Determination of a Coefficient and Kernel in a d -Dimensional Fractional Integro-Differential Equation Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2024, Volume 26, Issue 3, P. 86-111

YAK 517.95

DOI 10.46698/g9973-1253-2193-w

DETERMINATION OF A COEFFICIENT AND KERNEL IN A D-DIMENSIONAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

A. A. Rahmonov1,2

I V. I. Romanovsky Institute of Mathematics

of the Academy of Sciences of the Republic of Uzbekistan, 9 University St., Tashkent 100174, Uzbekistan;

2 Bukhara State University,

II M. Ikbol St., Bukhara 705018, Uzbeksitan E-mail: [email protected], [email protected]

Abstract. This paper is devoted to obtaining a unique solution to an inverse problem for a multidimensional time-fractional integro-differential equation. In the case of additional data, we consider an inverse problem. The unknown coefficient and kernel are uniquely determined by the additional data. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. The weak solvability of a nonlinear inverse boundary value problem for a d-dimensional fractional diffusion-wave equation with natural initial conditions was studied in the work. First, the existence and uniqueness of the direct problem were investigated. The considered problem was reduced to an auxiliary inverse boundary value problem in a certain sense and its equivalence to the original problem was shown. Then, the local existence and uniqueness theorem for the auxiliary problem is proved using the Fourier method and contraction mappings principle. Further, based on the equivalency of these problems, the global existence and uniqueness theorem for the weak solution of the original inverse coefficient problem was established for any value of time.

Keywords: fractional wave equation, Caputo fractional derivative, Fourier method, Mittag-Leffler

function, Bessel inequality.

AMS Subject Classification: 35R30, 35R11.

For citation: Rahmonov, A. A. Determination of a Coefficient and Kernel in a d-Dimensional Fractional Integro-Differential Equation, Vladikavkaz Math. J., 2024, vol. 26, no. 3, pp. 86-111. DOI: 10.46698/g9973-1253-2193-w.

1. Introduction and Setting Up the Problem

Fractional calculus plays an important role in mathematical modeling in many scientific and engineering disciplines. They are used in the modeling of many physical and chemical processes and engineering (see, e. g., [1-7]). A fractional integro-differential equation can be used to simulate a wide range of problems in the basic sciences, many scientists have focused their attention on presenting the solutions for these systems. That equation has played a significant role in finding solutions using diverse methods, which is in line with the rapid development in finding the answers to diverse problems originating from the basic sciences. The linear/nonlinear equations fractional integro-differential equation has various uses in fluid mechanics [8], Stokes flow [9], airfoil [10], quantum mechanics [11], integral models [12], mathematical engineering [13], nuclear physics [14] and the theory of laser [15].

© 2024 Rahmonov, A. A.

Other studies [16-21] demonstrate several interesting features of the fractional diffusion-wave equations, which represent a peculiar union of properties typical for second-order parabolic and wave differential equations. Fractional evolution inclusions are an important form of differential inclusions within nonlinear mathematical analysis. They are generalizations of the much more widely developed fractional evolution equations (such as time-fractional diffusion equations) seen through the lens of multivariate analysis. Compared to fractional evolution equations, research on the theory of fractional differential inclusions is however only in its initial stage of development. This is important because differential models with the fractional derivative provide an excellent instrument for the description of memory and hereditary properties, and have recently been proven valuable tools in the modeling of many physical phenomena (see, [22] and the references therein).

According to the fractional order a, the diffusion process can be specified as sub-diffusion (a € (0,1)) and super-diffusion (a € (1,2)), respectively. There is abundant literature on the studies of fractional equations on various aspects, such as physical backgrounds, weak solutions, and maximum principle and numerical methods (see, [23] and the references therein).

Practical needs often lead to problems in determining the coefficients, kernel, or the right-hand side of a differential equation from certain known information about its solution. Such problems have received the name inverse problems of mathematical physics. Inverse problems arise in various domains of human activity, such as seismology, prospecting for mineral deposits, biology, medical visualization, computer-aided tomography, the remote sounding of Earth, spectral analysis, nondestructive control, etc., (see [24-26]). In this paper, we discuss an inverse problem of determining a source term only depending on the time in a fractional-differential equation by the measurement data of time trace at a fixed point xj.

Let := Q x (0, T) for a given time T > 0, where Q be a bounded domain in Rd with sufficiently smooth boundary dQ and XT = dQ x (0,T). We consider a fractional integro-differential equation with a fractional derivative in time t:

dtau(x,t) + Au(x,t) = q(t)ut(x,t) + k * u(x,t) + f (x,t), (x,t) € QT, (1-1)

where 1 < a < 2 and d*u(x, t) is the left Caputo fractional derivative with respect to t and defined by [27]

J (t - r)m-a-1 v(m) (r) dr, m - 1 < a < m, m € N,

m

d?v(t) = { r(m-a) ' " "

u

v(m)(t), a = m € N,

r( ■) is the Gamma function and the operator A is a symmetric uniformly elliptic operator defined on D(A) = H2(Q) n H(Q) given by

d d ( d \ Av(x.,t) = - Y^ — laij(x) — v(x,t) J +c(x)v(x,t), (x.,t) € Qq,

i,j= 1 j j

in which the coefficient satisfy dij = ciji € C1(Q), c € C(Q), c(x) ^ 0, x € Q, and there exists a constant ¡i > 0 such that Ylij=i ^ I^YliLi I6I2 £ ii, ( £ and Laplace

convolution k * g(t) = Jq k(t — t)g(r) dr.

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

t

1

u(x, 0) = a(x), ut(x, 0) = b(x), x € Q,

(1.2)

and the zero boundary condition:

u(x, t) = 0, (x,t) € ST. (1.3)

If q(t), k(t), f (x, t), a(x) and b(x) are known, then problem (1.1)-(1.3) is called a direct problem. The inverse problem in this paper is to reconstruct q(t) and k(t) according to the additional data

u(x,t)= hi(t), t € (0,T), (1.4)

where h^t), i = 1,2, are given functions and xi € Q, i = 1,2, are given numbers. We investigate the following inverse problem.

Inverse problem. Find u € C([0,T]; D(AY+1/a)) n C 1([0,T]; D(AY)), q € C 1[0,T] and k € C[0, T] to satisfy (1.1)-(1.3) and the additional measurement (1.4), where D(AY) is a Hilbert space with some positive constant 7, see (1.6).

For the convenience of the reader, we present here the necessary definitions from functional analysis and fractional calculus theory.

For integers m, we denote Hm(Q) = Wm'2(Q) (see [28]) and H0m(Q) is the closure of C^(Q) in the norm of space Hm(Q). For a given Banach space V on (Q), we use the notation Cm([0,T]; V) to denote the following space:

Cm([0, T]; V) := {u : ||dju(t)||V is continuous in t on [0,T] V0 ^ j < m}.

We endow Cm([0, T]; V) with the following norm making it to be a Banach space: ||u||Cm([o,t];V) = Xj=0(max0^T lldju(t)||V). In addition, we define Banach space XT by XT := C([0, T]; D(AY+1/a)) nC 1([0,T]; D(AY)) with the norm ||u||XT := ||u||C([0;T];D(Ay+i/«)) + |u|ci([0,T];D(Ay)). Furthermore, we set YQT = XT x C 1[0, T] x C[0, T] endowed with the norm

||(u,q,k)||Y0T := ||u||XT + |q|Ci[0,T] + ||k||C[0,T].

It is well-known that the operator A is a symmetric uniformly elliptic operator, the spectrum of A is entirely composed of eigenvalues, and counting according to the multiplicities, we can set: 0 < A1 ^ A2 ^ ..., An = to. By en € H2(Q) n H0(Q),

we denote the orthonormal eigenfunction corresponding to An:

Aen — Anen, in Q, en = 0, on dQ.

It is well known that, if the coefficients aij-(x), c(x) are real-valued functions and aij-(x) = aj-i(x) € L^(Q), c(x) € L^(Q), then the eigenfunction sequence |en}neN is a orthonormal basis in L2(Q). Then for 7 € R we define a Hilbert space D(AY) by (see [29])

S ^O N <X

D(AY):= j u € L2(Q): ^ A^ |(u,e„)|2 < to}, AYu = ^ An(u,e„)e„,

equipped with the norm ||u||d(ay) = (Ail7 |(u, en)|2)1/2. We note that the norm ||u||d(ay) is stronger than ||u||L2(Q) for 7 ^ 0. Since D(AY) c L2(Q) for 7 ^ 0, identifying the dual of L2(Q) with itself, we have D(AY) c L2(Q) c (D(AY))' and D(A-Y) = (D(AY))', which consists of bounded linear functionals on D(AY). For u € D(A-Y) and ^ € D(AY), the value obtained by operating u to ^ is denoted by -Y(■, -)7. D(A-Y) is a Hilbert space with

the norm: ||^||d(a-y) = ( X^U A-27 |-7(u, e„)Y|2)1/2. We further note that _7(u,^)7 = (u, if u € L2(Q) and € D(AY) (see e. g., [30, Chapter V]).

Moreover, we introduce the Mittag-Leffler function in [27]: EP;jU(z) = J2kLo zk/r(pk + p), z € C, with Re(p) > 0 and p € C. It is known that Ep>jU(z) is an entire function in z € C.

Lemma 1.1. Let 0 < p < 2 and p € R be arbitrary and 0 satisfy np/2 < 0 < min{n, np}. Then there exists a constant c = c(p, p, 0) > 0 such that |EP;jU(z)| ^ c/(1+|z|), 0 ^ | arg(z)| ^n, and the asymptotic behavior of EP;jU (z) at infinity as follows

N —n

V^ z I r\( ~,-n-V\

For the proof, we refer to [31] for example.

Proposition 1.1 [27]. For A> 0, a > 0, ft € C and positive integer m € N, we have dm -

— EaA(-Xn = -\t«-mEa,a-m+1(-\n, t > 0,

d

- {t^Eaß{-\ta)) = t13-2Eaß_i(—Xta), t > 0,

da(E«;i(-Aia)) = -A£«;i(-Ata), t > 0.

Also, we mention

max —— = , O<0<1. (1.5)

y>0 1+y 1 + ^

We now give a similar definition of weak solution to (1.1)—(1.3), which is introduced by [32].

Definition 1.1. We call u a weak solution to (1.1)—(1.3) if (1.1) holds in L2(Q) and u(-,t) € H01(Q) for almost all t € (0,T), u, dtu € C([0,T]; D(A-Y)) and

Jim llu(',t) - a\\D(A-Y) = Jim0 llut(',t) - b||D(A-y) =0

with some 7 > 0. Here 7 > 0 may depend on a, b.

Throughout this paper, we set y0 > d/2 + 1, 7 > 0 and 1/a < e < 1 such that

maxj^ + 1, 70 + ^< 7 ^ 70. (1.6)

We make the following assumptions:

(C1) Ofhi € C 1[0,T], a € D (AY0+1/a), b € D (AY0), f € C 1([0,T]; D (AY));

(C2) hi(0)q(0) = dtahi(0) + Aa(xt) - /¿(0), where f (t) = f (xi,t), i = 1, 2;

(C3) a(xi) = hi(0), b(xi) = hi(0), i = 1, 2;

(C4) p(t) = hi(t)h2(0) - h'2(t)h1 (0) = 0 and p(t) € C 1[0,T] satisfies the following inequality: ||p||Ci[0,T] ^ 1/p0 > 0, where p0 is a given positive constant.

Remark 1.1. In (C1), dtah € C 1[0,T] implies hi € W2>1(0,T) ^ H 1(0,T) (see [26]) and from this we will be used in Lemma 2.7 below. Furthermore, if we also require that dfhi(0) = 0, then according to Remark 1.1 in [26], hi(t) € C2[0,T] is implied.

Remark 1.2. (C3) is the consistency condition for our problem (1.1)-(1.4), which guarantees that the inverse problem (1.1)-(1.4) is equivalent to (2.30) and (2.32) (see Lemma 3.3).

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

Theorem 1.1. Let (C1)-(C4) hold. Then, there exists a unique solution (u, q, k) € Y0T of the inverse problem (1.1)—(1.4) for any T > 0.

The outline of the paper is as follows. In Section 2, we give preliminary results in this paper, including the existence and uniqueness of the direct problem (1.1)-(1.3), and also an equivalent problem is presented. In Section 3, the local existence and global uniqueness of the solution of the inverse problem (1.1)-(1.4) is established by using the Fourier method and Banach fixed point theorem. Section 4 contains the proof of Theorem 1.1 (existence global in time). In Section 5, we give an example of the inverse problem (1.1)-(1.4).

2. Preliminary Results

This section presents some preliminary results, including the well-posedness for a fractional differential equation, an equivalent lemma for our inverse problem, and a technique result, which will be used to prove our main results.

Let Z2(t)n(x) = £~i(n,en) ¿£«,2(-A„ta) e„(x), (x,t) € QT, for n € L2(Q). We first consider the following initial and boundary problems:

'Sf u(x, t) + Au(x, t) = F(x, t), (x, t) € QT, u(x,t)=0, (x,t) € XT, (2.1)

ku(x, 0) = a(x), ut(x, 0) = b(x), x € Q.

Note that if a = 1 and a = 2, then equation (2.1) represents a parabolic equation and a hyperbolic equation respectively. Since we are interested mainly in the fractional cases, we restrict the order a to 1 < a < 2.

First split (2.1) into the following two initial and boundary value problems:

'Sf v(x,t) + Av(x,t) = 0, (x, t) € QT,

v(x, t) = 0, (x,t) € XT, (2.2)

v(x, 0) = a(x), vt(x, 0) = b(x), x € Q,

and

'dtaw(x,t) + Aw(x, t) = F(x,t), (x, t) € QT, w(x,t) = 0, (x,t) € XT, (2.3)

^(x, 0)=0, wt(x, 0)=0, x € Q.

Similarly to Theorem 2.3 in [32], it is easy to obtain the following assertion. Lemma 2.1. Let a € H2(Q) nH(Q) and b € Hq(Q). Let 7 > 0. Then for the unique weak solution v € C([0, T]; H2(Q) n H(Q)) n C 1([0,T]; D(A-Y)) to (2.2), there exists a constant c > 0 satisfying

l|v(-,t)||H2(n) + ||vt(■, t)||d(a-y) < c (HaHn2(Q) + ||b||Hi(Q)) . (2.4)

Then we have

iv(x,t) = Zi (t)a(x) + Z2(t)b(x), (x,t) € QT, (2 5)

|vt(x,t) = -Y(t)a(x) + Zi(t)b(x), (x,t) € QT, ( . )

using

Zi(t)n(x) = e„ )Ea,i(-A„ta)e„ (x), Y (t)n(x) = ^ A„(n, e„ )ta-1E«;«(-Ara ta)e„ (x),

n=1 n=1

the space in C([0,T]; H2(Q) n H1(Q)) n C 1([0,T]; D(A-Y)).

< The uniqueness of a weak solution is verified similarly to Theorem 2.1 in [32], but smoothness is taken in a different form. Therefore, here we show only (2.4) inequality. Using the Lemma 1.1 and (1.5), we have

œ œ

IK>t)|lH 2(0) = E An I (a, era)Ea,i(-A„ia)|2 + E An |(b,e„ )iE«;2(-A„ia)|2

n= 1 n= 1

< IMI^n) +c2t Ub, en? 2 fr2'*-

n=1 \ n )

Using An_2/° ^ a1_2/° , n = 1,2,..., we have

H2(0) ^ c2(llaNH2(0) + IHlH1 (0) . (2.6)

Further, as a second equality of (2.5), we have

œ œ

D(A-y ) = E A-2Y |AnÎa_1(a,en )Ea,a( —Ania) 1 + E A_27 |(b,en )Ea,1( —Anta) |

n=1 n=1

œ (a_1)/a\ 2 œ

^ /(\ ta)(a-1)/a\ ^

< E A2(«>e«)2 1 + ^ An2(7+1"1/a) + E ^e«)2 A«2(7+1/2)-

n=1 \ / n=1

In view of y > 0, we get A-2(Y+1-1/a) < A-2(Y+1-1/a) and A-2(y+1/2) < A-2(y+1/2) . Now, using Lemma 1.1, and (1.4), we have

llv M)l| D(A-y) ^ °2(||a||H2(Q) + IHlH1 (Q) . (2.7)

Thus the proof of Lemma 2.1 is complete. >

We introduce the following auxiliary lemmas to obtain the main results.

Lemma 2.2. Let F € C([0, T]; D(A1/a)). Then there exists a unique weak solution w € C([0, T]; H2(Q) n Hq(Q)) to (2.3) with dtaw € C([0,T]; L2(Q)). In particular, for any Y > 0, we have wt € C([0,T]; D(A-Y)), lim^o ||w(-,t)||H2(q) = lim^o ||wt(-^Hd^-y) = 0, Moreover, there exists a constant c > 0 such that

||w(',t)|H2(Q) + ||wt(',t)|D(A-Y) ^ c(t + 1)||F([0,t];D(A1/a)) (2.8)

and we have

t

w(x,i) = y A-1Y(t - s)F(x,s) ds, (x,t) € QT, (2.9)

o

the function (2.9) holds in the C([0,T]; H2(Q) n H(Q)) n C 1([0,T]; D(A-Y)).

< The uniqueness of the weak solution is proved similarly to Theorem 2.1 in [32]. Therefore, here we omitted it and we show only regularity, besides (2.8).

We first have

n=1

J(F(•, s), en)(t - s)a-1E«;a(-An(t - s)a) ds

<

^max |(A^en) |2

n=1

t

J A—1/a(t - s)a-1Ea;«(-An(t - s)a) ds

(2.10)

<

^max |(An/aF,en)|2

n=1

i\ c>a\ (a—1)/a (AraS } —A~lds

a n

1 + AnS

—2 I|rI|2

<cA—2 11F

I C([0,t];D (A1-7«))

t2

Furthermore, in a view of the condition of Lemma 2.2, for F € C([0, T]; D(A1/a)) and by Lemma 1.1, we have

II^M)^) = £A

n=1

I(F(., s), en)(t - s)a—1Ea,a(-An(t - s)a) ds

<E A2nl |(F(-,s),en)|2 ds / (t - s)2a—2 |Ea,a(-An(t - s)a)|2 ds <£ AnA—2/a (2.11)

n=1 n n n=1

x max |(A1/a[F],en)|2

0<s<t1v 71

(Ansa)(a—1)/a

1 + Ansa

ds A-(2a—2)/at < cII fI2

ds An < c llF Ilc([0,t];D(AV«))

t2.

By (2.3) and (2.10) we can estimate also ||dfw(-,i)||c ([o,T];L2 (n)) and we have limt^o ||w(-,t)||H2(n) = 0. Next apply Lemma 1.1, Proposition 1.1, and apply the Cauchy-Schwarz inequality, and for any 7 > 0, we have

IM-,t)||

2

D (A-y )

A

n=1

-2y

n

y (F (., s), en )(t - s)a—2Ea,a-1 (-An (t - s)a) ds

<

E A—27A—2/a max | (A1/a[F](, s), en) |2

n=1

/(t - s)a—2Ea)a—1(-An(t - s)a) ds

< E A«27" n^l {Al/a[F]{;s), en) f

n=1

0<s<t

ds

(sa—1Ea,a(-Ansa)) ds

< E A—2Y—2/a max |(A1/a [F](-,s),e^ |2 |ta—1Ea,a(-Anta)|2 < E A—2y

n=1

l—27—2/a

n

n=1

x max |(A1/a[F](-,s),en)|2

0<s<t1v 71

(Anta)(a—1)/a

1 + An ta

\ (2—2a)/a < ||F I2

An < cA1 llF II C([0,t];D(A1/a)).

Therefore limt^0 t) ND(A__^) = 0- Thus the proof of Lemma 2.2 is complete. >

By Lemma 2.1 and 2.2, we get the following assertion.

Lemma 2.3. Let a € H2(Q) n Ho0(Q), b € Ho0(Q) and F(x,t) € C([0,T]; D(A1/a)). Then there exists a unique weak solution u € C([0,T]; H2(Q)nH01(Q))nC 1([0,T]; D(A-Y)) to (2.1),

t

2

2

t

2

t

2

t

t

2

t

2

t

2

t

2

d

such that

lluM)llff2 (fi) + ll^t M^^A -7) ^ C[lalH2(fi) + llblH1 (fi) + (t + 1)lFlC([0,i];D(A1/a))] (2-12)

for all t € [0, T], where the constant c is dependent on a, Q and the coefficients of A, but does not depend of T. Furthermore, we have

t

u(x, t) = Zi(t)a(x) + Z2(t)b(x) + J A-1Y(t - s)F(x, s) ds, (x, t) € QT, (2-!3)

where Zj (i)[-j (j = 1,2) and Y(i)[-j are defined above.

The next two lemmas are regularity results of the solution u of the problem (2.1). Lemma 2.4. Let a € D(AY+1/a), b € D(AY) and F € C([0, T]; D(AY)). Let 1/a < e < 1. Then u € X0T such that

llu(->t)lD(AY+1/a ) + lut(',t)lD(A7 ) < c(llallD(A7+i/«) + II&IId(AY) + (ta-1 + ta(1-£)) llFllc([0,t];D(AY)))•

< By Lemma 1.1 and the Cauchy-Schwarz inequality, we have

(2.14)

R,i)||D(AY+i/«) = E An7+2/a I (a, era)Ea,i(-A„ia)|2 + E AnY+2/a t2

n=1

n=1

x |(6,e„)E«;2(-Araia)|2+ EAnY+2/c

n=1

y (F (■, s), en )(t - s)a-1F«;«(-A„(t-s)a) ds

< c2 E a27+2/" I(«»!" + E Xnf(b, e„)2 2

n=1 n=1

(2.15)

^max |(AY[F](-,s),e„)|

n=1

From Lemma 1.1, we have

t

J An/a(t - s)a-1 F«,«(-A„(t - s)a) ds

F«,«(-A„(t - s)a) <

1 + A„(i - s)c

< cA-£(i - s)"

(2.16)

for any 0 < e < 1. Let 1/a < e < 1. Because of these inequalities, rewrite the inequality (2.15) as follows

|U(', t) ||D(AY + 1/a) < C ||a|D(AY + !/a) + C ||b

■2||b||2 a) 1 ^ IrllD(Ay)

+ c2V max |(AY[F](-,s),e„)|

' Q<s<i 1

n=1

f A/a-e (t - s)a

Q

/a-^ s)a-a£-1 ds

2

< C2||a||2 + c2 | b |2 + c2A2/a-2^ t2a(1-e)||F |

< c Irll D(AY+1/a) + c ||b|| d (Ay ) + c A1 b llF ll C([Q,i];D (Ay )) •

As a result, we get

||u(,,t)||D(Ay+i/a) < c(a)(||a||D(AY+i/a) + ||b||D(AY) + ta(1-£)||F||C([Q,i];D(Ay))) . (2.17)

oo

2

2

2

c

t

2

Furthermore, by Lemma 2.3, we have

ro

ut(x, t) = E {-Anta-1(a, en)Ea>a(-A„ta) + (b, e„)Ea>1 (-Anta)} e„(x)

n=1

+ j(F(■, s), e„)(t - s)a-2Ea,a-1 (-A„(t - s)a) ds j e„(x). Therefore, applying (1.5), Lemma 1.1 again, and An = O(n2/d), we have

ro ro

11ut(■, t)||D(ay) =E An7An |(a,en)|2 |ta-1E«;„(-Anta)|2 + E A^|(b,en)|2

n=1 n=1

(2.18)

ro

x |Ea,1(-Anta)|2 + E A^

n=1

t

J(F(•, s), en)(t - s)a-2Ea>a-1 (-An(t - s)a) ds

ro ^(Anta)(a-1)/n2 , ^ (2.19)

< c2 E A27+2/>>e«)2 'TA ya + ■c2 E A27(&> e«)2

X 1+ Ant^ / n=1 n=1

ro

^EmfX |(AY[F](',s),en)|2

n=1

t

[ sa~2-?-ds

J 1 + Ansa 0

2

Thus,

^ c ||a||D (AY+1/a) + c ||b||D (AY) + ct( )||F ||C([0,t];D(AY)).

||ut(,t)||D(AY) < c(||a|D(AY+i/a) + ||b||D(AY) + ta 1|F||C([0,t];D(AY))) (2.20)

for all t € [0,T]. Then we immediately obtain the desired estimate (2.14). This completes the proof of this lemma. > It is easy to see that

roro

Au(xi ,t) = E An (a, en)Ea,1(-Anta)en(xi) + E An(b, en )tEa, 2(-Anta)en(xi) n=1 n=1

+ E An ^ /(F(■, s), en)(t - s)a-1Ea;«(-An(t - s)a) d^ e^x*), i = 1,2.

The following lemma is valid.

Lemma 2.5. Let a € D(AY0+1/a), b € D(AY0) and F € C([0,T]; D(AY)). Then there exists a positive constant c such that

||Au(x, ■)||c[0,t] < c(|a|D(AY0+i/a) + ||b|D(AY0) + Ta/2 ||F||c([0,T];D(AY))), i = 1,2, (2.22) and

|Aut(xi, ■)||c[0,T] < c(|a|d(AY0+i/a) + ||b|D(AY0) + Ta-1||F||C([0,T];D(AY))), i = 1,2, (2.23) where c is dependent on Q, a, 7, 70, d, A1.

2

< The estimate (2.22) may get similarly as in [33]. However, the smoothness differs from the given ones, so we provide the above inequality (2.22) in detail.

We note that A defines the fractional power A^ with ft € R and ||u||H2^(Q) ^ c|Au||L2(q) (see, [29]).

Let e0 = min{e01, e02} with 2e01 = y0 + 1/a - 1 - d/2 > 0 and 2e02 = 7 - d/4 - 1/2 > 0. According to the Sobolev embedding theorem H2^(Q) c C(Q) for ft = d/4 + e0, we have

|en||c(Q) < c(Q)|enH^^ < c(Q)|Aenl^ < c(Q)A^ (2.24)

For simplicity, we study Au(xi,t) in three parts, namely Au(xi,t) := I1 +12 +13. For I1, by Lemma 1.1, and noticing that An = O(n2/d), we have

ro

+a\l l„ (x.)l ^ „^n „A ^ " \70 + 1/a|

|Ii| ^ A„|(a,e„)| |E«;i(-A„ta)| |e„(x)| < c(fi,a) £ An°+1/a|(a,e„)|A-(Y°+1/a-^-1)

n=1 n=1

/ œ \ 1/2 / œ \ 1/2

< c(n,a) E AnY0+2/a|(a,en)|2 EA-2(7°+1/a-/?-1)

\n=1 J \n=1 J

/ œ \ 1/2

< c(n,a)|a|D(AY°+i/a) E n-4(Y°+1/a-^-1)/d .

/a)

n=1

By the choice of ft, we have 4(y0 + 1/a - ^ - 1)/d = (d + 8e01 - 4e0) > 1, which implies Ero=1 n-4(Y0+1/a-^-1)/d < c(70,a,d). So, we obtain

1111 < c(Q,a,70,d)||a||D(AY0+i/«). (2.25)

Further, by Lemma 1.1 and (1.5), we have the following estimate for I2:

œœ

Ant

N < EA* e™)ii Mx*)i < «) E K6' e«)i rrk^

n=1 n=1

n1' n,1-

1/2

œ (A ta)1/a œ \ 1/2

< c(0, a) E ^|(6, en)\ \ n / A-(7o+i/«-/3-i) ^ c(f)> a) ^ \^\(b,en)\2 ) (2.26)

, 1 + Anta .

n=1 n=1

1/2

x E A-2(Y0+1/a^-1)J < c(Q,a,70,d) ||b||D(AY0).

Next we calculate I3. Here the estimate for I3 as the same as [33] for 7 - ft -1/2 = 2e02 - e0 > 0, and we have

|I3| < c(Q, a)ta/2 ||F11c([0,t];D(AY)) (Vt € [0,T]). (2.27)

According to (2.25)-(2.27), we obtain (2.22).

By directly differentiating (2.21) concerning the variable t and taking into account Proposition 1.1, we obtain

d ro ro

—Au(xi,t) = en)ta~l £a,a(-Anf)en(xi)+^A„(6,en)£aii(-A,if)e„(xi)

n=1 n=1

ro / t X (2.28)

+ E A^ /(F(■, s), en)(t - s)a-2Ea;«-1(-An(t - s)a) dsj en(x.) := I1 +12 + I3.

œ

Let e1 = min{e1o, e11} with 2e1o = 70 — 1 — d/2 > 0 and 2e11 = 7 — d/4 — 1 > 0. By the asymptotic property of the eigenvalues An = O(n2/d), for i1, by Lemma 1.1 and (1.5), we have

I11 m E An |(a, e„)| ta-1 |E«,«(—A„ia)| |e„(xt)| m c(fi,a)£ An0+1/a |(a,en)|

n=1

/ ro

1/2

ro

n=1

(A „i")^"1)/« 1 + A nta

1/2

A-(70-^-1) m a) £ AnY0+2/a |(a, en)|2 £ A-2(Y0

> n=1

n=1

m

n=1

1/2

By choice of ¿5, we have 4(y0 — ft — 1)/d = (d + 8e10 — 4e1)/d > 1, which implies En=1 n-4(70-^-1)/d < c(Yo,d). So, we obtain

|111 m C(n,a,7o,d)||a||D(AY0+i/«)-Similarly, we have the following estimate for I 2:

ro ro

| I 21 An|(b,en)||Ea>1 (—Anta)||en(x*)| m c(Q,a)£ An0|(b,en)|

(2.29)

n=1

1/2

n=1 1/2

A-(70-^-1) An

1 + A nta

(2.30)

m c(fi,a) £ AnY0|(b,en)|M n-4(70-^-1)/d m c(Q,a,7o, d)||b||D(AY0).

\n=1 J \n=1 J

Further, we estimate I3. By Lemma 1.1 and 7 — ft — 1 = 2e11 — e1 > 0, we have

Is|2 m £

n=1

t

An / (F(■, s), en)(t — s)a-2E«;«-1(—An(t — s)a) ds ■ e^x*)

m c(Q,a)£ A?y omax |(F(■, s), en)

n=1

„a—2

(1 + Ansa)2

ds

, ,-2(7-8-1)

/\rri

m c(Q,a)V A?7 max |(F(-,s),en) n omsmt

n=1

sa-2 ds

■ A1

-2(7-3-1)

So that | |

|Is| m c(Q, a, A1) ||F||c([o,t],D(AY))ta-1 (Vt € [0, T]). (2.31)

Finally, by (2.29)-(2.31), we get (2.23) and so complete the proof of this lemma. >

To study the main problem (1.1)-(1.4), we consider the following auxiliary inverse initial and boundary value problem.

Lemma 2.6. Let (C1)-(C5) beheld. Then the problem of finding a solution of (1.1)—(1.4) is equivalent to the problem of determining the functions u(x,t) € , q(t) € C 1[0,T] and k(t) € C[0,T] satisfying

(dfu)(x, t) + Au(x, t) = q(t)ut(x, t) + (k * u)(t) + f (x, t), (x, t) € QT, u(x, 0) = a(x), ut(x, 0) = b(x), x € Q,

(2.32)

u(x, t) = 0,

(x,t) € ST,

ro

ro

x

ro

ro

2

ro

t

2

1

2

t

2

2

and

q(t) = ^{h2(0),Ai[u,l](t)-h1(0)^2[u,l](t)), 0<t<T, (2.33)

k(t) = Dt

1

Lp(t)

where Dt := (d/dt), Nj, i = 1,2, are defined by (2.39) below and

0 ^ t < T, (2.34)

t

l(t) = J k(T) dr. (2.35)

On the other hand, if (2.32)-(2.34) has a solution and the technical condition (C1)-(C4) holds, then there exists a solution to the inverse problem (1.1)-(1.4).

Remark 2.1. From Lemma 2.6, we know that (2.32)-(2.34) is an equivalent form of the original inverse problem (1.1)-(1.4). So, in the next sections, we discuss (2.32)-(2.34), other than the original one.

< The solution (u(x,t),q(t),k(t)) € Y0T of our inverse problem (1.1)-(1.4) is also a solution to the problem (2.32) in Y0T. Because the problem (2.32) is the same as (1.1)-(1.3). Therefore, we should show only (2.33) and (2.34). Let the three {u(x, t), q(t), k(t)} functions be a solution of problem (1.1)-(1.4). Taking into account the conditions of Remark 1.1 and implies h € C 1[0,T], and fractional differentiating both sides of (1.4) respect to t gives

(d»(xi,t) = (dahi)(t), ut(xi,t) = hi(t), 0 < t < T. (2.36)

Setting x = xj in Equation (1.1), the procedure yields

t

dtau(xj,t)+ Au(xj ,t) = q(t)ut(xj,t) + J k(t - r)u(xj ,t) dr + /(xj,t), i = 1, 2. (2.37)

o

We note that l(t) = /0 k(r)dr. Then by integration by parts, we get the following equality:

t t J k(r)hi(t - t) dr = hi(0)1(t) + ^ 1(t - t)hj(r) dr. (2.38)

oo

With the help of (2.36) and (2.38), we can rewrite (2.37) as hj(t)q(t) + hj(0)1(t) = dtahj(t)+ Au(xj,t) - (l * hj)(t) - /¿(t) := Nj[u, l](t), i = 1, 2. (2.39) Due to (C4), we can solve this system to get (2.33) and

l(t) = ^ (2-40)

Furthermore, by differentiating (2.40) concerning t, we get (2.34).

Now we assume that (u, q,k) satisfies (2.32)-(2.34). In order to prove that {u,q,k} is the solution to the inverse problem (1.1)-(1.4), it suffices to show that {u, q, k} satisfies (1.4). Setting x = xj to the Equation in (2.32), we have

(dtaut)(xi,t) + Au(xj,t) = q(t)ut(xj,t) + (k * u)(t) + /(t). (2.41)

On the other hand, from (C2), we easily see that ^{h'^fyJQu, l](0) - h'2{0)Ai[u, Z](0)) = 0. We get (2.40) by integrating (2.34) over [0, t]. From (2.33) and (2.40), we conclude that

hi(t)q(t) = -hi(0)1(t) + Sahi(i) + Au(xi,t) - (l * hi)(t) - fi(t) = da hi (t) + Au(xi, t) - (k * hi)(t) - /¿(t)

or

fi(t) = -hi(t)q(t) + dahi(t) + Au(xi,t) - (k * hi)(t). (2.42)

Then substituting (2.42) into (2.41), and using (C3), we have that Pi(t) := u(xi,t) - hi(t), i = 1,2), satisfy

f SfPi(t) = q(t)p'(t) + (k * Pi)(t), t > 0, \Pi(0)= P/(0) = 0.

Then, the fractional initial value problem (2.43) is equivalent to the integral equation (see, [27, p. 199])

t , t , W) = -^y J ( J(t ~ T)a~lk{T - s) drU(s) ds

t 0 s t (2.44)

1

J(t- s)a" V(s)fi(s) ds + r(aX_ 1} fit- s)a~2 q{s)Pi{s) ds, ¿ = 1,2.

r(a) J v J H K J lK J r(a - 1) 00

This is a weakly singular homogeneous integral equation, and it has only a trivial solution for q(t) € C:[0,T] and k(t) € C[0,T] (see, [27, p. 205]). Then, u(x,t) - hi(t) =0, 0 < t < T, i. e., the condition (1.4) is satisfied. This completes the proof of Lemma 2.6. >

At the end of this section, we give a lemma that will be used to estimate q and k. Lemma 2.7. Let (C1) hold. Then for all (u,q,k) € Y0T and I € C*[0,T], there exists a constant c > 0 depending on f, a, b, hi, but independent of T, such that

M]||c1[0,T] < C [! + (Ta/2 + T"-^ (1 + llqNc[0,r] Nut ||C([0,T];D(Ay))) + (Ta/2+1 + Ta)NfcNc[0;T]NuNc([0;T];D(AY+i/a)) + T1/2 ||1||ci [0,T]] ,

(2.45)

where Ni, i = 1,2, are the same as those in (2.39) and l(t) is in (2.35). < By Lemma 2.5 and condition (C1), we see that

||Ni[U, 1]||C[0,T] ^ llSahi||C[0>T] + ||Au(xi,t)NC[0,T] + 111 * hi|c[0,T] C[0,T] ^ ||dt ||C[0,T]

+ c(11 a 11 D( Ayo+i/a ) + ||b||D(AY0 ) +T a/2|F ^([O.T ];D (Ay ))) +T 1/2 ^^[O.T ]|hi|L2(0;T ) + ||/i|c[0,T ].

By the definition of F, the last inequality becomes

||N£M]||C[0>T] ^ ||dahi|C[0,T] + c [11 a 11D(Ayo+i/a ) + ||b||D(AY0) + T"/2(^^[OjT] |ut |C([0,T];D(AY)) + A- 1/aT ||k||c[0,T] ||u||C([0,T];D(AY+1/«)) (2.46)

+ ||f ||C([0,T];D(Ay)))] + T1/2 ||1 |C[0,T] ||hi H L2(0,T) + ||fi ¡cp.T],

where we have used

lv|D(AY) = 5-/ An7+ / (v,en) A- / ^ / ||v|D(AY+I/a)-

n=1

On the other hand, direct calculations imply

ANiM](t) = (3*h)' + Aut(xi,t) - (1' * hi)(t) - fi(t). (2.47)

Here we note that 1(0) = 0. By Lemma 2.5, we have

||DtNi[u,1]|C[0)T] ^ ||(dahi)'|C[0,T] + c[||a||D(AY0 + 1/«) + (AY0)

+ T"-1(||qyc[0,T]yuiyC([0,T];D(AY)) + A- 1/a T ||k||c[0,T] IMIc([o,T];0(Ay+i/*)) (2.48)

+ ||f||C([0,T|;D(AY)))] + T1/2 ||l'|c[0,T]|hi|L2(0>T) + ||fi Lp.ïy

(2.46) and (2.48) bring the desired estimate (2.45). This is complete proof of this lemma. >

3. Existence of the Solution to an Inverse Problem

We can now prove the existence of a solution to our inverse problem, i. e., Theorem 1.1, which proceeds by a fixed point argument. First, we define the function set

Bp,T = { (u, fc, fc) € Y0T : u(x, 0) = a(x), ut(x, 0) = b(x), u(x, t) = 0,

(x,t) € , + ||fc||Ci[0,T] + ||fc||C[0)T] ^

Here r is a large constant depending on the initial data a, b, f measurement data h. For given (u, fc, fc) € Bp,T, we consider

'(d*u)(x, t) + Au(x,t) = F(x, t), (x, t) € QT, u(x, 0) = a(x), ut(x, 0) = b(x), x € Q, (3.1)

ku(x, t) = 0, (x, t) € ST,

where F(x, t) = fc(t)ut(x, t) + (fc * u)(t) + f (x, t), and

g(i) = (h2(0)^i[u,i](i) (3.2)

d /hi(t)N2[u, fc](t) - h2(t)Ni [u, fc](t)

to generate (u, q, k), where Z(t) = f0 fc(r) dr, Nj, i = 1,2, are the same as those in (2.39). By Holder's inequality, we have

t

* U)(t)||!(AY ) ^y"|fc(t - T )|2d^ ||u(',T )|D(Ay )dT < A-2/at2|fc|C [0>t] ||u|D(AY+1/a ) (3.4) 00

which implies ||(fc * u)(t)||C([ot];d(Ay)) ^ A-1/ap2T. Furthermore

(fc(t)ut(-,t),e„ )

2

lq Utllc([0,T];D(AY)) = o™|t

n=1

^ ||fc|C[0,T]|ut|C([0,T];D(AY )) ^ P . (3.5)

t

Using these results together with f € C 1([0, T]; D)), we have fc(t)ut(x, t) + k*u + f (x, t) € C([0, T]; D(AY)). By Lemma 2.4, the unique solution u € of the problem (3.1), given by (2.13) satisfies

IMIxT < c(||a|Id(AY+1/a) + II^Nd(AY) + (Ta-1 + Ta(1-£))||F||c([0,T];D(AY))). (3.6)

Further, (3.2)-(3.3) define the functions q(t) and k(t) in terms of u. Furthermore, by Lemma 2.7, we have

(|hi(0)| + |h2(0)| + ||hiNci[o,T ]

C1[0,T ]

NqNCi[0,T] + ||k||C[0,T] < c + I|h'2Nci[0,T]^ 1 + (Ta/2 + Ta-1) (1 + ||fc Nc[0,T] IH|C([0;T];D(AY))) (3J)

+ (T"/2+1 + T") ||k||C[0;T] ||u||C([0,T];D(AY+Va)) + T1/2 ||fc Hci[0,T]) . Note l(t) = /J k(r) dr. We obtain

r |C1[0,T]

+ ||k||C[0,T] ^ (1 + T)|k|C[0,T]. (3.8)

C[0,T ]

J k(T) dT

o

Substituting (3.8) into (3.7) yields

NqNCi[0,T] + ||k||C[0,T] ^ c(T)[1 + ||(?NC[0,T] ||ut NC([0,T];D(AY)) + ||k||C[0,T] ||u||C([0,T];D(AY+i/a) + P^cp.T]] .

(3.9)

This implies that q(t) € C 1[0,T] and k(t) € C[0,T]. Thus the mapping

Z : Bat ^ Y0t, (u,fc,fc(u,q,k) (3.10)

given by (3.1)-(3.3) is well defined.

The next lemma shows that Z is a contraction map on for sufficiently small T > 0. More precisely, we have the following result.

Lemma 3.1. Let (C1)-(C5) be hold. For (u,fc,k), (U,Q,K) € Br>T, define (u, q, k) = Z(u, fc, k), (U, ) = Z(U,Q,K). Then for properly small t > 0, we have ||(u, q, k)||yT ^ p and

||(u-f/,q-Q,k-K)|LT < J ||(u-[/,q-Q,k-ii)|LT (3.11)

ii ii y0 2 11 v 11 Yo

for all T € (0, t].

Everywhere the following proof, we use Cj to denote a constant which depends on Q, a, 7, 7o, A1 and the known functions a, b, f and measurement data hj, i = 1,2, but independent of p and T.

< First we prove that the operator Z(Bp,t) c for sufficiently small T and suitable

larger p. To simplify the calculations, we restrict T € (0,1]. From Lemma 2.4, (3.4)-(3.6), we have ( ) ( )

I|u|Ix0T < cA-(7°-Y)(NaND(AYo+i/a) + ||b||D(AYO)) + c(Ta-1 + Ta(1-£)) x (||fc(t)ui NC([0,T],D(Ay)) + || (k * u) |C([0)T])D(AY)) + Nf NC([0,T],D(AY))) (3.12)

< C1[1+ (Ta-1 + Ta(1-£))(1 + p + pT)].

t

On the other hand, by (3.2)-(3.3), together with Lemma 2.7 and (3.8), we have

l|q||cl[0,T] + ||k||C[0,T] < c2 (||N1 [uJ] ||C1[0>T] + ||N2 K I] ||C1[0>T]) < cs(1+ Ta/2 + Ta-1 + p(Ta/2 + Ta-1)lutlc([0,T];D(A,)) + p(Ta/2+1 + Ta) ||u(c([0,T];D(Ay+1/*)) + pT 1/2(1 + T)) ( . )

< C3 (1 + Ta/2 + Ta-1 + p(Ta/2 + Ta-1) ||u||xt + pT 1/2(1 + T)),

where we have used the assumption T € (0,1]. Then, adding up (3.12) and (3.13) leads to

||(u,q,k)|Lt < C4(1+ Ta/2 + Ta-1) + C4p(Ta/2 + Ta-1)

(3.14)

x (1 + Ta-1 + Ta(1-£) + p(1 + T)(Ta-1 + Ta(1-£)) + T1/2 + T3/2). We choose sufficiently small r1 such that

no—1\ i „ „ (rjia/2 \ -Tia-1\

(Q 1

x (1 + Ta-1 + Ta(1-£) + p(1 + T) (Ta-1 + Ta(1-£)) + T1/2 + T3/2) < p, and therefore, for all T < min{1,r1} we have

||(u,q, fc)^ < p. (3.16)

That is, Z maps Bp,T into itself for each fixed T € (0, min{1,r1}].

Next, we check the second condition of contractive mapping Z. Let (u, q,k) = Z(u, q, k) and (U, Q, K) = Z(U, Q, K). Then we obtain that (u - U, q - Q, k - K) satisfies that

t

u(x,t) - U(x,t) = y A-1Y(t - s)F(x,s) ds, (x,t) € QT, (3.17)

0

and

q(i)-Q(i) = (3.18)

, m d (h WW K ^^ (t))-h'2(t)(Jj [u, I] {t)-<A1 [[/, L] (¿)) ^

mK[t)~dt{ p(t) J' ^

where L(t) = /0 K(r)dr and F := q(u - Ut) + (q - Q)Ut + k * (u - U) + (k - K) * U. Using Lemma 2.4, (3.5) and (3.6), we get

||u - U||xT < c(Ta-1 + Ta(1-£))[|(fc-Q)ut|c([0,T],D(AY)) + ll(ut-Ut)q1c([0;T];D(Ay))

+ H (k - K) * u||c([0,T],D(AY)) + ||k * (u - U) ||C([0,T],D(Ay))] < c(T"-1 + T"(1-£)) X - Q||C[0,T]||ut||C([0,T];D(Ay)) + ||ut - Ut ||C([0,T],D(Ay))||fc !C[0,T] + ^ (3.20)

X ||k-K||C[0,T] ||u|C([0,T],D(AY+1/a)) + / ||u-U||c([0,T],D(AY+1/a))|k|C[0,T]

< pc(Ta +Ta(1-£)) max {1,T2A-1/a}[||q - ^^] + ||u - UH^t + ||k - K||c[0,t] .

(3.21)

Similarly, by (3.18)-(3.19) and Lemma 2.7, we have

||q - QNci[o,T] + ||k - KNC[o,T] < pc(Ta/2+1 + Ta-1) max {1,T2A-1/a,T3/2 } x ||q - QNc[o,T] + ||u - U+ ||k - K|C[0,T] .

Therefore, by (3.20) and (3.21), we have

|| (u - U, q - Q,k - K) H^t < cp (Ta + Ta(1-e)) max {1, T2A-1a} + (T«/2+1 + Ta-1) max {1, T2A-1/a, T3/2}] H(u - U, fc - Q,k - K)1 Hence we can choose sufficiently small t2 such that

"(T"+T"(1-£))max{l,T2A-1/a} + (T"/2+1 + T"-1)max{l,T2A-1/a,T3/2}l < - (3.23)

(3.22)

lY T. 1 Yo

Cp

for all T € (0, t2] to obtain

2

\\(u-U,q-Q,k-K)\\YT < i ||(u-[/,q-Q,k-ii)|LT. (3.24)

ii ii y° 2 Yo

Estimates (3.16) and (3.24) show that Z is a contraction map on for all T € (0,t], if we choose t ^ min{1, T1, t2}. >

To prove the main result, we should prove the following assertion.

Lemma 3.2. Under conditions (C1)-(C5), for given measurement data hj(t) for i = 1, 2 in (1.4), if the inverse problem (1.1)—(1.4) has two solutions (uj, qj, kj) € YJT (j = 1, 2) for any time, then (u1,q1,k1) = (u2,q2,k2) in [0,T].

According to Remark 2.1, we know that (2.32)-(2.34) is equivalent to (1.1)-(1.4). So, in Lemma 3.2 we discuss the global uniqueness of the inverse problem (2.32)-(2.34).

< Given any time T, let (ui,qi,ki), i = 1,2, be two solutions to the inverse problem (2.32)-(2.34) in [0,T] with the regularity (ui,qi,ki) € Y0t. This implies

||uj,qi,kj||Y°o ^ C*, i = 1, 2, (3.25)

where C* is depending on a, T, initial data ^ and the known function f and measurement data hi.

Let u = u1 - u2, q = q1 - q2, k = k1 - k2. Then (u, k, k) satisfies

and

dfu + Au = q1ut + ku2t + k1 * u + k * u2, (x, t) € ,

u(x, 0) = ut(x, 0)=0, x € Q, (3.26) ku(x,t) = 0, (x, t) € XT,

q(i) = -!- (h2(0)Au(xi,i) -hi(0)Au(x2,i) (3.27) p(t)

= ^h'i(t)(Au(yL2,t) - U h'2) - h'2(t)(Au(yLi,t) - U ^ ^

where T(t) = 11 - 12 and the functions Zj, i = 1,2, satisfy ¿¿(t) = f^ k(s)ds. We have to show

||(u,q,k)||YT = 0. (3.29)

II nr°

Define

a = inH t € (0, T] : || (u, k, k) ||yt > 0

(3.30)

It suffices to prove that a = T .If (3.30) is not true, then it is obvious that a is well-defined and satisfies a < T. Choose e such that 0 < e < T - a. Further, by (2.13), we can write the solution u as

u(x, t) = I A-1 Y(t - s)F(x, s) ds, (x, t) € Q\

(3.31)

where

F(x, t) = q1ut + qu2t + k1 * u + k * u2. Then similar to the proofs of Lemma 2.4 and 2.5, we have

u I

< C5(ea-1 + ea(1-£))||F|

C([o-,CT+e];D (Ay )),

and

< c?ea-1 || F||

e«/2 11F

' ' 11 C [<T,<T+£] '

|Aut(xj

From the definition of a, we see that

u = k = k = 0 in [0, a]. By the definition of F, and using (3.4), (3.5) and (3.25), we have

Due to k(a) = 0, then implies

Ilkllc

k'(s) ds

< e |k||Cl[a)CT+e].

Substituting (3.36) into (3.35), we have

11 uk | IxCT+e

< CsC* (ea-1 + ea(1-£)) max {1, e, A-1/ae} ^ (u, k, k) |

Y

<7 + e .

(3.32)

(3.33)

(3.34)

||u||X£+E < Cs(e" 1 + e"(1 (|k1ut|C([CT,CT+e];D(AY)) + |ku2t|C([CT,CT+e];D(AY))

+ ||k1 * uyC([a,a+e];D(ay)) + ||k * u2|C([CT;CT+£];D(AY))) < csC*(ea-1 + ea(1-£)) (3.35)

CT,CT+e];D(AY))+ ||k||C[o-,CT+e]+ A1 / e||u||C([o-,CT+e];D(AY+1/a))+ A1 / e||k||C[o-,CT+e]

(3.36)

(3.37)

Note |k|C1[0,CT] = llkllc[0,o-] = 0. On the other hand, by (3.27), and using (3.33), we have the following estimate for qk

|C1[CT,CT+e] < c<H e +e )

h2(0)

p(t)

+

C 1[CT,CT+e]

h2(0)

p(t)

f|

C M^o+e]

C([o-,CT+e];D(AY ))

+ e ||P|C[a,a+e] ||k|c[a,a+e] < CsC(hKe"^ + e" ^ (J|ut|C([a,CT+e];D(AY)) + e |k||C1[a,CT+e] + Ai1/0e |u|C([CT,CT+e];D(AY+1/a))) + C) e3/2 11 k 11 c[CT,CT+e],

(3.38)

t

t

where we have used that

= max

G[o-,CT+e] o^t^o+e

k(s) ds

^ e llk|

C [o-,CT+e].

Similar to (3.38), by (3.28) we can easily estimate for k as follows

||k||CKa+e] ^ C(hi) C10 (e°/2 + ea-1) (NutNC([CT)CT+e];D(AY)) + e ||k||C1[ff,ff+e] + A!1/ae ||u||c([CT,CT+e];D(AY+1/a))) + e3/2 HkHC[a,CT+e] .

From (3.37)-(3.39), we obtain

H (u,k,k) HY , ^ C(hj, C*)n(e)H (u,k,k) HY

(3.39)

(3.40)

with lime^+0 n(e) = lime^+0(ea/2 + 2ea-1 + ea(1-e)) max{1, e, A-1/ae, e3/2} = 0, and implying ||(u, k, k)||Y(T CT+e = 0 for some sufficiently small positive constant e. This means that (u1 - u2,q1 - q2, k1 - k2) vanishes in [0, a + e], which contradicts with the definition of a. Therefore (3.29) is proved. From here, we can conclude that (u1,q1,k1) = (u2,q2,k2) in [0, T] for any time T. >

4. Proof of the Main Result

In this section, we give proof of the global solubility of the solution to our inverse problem, i. e., Theorem 1.1.

Lemma 3.1 ensures that there exists a unique solution (u, q, k) € Y0T of the inverse problem (2.32)-(2.34) for sufficiently small t > 0. In this section, we show that the unique solution (u, q,k) in [0,t] can be extended to a large time interval [0, 2t].

To do this, we consider

T

(dtav)(x, t) + Av(x, t) = y(t)vt(x, t) + J k(t - s)u(x, s) ds

o

t

+ J r(t - s)v(x, s) ds + f (x, t), (x, t) € QT,

(4.1)

and

v(x, T) = u(x, T), vt(x, T) = ut(x, T), v(x,t) = 0,

1

x € Q,

(x, t) € XT,

= (h2(0)Ni [v, IT] (t) - hi (OP* [v, r ] (t)), r^t^T,

r(t) =

d /h1(t)N2 [v, ZT] (t) - h'2(t)N1 [v, ZT] (t)

dt

p(t)

t < t < T,

(4.2)

(4.3)

where

T t

N[v, fT ] (t) := dtahj(t) + Av(xj ,t) -J Z(t - s)hj(s) ds -J fT (t - s)hj (s) ds - /¿(t), (4.4)

t

and fT(t) = /T r(s)ds. Obviously, if we prove that there exists a solution (v,y,r) € YTT with some T ^ 2t, then (u, q, defined by

(ûôjfeï ^(u,q,k), t € [0,t], (45) \(v,y,r), t € [t, 2t], V ' '

is a solution of the inverse problem (4.1)-(4.3) on the larger interval [0, 2t].

We repeat a similar fixed-pointed argument to prove the existence of (v,y,r). Define an operator

K : Bp? ^ YTT, (v,y,f) ^ (v,y,r) (4.6)

with (f , y, f) € Bp,T, where

-Bp,T = {(v , y , f) € YTT : v (x, t) = u(x, t), vt(x, t) = ut(x, t), x € Q,

f (x, t) = 0, (x,t) € , HfNxT + lly 11C1 [ t,T ] + ||f IIc[t,T] < p} Here v is the solution to the initial and boundary value problem

((Sfv) (x, t) + Av(x, t) = F(x, t), (x, t) € QT,

<v(x, t)= u(x, t), vt(x, t)= ut(x, t), x € Q, (4.7)

[v(x, t) = 0, (x, t) € ST,

where

.F(x, t) = y (t)ft(x,t)+(k * u)(t) + (f * v )(t + t)+ f (x, t), (x, t) € QT. (4.8)

Furthermore, y is the solution of (4.2) in terms of v and r is (4.3). Additionally, we have u(-, t) € D(A70+1/a) and ut(-, t) € D(AY0). Indeed, in view of (2.13), u(x, t) can be written as

T

u(x, t) = Zi(t)a(x) + Z2(t)b(x) + J A-1Y(t - s)F(x, s) ds (4.9)

0

with F (x, t) = q(t)ut(x,t) + (k * u)(t) + f (x, t) € C ([0,t]; D(Ay )) such that ||F ||c([0,t];d(ay )) < c5(p, t, A1, f). Then, by Lemma 1.1, we have

œ

•,t)||D(ayo+i/«) = E AnY0+2/a |(a,en)|2 |E«;i(-A„Ta)|2

n=1

œ œ

+ E An70+2/a |(b, en)|2 |TEa,2(-A„Ta)|2 + E A^0+2/a

n=1 n=1

xE«,a(-A„(T - s)a) ds

T

|(F(-,s),e„)(t - s)a—1

^ c21 ||a|D(AY0 + 1/a) + c22 E An70 |(b, en)|

(A„T a)1/a

œ

^E^X |(AY[F](-,s),e„)|2

n=1

T

/(T - s)

1 1+ AnTa

n=1

2

ds A-2(Y-Y0— 1/a+e).

2

a—ae-1

By y > y0 + 1/a — e, together with (1.4), we get

||u(',T )|D(AY0+1/a) ^ c14 (|a|D(AY0 + 1/a) + ||b||D(AY0 ) + A—Y+Y0+1/a—eTa(1—e)|F ||C([0,T];D(AY )) ).

According to the (2.18), we have

ut(x,r) = E {-A„ra 1(a,e„)Ea;a(-Arara) + (6,e„)Fa,1(-AnTa)} e„(x)

n=1

+ E / (F(■, s), e„)(r - s)a-2Ea;a-1(-A„(r - s)a) ds e„(x).

n=1

Then, by Lemma 1.1 and applying (1.4) again, we have

(4.11)

||ut(-,r)|||(AY0) = E A^0 An I(a,e„)|2 |r«-1Ea;a(-A„ra)|2 + E A^0 |(b,en)|2

n=1

TO

n=1

x |E«,1(-Anra)|2 + E A^0

n=1

I(F(•, s), en)(r - s)a-2Fa;a-1(-An(r - s)a) ds

0

^ /(\ ra)(a-1)/a\ 2 ^

< EA270+2/>>r n1+\nTa ) + EA270(&>

n=1 ^ ' n=1

2

+ E0n?a<xr |(A[F](-,s),en)|

n=1

l

J sa-2Fa;«-1(-Ansa) ds

2

\-27+270 A

(4.12)

< c26 ||a|D(AY0 + 1/a) + CU ||b||D (AY0 ) + E (ma<XT |(AY [F](',s),en)|

2 | | 2

n=1

|r° 1 Ea,a( Anra) | A-2y+270 < C26|a|D(AY0 + 1/a) + c21||b||D(AY0)

+ E0ma<T KA7[F](',s),en)|2

n=1

(AnT «)(«-1)/o

1 + An T0

A-27+270+2/a-2

An,

where we have used

By y > y0 + 1/a - e for 1/a < e < 1, we have 2y - 2y0 - 2/a + 2 > 0. Thus,

||ut(-

, T)|D(AY0) < c17(|a|D(AY0+1/a) + ||b||D(AY0) + ||F||c([0,t];D(ay))). (4.13)

Moreover, by (3.12) we have

a-1

llvlxT < C18A-(Y0-y)(||u(-,T)|d(AY0+1/a) + ||ut(',T)|d(AY0)) + C19((T - T) + (T - T)a(1-£)) (Hy(t)vt |C([T,T],D(AY)) + H (k * u) ||C([0,T],D(AY)) + y * u) ||c([t,t],D(ay)) + ||f Hc([t,T],D(Ay))) < C20A-(Y0-Y^||u(- , T)Hd(Ay0+1/") + l|ut(-,T)Hd(AY0)) + C21 ((T - T)a-1 + (T - T)a(1-£) )(p + A-1/apr + A-1/ap(T - r) + C1).

(4.14)

2

T

2

2

x

t

t

On the other hand, by (4.2), and using (3.33), we have the following estimate for y

IMIcMr,T] < Mo)| ||p-1||cl[T)T]|lNi[v,ir]||cl[T)T] + Mo)|||p-1IU[r>r] 2

x||N [v T ]|C1[T)T] < E [C M1 + H'^ )||D(AY0+i/«) + |M',T) ||d(AY0 ))

i=1

+ C(hi, fi) + C(hi) ((T - T)a/2 + (T - T)a-1) ||y ||c[t,T] ||c([t,T];D(AY)) + C(hi) ((T - T)a/2+1 + (T - T)a ) ||f ||c[t,T] |V|C([T,T];D(AY+1/a))

+ C(hi, f) ((T - T)a/2 + (T - T)a-1) + ||fT * hi|ci[T)T] + C(hi)C*" < C22 (||u(-, T) ||D(AY0+i/a) + |M-, T) ||D(AY0)) + C23 + C24 ((T - t)a/2 + (T - T)a-1) + C25 ((T - T)a/2+1 + (T - T)a) + C26 (T - T + (T - T)3/2). The last term becomes from

(4.15)

11^ * h 11C1 [t,T] ^ lhi(0)H1 llc[r ,T ] + II1 llc[r ,T] llhi IIl!( r,T ) + (T T>' II1 IIC[r,T]ir II L2(r,T) '

here we notice that, by the Sobolev embedding theorem, we have ||hi||w2,i(T,T) ^ c|dfhi|ci[o,T] (see Remark 1.1). Similarly, we have

|r|C[T,T] < M|u(',T)|D(AY0 + i/a) + ||ut(-,T)|D(AY0)) + ¿23 + C24 ((T - t)"/2

. . (4.16)

+ (T - Tr-1) + ¿25 ((T - Tr/2+1 + (T - T)a) + C2^T - T + (T - T)3/2).

We set T - t ^ 1. Combining the estimates (4.14)-(4.16), as a result we have

||(v,y,r)|YTT < c27 (|u(',T)|D(AY0+i/a) + ||ut(-,T)|D(AY0)) + c28 ((T - t)"-1

+ (T-T)a(1-£))(1+t) + C29((T-T)a + (T-t)a(1-£)+1) + C3o((T-T)a/2 + (T-T)a-1) (4.17) + C31 ((T - T)a/2+1 + (T - T)a) + C32 (T - T + (T - T)3/2) + C33.

Moreover, using (4.10) and (4.13), by similar calculations to (3.22), we have

¡K(V1,y1,f1) - K(V2,y2,r2)!YT

< C34 [((T - t)a-1 + (T - T)a(1-£))(1 + t) + (T - T)a + (T - T)a(1-£)+1 (4.18)

+ (T - T)a/2 + (T - T)a-1 + T - T + (T - T)3/2] ||V1 - V2, y1 - y2, r1 - r2|yT.

We choose p such that p ^ p and C27(||«(-,t)|D(Ay0+i/«) + |K(-,t)||d(ay0)) + C33 ^ p/2. It is easy to see that if we choose p larger, then we could get larger T - t to satisfy

C28 ((T - T)a-1 + (T - T)a(1-£)) (1 + T) + C29((T - t)a + (T - T)a(1-£)+1) + C3o((T - T)a/2 + (T - T)a-1) (4.19)

+ C31 ((T - r)"/2+1 + (T - rT) +c32(T-t + (T- rf'2) < |.

Furthermore noticing that (4.19) and (3.15) have the same structure, we can choose T - t = t to satisfy (4.19), which yields ||K(v,y, r)||yT ^ p, i. e., K(B?p,T) C -Bp,T. Additionally,

11^(^1, Vi, ri) -K(v2, V2, r2)\\YT < \ II(vi -«2,2/1 -V2,ri -r2)\\YT- (4.20)

Hence we prove that K is a contraction operator on for T = 2r.

Repeating the extension process limited times, we could obtain a solution (u, q, k) € Y0T of the inverse problem (2.32)-(2.34) for any T. Lemma 2.6 shows that the inverse problem (2.32)-(2.34) is equivalent to our inverse problem. Consequently, the inverse problem (1.1)-(1.4) also admits a unique solution (u, q, k) in the space x C 1[0, T] x C[0, T] for any T.

5. Example

In this section, as an illustration, we give an example of the inverse problem (1.1)-(1.4) when d = 2. In this case, we assume that A = -A := -dX - c^. Let Q = (0,1) x (0,1) be open rectangular. Then in the domain := {(x,y,t) : (x,y) € Q, 0 < t < T} we have the following problem:

t

fl?u(x,y,t) - Au(x,y,t) = q(t)ut(x, y,t) + j k(t - s)u(x,y,s) ds - Aa(x,y)

(5.1)

- tAb(x,y) + 2(1 - e-t + (15 + n2)t)b(x,y) - (1 - e-t)a(x,y), (x,y,t) € Qj,

(5.2)

with initial

{u(x, y, 0) = a(x, y) := sin 2nx sin 2ny, (x, y) € Q,

ut(x, y, 0) = b(x, y) := (10 - 32x2)y sin nx sin ny, (x, y) € Q,

and boundary conditions

u(0,y,t) = u(1, y, t) = 0, u(x, 0,t)= u(x, 1,t)=0, t € (0,T). (5.3)

In the inverse problem, it is required to find the functions q(t) k(t), if there are additional information regarding the solution of the direct problem (1.1)—(1.3):

O^i^T. (5.4)

It is not difficult to check that all given data satisfy conditions (C1)-(C4). Then, by Lemma 2.6 the solution of the inverse problem (1.1)-(1.3) is of the form

u(x,t) = sin2nxsin2ny + t( 10 - 32x^ysinnxsinny,

k(t) = e-t, q(t) = e-t - 1 - (31 + 2n2)t. (5.5)

Of course, the solution of the inverse problem (5.1)-(5.4) also satisfies the conditions of Theorem 1.1.

Conclusion. The weak solubility of a nonlinear inverse boundary value problem for a d-dimensional fractional diffusion-wave equation with natural initial conditions was studied in the work. First, the existence and uniqueness of the direct problem were investigated. The considered problem was reduced to an auxiliary inverse boundary value problem in a certain sense and its equivalence to the original problem was shown. Then, the local existence and uniqueness theorem for the auxiliary problem is proved using the Fourier method and contraction mappings principle. Further, based on the equivalency of these problems, the global existence and uniqueness theorem for the weak solution of the original inverse coefficient problem was established for any value of time.

Acknowledgements. The author wishes to express his sincere gratitude to the referee for the useful suggestions, which helped them to improve the paper.

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33. Wu, B. and Wu, S. Existence and Uniqueness of an Inverse Source Problem for a Fractional Integro-Differential Equation, Computers and Mathematics with Applications, 2014, vol. 68, no. 10, pp. 1123-1136. DOI: 10.1016/j.camwa.2014.08.014.

Received January 22, 2024 Askar A. Rahmonov

V. I. Romanovsky Institute of Mathematics of the Academy

of Sciences of the Republic of Uzbekistan,

9 University St., Tashkent 100174, Uzbekistan,

Doctoral Student;

Bukhara State University,

11 M. Ikbol St., Bukhara 705018, Uzbeksitan,

Associate Professor of the Department of Differential Equations

E-mail: [email protected], [email protected]

https://orcid.org/0000-0002-7641-9698

Владикавказский математический журнал 2024, Том 26, Выпуск 3, С. 86-111

ОПРЕДЕЛЕНИЕ КОЭФФИЦИЕНТА И ЯДРА В d-МЕРНОМ ДРОБНОМ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОМ УРАВНЕНИИ

Рахмонов А. А.1

1 Институт математики им. В. И. Романовского Академии наук Республики Узбекистан, Узбекистан, 100174, Ташкент, ул. Университетская, 9;

Бухарский государственный университет, Узбекистан, 705018, Бухара, ул. М. Икбол, 11 E-mail: araxmonov@mail. ru

Аннотация. Настоящая работа посвящена получению однозначного решения обратной задачи для многомерного дробно-временного интегро-дифференциального уравнения. В случае дополнительных данных рассмотрим обратную задачу. Неизвестный коэффициент и ядро однозначно определяются дополнительными данными. Используя теорему о неподвижной точке в подходящих пространствах Соболева, получены глобальные во времени результаты существования и единственности этой обратной задачи. В работе исследована слабая разрешимость нелинейной обратной краевой задачи для d-мерного дробного диффузионно-волнового уравнения с естественными начальными условиями. Сначала исследовались существование и единственность прямой задачи. Рассматриваемая проблема заключалась в сведена к вспомогательной обратной краевой задаче в определенном смысле и показана ее эквивалентность исходной задаче. Затем с использованием метода Фурье и принципа сжимающих отображений доказывается локальная теорема существования и единственности вспомогательной задачи. Далее на основе эквивалентности этих задач была установлена глобальная теорема существования и единственности слабого решения исходной обратной коэффициентной задачи для любого значения времени. Далее на основе эквивалентности этих задач была установлена глобальная теорема существования и единственности слабого решения исходной обратной коэффициентной задачи для любого значения времени.

Ключевые слова: дробное волновое уравнение, дробная производная Капуто, метод Фурье, функция Миттаг-Леффлера, неравенство Бесселя.

AMS Subject Classification: 35R30, 35R11.

Образец цитирования: Rahmonov, A. A. Determination of a Coefficient and Kernel in a d-Dimensional Fractional Integro-Differential Equation // Владикавк. мат. журн.—2024.—Т. 26, № 3.—C. 86-111 (in English). DOI: 10.46698/g9973-1253-2193-w.