfies that

    {(i)bijL(Ω),bij=bji,(ii)B(x)ηην|η|2,|B(x)η|μ|η|,x¯Ω,ηRd, (2.6)

    for 0<ν<μ. From [13,15], we know that there exists a weak solution uL2(0,T;H10(Ω)) satisfying uu0Hα(0,T;H1(Ω)), and

    αt(uu0),ϕH1(Ω),H10(Ω)+(Bu,ϕ)L2(Ω)=f,ϕH1(Ω),H10(Ω) (2.7)

    for a.e. t(0,T) and ϕH10(Ω).

    For Y=(0,l1)××(0,ld), we say that a function f(x) is Y-periodic if

    f(x)=f(x+kliei),a.e. xRd,i=1,,d,kZ.

    Theorem 2.1. [15] For Bε(x)=B(xε), assume that B(y)=(bij(y))d×d is Y-periodic and satisfies Eq (2.6), uε0L2(Ω), and fεL2(0,T;H1(Ω)). If

    {uε0u0weakly in L2(Ω),fεf0in L2(0,T;H1(Ω)), (2.8)

    and uε is the weak solution of IBVP (1.1), then

    {uεu0weakly in L2(0,T;H10(Ω)),uεu0in L2(0,T;L2(Ω)), (2.9)

    where u0 is the weak solution of the homogenized problem (1.2), and B0 is the homogenized coefficient matrix. Furthermore, for the layered material, that is, B(y) is a diagonal matrix

    B(y)=diag{b11(y1),,bdd(y1)},

    then

    B0=diag{1M(0,l1)(1b11),M(0,l1)(b22),,M(0,l1)(bdd)}, (2.10)

    where MΩ(b)=1|Ω|Ωb(x)dx.

    For the diagonal matrix Bp=diag{p1,,pd}, pi are constants and νpiμ, denote a vector p=(p1,p2,,pd)Rd and an operator Bp()=div(Bp), we consider an eigenvalue problem of the operator Bp on Ω=di=1(0,δi).

    Bpφ=λφ,φH2(Ω)H10(Ω). (2.11)

    According to the domain Ω, we consider the sub-eigenvalue problems

    φi(xi)=σiφi(xi),φiH2(0,δi)H10(0,δi),i=1,,d. (2.12)

    Then, we can verify that φ(x)=di=1φi(xi) is a solution of eigenvalue problem (2.11) with λ=di=1piσi, i.e., Bpφ=(di=1piσi)φ. Denote by φkii the kith simple eigenvalue of the ith sub-eigenvalue problem, that is, d2dx2iφkii(xi)=σkiiφkii(xi). It is known that φi(xi) are the sine functions. Since the eigenfunctions {φki}k=1 are an orthonormal basis of L2(0,δi), so {di=1φkii(xi)}k1,,kdN is an orthonormal basis of L2(Ω). Then, we can prove that the all eigenvalues λ of Bp have following form:

    λ=di=1piσkii,k1,,kdN.

    In fact, for Bpφ=λφ, φ0, there exists di=1φkii such that (φ,di=1φkii)L2(Ω)0. Taking inner product of L2(Ω) with respect to di=1φkii on both sides of equation (2.11) and integration by parts, we can complete the proof.

    For example, for d=3,pi=1,i=1,,d, we have Bp=Δ. Taking δ1=δ2=δ3, we have 0<σ1i<σ2i<+, i=1,2,3, σk1=σk2=σk3, and φk1=φk2=φk3, kN. Then we can write the eigenvalue of Eq (2.11) as

    λl=σn1+σm2+σk3,l=n+m+k2,n,m,kN

    and the corresponding eigenfunctions are

    φnmk=φn1(x1)φm2(x2)φk3(x3),φnkm=φn1(x1)φk2(x2)φm3(x3),φmnk=φm1(x1)φn2(x2)φk3(x3),φmkn=φm1(x1)φk2(x2)φn3(x3),φknm=φk1(x1)φn2(x2)φm3(x3),φkmn=φk1(x1)φm2(x2)φn3(x3).

    Thus, we know that some eigenvalues of problem (2.11) have more than one geometric multiplicity, which is different from the eigenvalues of problem (2.12) such that all eigenvalues are simple.

    Returning to the general operator Bp on a bounded domain ΩRd, we rearrange the eigenvalues of Bp without multiplicity, 0<λ1<λ2<+, and rearrange the eigenvalues {σki}k=1(i=1,,d) such that

    λjn=di=1σnji,j=1,,mn,nN,

    where mn is the multiplicity of the eigenvalue λn=λ1n=λ2n==λmnn. Note that λ1 is simple, i.e. m1=1 and σ11i<σnji,n>1,j=1,,mn. Set

    φnj(x)=di=1φnji,j=1,,mn,nN,

    where φnji(xi) is the eigenfunction corresponding to eigenvalue σnji, and {φnj}dnj=1 is the orthonormal basis of ker(BpλnI).

    We introduce a projection operator Pn:L2(Ω)ker(BpλnI) such that

    Pnv=mnj=1(v,φnj)φnj,vL2(Ω),

    is an eigenprojection. We note that the eigenfunctions of Δ and Bp are identical indeed, but their eigenvalues are not identical.

    From [17], the Mittag-Leffler function is defined by

    Eα,β(z)=n=0znΓ(αn+β),α,β>0,zC,

    which is an entire function in the complex plane.

    When α=β=1, Eα,β is precisely an exponential function. What is more, we have the asymptotic expansion and estimate

    Eα,1(z)=Kk=1zkΓ(1αk)+O(|z|1K),|z|,θ|argz|π, (2.13)
    Eα,1(z)C1+|z|,θ|argz|π, (2.14)

    where KN, 0<α<2, and πα2<θ<min{π,απ}.

    We now consider problem (1.1) on the domain Ω=di=1(0,δi),δi>0,d<4. In order to state the main results, we first give a definition.

    Definition 13.1. For the matrices A=(aij)N×M and B=(bij)N×M, we say AB if

    aijbij,i=1,,N,j=1,,M.

    We say A>B if AB and there exists an index i or j such that aij>bij.

    We consider the IBVP with a periodic structure

    {αtuεp(x,t)div(Bεp(x)uεp(x,t))=0,xΩ,t(0,T),uεp(x,t)=0,xΩ,t(0,T),uεp(x,0)=uε0,p(x),xΩ, (3.1)

    with unknown initial function uε0,p and unknown diffusion matrix Bεp with layer structure

    Bεp(x)=Bp(xε)=diag{p1(x1ε),p2(x1ε),,pd(x1ε)} (3.2)

    satisfying

    pi(y1) is l_1 -periodic, piL(0,l1),νpi(y1)μ,y1[0,l1],i=1,,d. (3.3)

    Due to Theorem 2.1, we get uεpu0p in L2(0,T;L2(Ω)), where u0p(x) is a weak solution of the homogenized equation

    {αtu0p(x,t)div(B0pu0p(x,t))=0,xΩ,t(0,T),u0p(x,t)=0,xΩ,t(0,T),u0p(x,0)=u0,p(x),xΩ, (3.4)

    where u0,p(x) is the L2(Ω) limit of uε0,p(x), B0p=diag{p01,p02,,p0d} and

    p01=1M(0,l1)(1p1(y1)),p02=M(0,l1)(p2(y1)),,p0d=M(0,l1)(pd(y1)) (3.5)

    satisfying νp0iμ,i=1,,d. By Eq (3.5) and simple calculation, we have

    |p0iq0i|CpiqiL1(0,l1),i=1,,d, (3.6)

    with C=C(ν,μ,l1)>0. Moreover, if

    Bp(y)Bq(y)a.e.yY, (3.7)

    we have

    C1piqiL1(0,l1)p0iq0iCpiqiL1(0,l1),i=1,,d, (3.8)

    with C=C(ν,μ,l1)1. These can be seen in the proof of [15, Lemma 3.11].

    For simplicity of notation, we set p=(p1,,pd)Rd+. We first consider several inverse problems of determining the diffusion matrix of the following IBVP:

    {αtup(x,t)div(Apup(x,t))=0,xΩ,t(0,T),up(x,t)=0,xΩ,t(0,T),up(x,0)=u0,p(x),xΩ, (3.9)

    where Ap=diag{p1,,pd} and νp1,,pdμ.

    Inverse problem Ⅰ: Let x0Ωt0(0,T). We will determine the diffusion matrix Ap by the single data point up(x0,t0) of problem (3.9).

    Theorem 3.1. Let u0,p=u0,q=u0H2(Ω)H10(Ω) and up(x,t) be a solution of problem (3.9). Assume pq and

    u00,xΩ,21u00,,2du00,a.e.xΩ. (3.10)

    Then there exists a constant C(ν,μ)>0 such that

    Ni=1|piqi|C|up(x0,t0)uq(x0,t0)|. (3.11)

    Inverse problem Ⅱ: Let ωΩI(0,T). We will determine the diffusion matrix Ap by the data Iωup(x,t)dxdt of problem (3.9). Note that the measurement data is an integral expression. Thus, it is more useful for applications.

    Theorem 3.2. Let u0,p=u0,q=u0H2(Ω)H10(Ω) and up(x,t) be a solution of problem (3.9). Assume pq and

    u00,xΩ,21u00,,2du00,a.e.xΩ. (3.12)

    Then there exists a constant C(ν,μ)>0 such that

    Ni=1|piqi|C|Iωup(x,t)dxdtIωuq(x,t)dxdt|. (3.13)

    Inverse problem Ⅲ: Let x0Ωt1(0,T). We determine the diffusion matrix Ap by the time trice data up(x0,t),0<t<t1 of problem (3.9). We can only get uniqueness here.

    Theorem 3.3. Let up(x,t) be a solution of problem (3.9) and u0,p,u0,qH2(Ω)H10(Ω) such that

    P1u0,p(x0)0andP1u0,q(x0)0. (3.14)

    If up(x0,t)=uq(x0,t),0<t<t1, then Ap=Aq.

    Remark 3.1. Let Ω=(0,π)×(0,π). The eigenfunction corresponding to λ1=2 is φ1=2πsinxsiny and the eigenfunction corresponding to λ2=5 is φ2=2πsin2xsiny and φ3=2πsinxsin2y. We take u0,p=φ1 and u0,q=φ2. We have

    up=Eα,1((p1+p2)tα)φ1,uq=Eα,1((4q1+q2)tα)φ2

    and

    P1u0,p=(u0,p,φ1)φ1,P1u0,q=0.

    If p=(4,1),q=(1,1), and (x0,y0)=(π3,π2), we have up(x0,y0,t)=uq(x0,y0,t),t>0, but ApAq. Thus, condition (3.14) is necessary.

    We will present the proof of these theorems later in Section 4.

    Following Theorems 3.1–3.3 and the estimates (3.8), we can immediately obtain the following corollaries of the inverse problem determining the diffusion matrix between different structures to problem (3.1) and problem (3.4). First, we present the inverse problem of determining the period coefficient matrix by the homogenized data.

    Corollary 3.1. Let u0p(x,t) be a solution of problem (3.4) and u0,p=u0,q=u0H2(Ω)H10(Ω). Under the condition (3.7) and

    u00,xΩ,21u00,,2du00,a.e.xΩ.

    Then there exists a constant C(ν,μ,l1)>0 such that

    Ni=1piqiL1(0,l1)C|u0p(x0,t0)u0q(x0,t0)|.

    We see that the condition for u0 is from Condition 3.10 and p(y),q(y) are vector-valued functions over (0,l1) portraying the period structure. Further, we must guarantee uϵ0,p and uϵ0,p have the same limit u0. Similarly, the following result also follows.

    Corollary 3.2. Let u0,p=u0,q=u0H2(Ω)H10(Ω), u0p(x,t) be a solution of problem (3.4) and up(x,t) be a solution of problem (3.9). Under condition (3.7) and

    u00,xΩ,21u00,,2du00,a.e.xΩ.

    Then there exists a constant C(ν,μ,l1)>0 such that

    Ni=1piqiL1(0,l1)C|Iωu0p(x,t)dxdtIωu0q(x,t)dxdt|.

    Corollary 3.3. Let u0p(x,t) be a solution of problem (3.4) and u0,p,u0,qH2(Ω)H10(Ω) such that

    P1u0,p(x0)0andP1u0,q(x0)0.

    Then if u0p(x0,t)=u0q(x0,t),0<t<t1, we have Bp(y)=Bq(y)a.e.yY.

    Limited by our approach, as in Theorem 3.3, we can only obtain the uniqueness of this inverse problem. We can also use the periodic structure data to determine the homogenized coefficient matrix as the following result of asymptotic stability.

    Corollary 3.4. Let u0,p=u0,q=u0H2(Ω)H10(Ω), uεp(x,t) be a solution of problem (3.1) and u0p(x,t) be a solution of problem (3.4). Under the condition (3.7) and

    u00,xΩ,21u00,,2du00,a.e.xΩ.

    Then there exists a constant C(ν,μ,l1)>0 such that

    Ni=1|p0iq0i|C|Iωuεp(x,t)dxdtIωuεq(x,t)dxdt|+θ(ε),

    where θ(ε)0 as ε0.

    In this section, we give the proof of Theorems 3.1–3.3.

    Proof of Theorem 3.1. We split the proof into the following three steps, and prove separately for each step.

    Step 1. We first prove that pq implies that up(x,t)uq(x,t). Set y=upuq, then

    {αty(x,t)div(Apy(x,t))=di=1(piqi)2iuq,xΩ,t(0,T),y(x,t)=0,xΩ,t(0,T),y(x,0)=0,xΩ. (4.1)

    Denote v=di=1(piqi)2iuq=div(Apquq):=Apquq. Since

    uq=n=1mnj=1Eα,1((q1σnj1++qdσnjd)tα)(u0,φnj)φnj,

    we get

    Apquq=n=1mnj=1Eα,1((q1σnj1++qdσnjd)tα)(u0,φnj)Apqφnj=n=1mnj=1Eα,1((q1σnj1++qdσnjd)tα)(Apqu0,φnj)φnj.

    Thus, by [12, Theorem 2.1], we know that v is a weak solution of the following problem

    {αtv(x,t)div(Apv(x,t))=0,xΩ,t(0,T),v(x,t)=0,xΩ,t(0,T),v(x,0)=Apqu0,xΩ. (4.2)

    By [18, Theorem 2.1] and di=1(piqi)2iu00, we have v0. Applying [18, Theorem 2.1] to Eq (4.1), we get y0, i.e., upuq.

    Step 2. For p=(p1,,pd), we prove the analyticity of up(x0,t0) with respect to every pi>0. Observe that

    Apup=n=1mnj=1(p1σnj1++pNσnjN)Eα,1((p1σnj1++pNσnjN)tα)(u0,φnj)φnj,

    hence

    Apup2L2(Ω)Cn=1mnj=1(u0,φnj)2L2(Ω)((p1σnj1++pdσnjd)tα1+(p1σnj1++pdσnjd)tα)2t2αCu02L2(Ω)t2α.

    By the Sobolev embedding H2(Ω)C(¯Ω), we have

    |up(x,t)|Ctαu0L2(Ω).

    Thus, we get the convergent series

    up(x0,t0)=n=1mnj=1Eα,1((p1σnj1++pdσnjd)tα0)(u0,φnj)φnj(x0), (4.3)

    and since Eα,1(z) is holomorphic in the complex plane, we see that h(p)=up(x0,t0) is analytic with respect to pi,i=1,,d.

    Step 3. We prove that p>q means h(p)>h(q). First, 2iu00,i=1,,d, so Δu00. Since u0H2(Ω)C(¯Ω) and u0H10(Ω), by the strong maximum principle [20], we have u00. On the basis of [19, Theorem 9], we know that up(x0,t0)<0 for all pRd+.

    If there exist p0,q0 such that p0>q0 and up0(x0,t0)=uq0(x0,t0), then there is i{1,,d} such that p0i>q0i. Therefore, when q0i<s<p0i,

    h(p01,,p0i,,p0d)=h(p01,,s,,p0d).

    Since h(p) is analytic with respect to pi, we have

    h(p0)=h(p01,,s,,p0d),s>0.

    Moreover,

    |h(p0)|=|n=1dnj=1Eα,1((p1σnj1++sσnji++pNσnjN)tα0)(u0,φnj)φnj(x0)|Cn=1dnj=11p1σnj1++sσnji++pNσnjN|(u0,φnj)|tα0Ctα01p01σ111++sσ11i++p11N, (4.4)

    where we use the fact that

    |(u0,φnj)|=|(u0,21φnjσnj1)|=|(21u0,φnj)|σnj121u0L2(Ω)σnj1,

    and the series n=1mnj=1σnj1 converges. Passing to the limit as s in Eq (4.4), we have up0(x0,t0)=0, a contradiction. Therefore, p>q means that h(p)>h(q).

    Step 4. By the last step, we have ih(p)>0 for all pRd+,i=1,,d. Set g(t)=h(q+t(pq)), then g(t)=h(q+t(pq))(pq). By the mean value theorem we get

    |h(p)h(q)|=h(q+t(pq))(pq)=di=1(piqi)ih(q+t(pq))Cdi=1(piqi),

    where

    C=min1idinfνpiμih(p)>0.

    This ends the proof of Theorem 3.1.

    Proof of Theorem 3.2. Referring to Theorem 3.8(ⅱ) in [15], we can similarly verify that the function H(p)=Iωup(x,t)dxdt satisfies iH(p)>0 for all pRd+,i=1,,d. Thus, we can complete the proof similarly to that of Theorem 3.1.

    Proof of Theorem 3.3. From up(x0,t)=uq(x0,t) for 0<t<t1 and the analyticity of up(x,t) with respect to t, we have up(x0,t)=uq(x0,t) for t>0. Since

    uq(x0,t)=n=1mnj=1Eα,1((q1σnj1++qdσnjd)tα)(u0,φnj)φnj(x0)

    and by the asymptotic expansion (2.13), we have that

    Kk=1(1)k+1Γ(1αk)tαkn=1mnj=1(u0,q,φnj)φnj(x0)(q1σnj1++qdσnjd)k=Kk=1(1)k+1Γ(1αk)tαkn=1dnj=1(u0,p,φnj)φnj(x0)(p1σnj1++pdσnjd)k+O(1tα(K+1))

    holds for KZ+. We equate the coefficients of tαk, which yields

    n=1mnj=1(u0,q,φnj)φnj(x0)(q1σnj1++qdσnjd)k=n=1mnj=1(u0,p,φnj)φnj(x0)(p1σnj1++pdσnjd)k,kZ+.

    If q>p, we have

    (u0,p,φ11)φ11(x0)+n=2mnj=1(p1σ111++pdσ11dp1σnj1++pdσnjd)k(u0,p,φnj)φnj(x0)=n=1mnj=1(p1σ111++pdσ11dq1σnj1++qdσnjd)k(u0,q,φnj)φnj(x0),kZ+. (4.5)

    We observe that

    p1σ111++pdσ11dp1σnj1++pdσnjd<1,n2,j=1,,mn,p1σ111++pdσ11dq1σnj1++qdσnjd<1,n1,j=1,,mn.

    Taking k in Eq (4.5), we get

    P1u0,p(x0)=(u0,p,φ11)φ11(x0)=0,

    this yields a contradiction. Thus, qp. Similarly, we can prove pq. This ends the proof of Theorem 3.3.

    We should mention that our work is done under some constraints. We require the diffusion coefficient matrix to be diagonal and the area considered to be a rectangular area. This is because our method relies on the expansion of the eigenfunction, and only under these constraints can we clarify our eigenfunction, which is very advantageous for our proof. In addition, we only consider the case of layered matter, that is, the diffusion matrix only depends on one variable. Only in this way can we obtain formula (3.8). In order to break through these limitations, we believe that we can only seek other more difficult methods. Besides, we can also consider other more general problems. For example, we can consider the inverse problem of the variable-order time-fractional equation [21] in the current frame, but we do not discuss these problem in this paper. Moreover we also consider the inverse problem of determining the variable order and diffusion matrix simultaneously, such as in article [22], which investigates this problem in one space dimension. We also hope to expand their results to the situation of high-dimensional situations in the future.

    We would like to thank the Editor and Referees for valuable comments and contributions. The research is supported by the National Natural Science Foundation of China (No. 12171442). Feiyang Peng carried out the inverse problem and the homogenization theory of fractional diffusion equations, and Yanbin Tang carried out the reaction diffusion equations and the perturbation theory of partial differential equations. All authors carried out the proofs and conceived the study. All authors read and approved the final manuscript.

    The authors declare that there is no conflict of interest.



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