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Journal of Siberian Federal University. Mathematics & Physics 2021, 14(4), 463-474
DOI: 10.17516/1997-1397-2021-14-4-463-474 УДК 517.95
On some Inverse Parabolic Problems with Pointwise Overdetermination
Sergey G. Pyatkov* Vladislav A. Baranchuk
Yugra State University Khanty-Mansiysk, Russian Federation
Received 10.03.2021, received in revised form 05.04.2021, accepted 20.05.2021 Abstract. We examine well-posedness questions in the Sobolev spaces of inverse problems of recovering coefficients depending on time in a parabolic system. The overdetermination conditions are values of a solution at some collection of points lying inside the domain and on its boundary. The conditions obtained ensure existence and uniqueness of solutions to these problems in the Sobolev classes. Keywords: parabolic system, inverse problem, pointwise overdetermination, convection-diffusion. Citation: S.G. Pyatkov, V.A. Baranchuk, On some Inverse Parabolic Problems with Pointwise Overdetermination, J. Sib. Fed. Univ. Math. Phys., 2021, 14(4), 463-474.
DOI: 10.17516/1997-1397-2021-14-4-463-474.
Dedicated to Yu. Ya. Belov
Introduction
We consider inverse problems with pointwise overdetermination for a parabolic system of the form
Lu = ut + A(t, x, D)u = f (x, t), (t, x) £ Q = (0, T) x G, G c Rn, (1)
where
n n
A(t,x,D)u = -^2 aij(t,x)uXjXj +^2 ai(t,x)uXi + a0(t,x)u, i,j = l i=l
G is a bounded domain with boundary Г £ C2, aij, ai are matrices of dimension h x h, and u is a vector of length h. The system (1) is supplemented by the initial and boundary conditions
u|t=o = uo, Bu\s = g, S = (0, T) x Г, (2)
n
where Bu = ^ Yi(t,x)uXi + j0(t,x)u. The overdetermination conditions are as follows:
i=1
<u(xi,t),ei >= ^i(t), i = 1, (3)
where the symbol < •, • > stands for the inner product in Ch, {ei} is a collection of vectors of unit length and among the points {x^} as well as the vectors {ei} can be coinciding points and
*s_pyatkov@ugrasu.ru http://orcid.org/0000-0002-7238-9559 t vladinho@mail.ru
© Siberian Federal University. All rights reserved
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vectors. The right-hand side is of the form f = У fi(x,t)qi(t) + f0(x,t). The problems is to
i=1
find the unknowns qi(t) occurring into the right-hand side and the operator A as coefficients and a solution u to the system (1) satisfying (2) and (3). The conditions (3) generalized the conventional pointwise overdetermination conditions of the form u(xi,t) = ф(). In particular, it is possible that only part of the coordinates of the vector u at a point xi is given. These problems arise of describing heat and mass transfer, diffusion, filtration, and in many other fields (see [1-3]) and they are studied in many articles. First, we should refer to the fundamental articles by A. I. Prilepko and his followers. In particular, an existence and uniqueness theorem for solutions to the problem of recovering the source f (t, x)q(t) with the overdetermination condition u(x0,t) = ф(t) (x0 is a point in G) is established in [4,5]. Similar results are obtained in [6] for the problem of recovering lower-order coefficient p(t) in the equation (1). The Holder spaces serve as the basic spaces in these articles. The results were generalized in the book [7, Sec. 6.6, Sec. 9.4], where the existence theory for the problems (1)-(3) was developed in an abstract form with the operator A replaced with -L, L is generator of an analytic semigroup. The main results employ the assumptions that the domain of L is independent of time and the unknown coefficients occur into the lower part of the equation nonlinearly. Under certain conditions, existence and uniqueness theorems were proven locally in time in the spaces of functions continuously differentiable with respect to time. We note also the article [8], where an existence and uniqueness theorem in the problem of recovering a lower-order coefficient and the right-hand was established with the overdetermination condition u(xi,t) = ф(t) (xi are interior points of G, i = 1, 2). There are many articles devoted to the problems (1)-(3) in model situations, especially in the case of n = 1 (see, for instance, [9-14]). In these articles different collections of coefficients are recovered with the overdetermination conditions of the form (3), in particular, including boundary points xi. In this case the boundary condition and the overdetermination condition define the Cauchy data at a boundary point. Many results in the case of n =1 are exhibited in [15]. Note the book [16], where the solvability questions for inverse problems with the overdetermination conditions being the values of a solution on some hyperplanes (sections of a space domain) are studied. The problems (1)-(3) were considered in authors’ articles in [17,18], where conditions on the data were weakened in contrast to those in [7, Sec. 9.4] and the solvability questions were treated in the Sobolev spaces. In contrast to the previous results, we examine the case of the points {xi} lying on the boundary of G as well and the special overdetermination conditions (only some combinations of the coordinate of a solution are given). These overdetermination conditions also arise in applications (see [3]). Note that numerical methods for solving the problems (1)-(3) have been developed in many articles (see [2,3,19]).
1. Preliminaries
First, we introduce some notations. Let E be a Banach space. Denote by Lp(G; E) (G is a domain in R") the space of strongly measurable functions defined on G with values in E and the finite norm ||||u(x)||£\\lp(g) [20]. We employ conventional notations for the space of continuously differentiable functions Ck (G; E) and the Sobolev space Wp,(Q; E), Wp,(G; E), etc. (see [20,21]). If E = C or E = C" then the latter space is denoted simply by W^(G). Therefore, the membership u G Ws(G) (or u G Ck(G)) or a G Wps(G) for a given vector-function u = (u1, u2,..., uk) or a matrix function a = {aij}k i=1 mean that every of the components ui (respectively, an entry aij) belongs to the space W£(G) (or Ck(G)). Given an interval J = (0, T),
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put Wp,r (Q) = Wp (J; Lp(G)) n LP(J; Wp (G)), Respectively, we have Wp,r (S) = Wp(J; LP(T)) П Lp(J; Wp (Г)). The anisotropic Holder spaces Ca,e (Q) and Ca,e (S) are defined by analogy.
The definition of the inclusion Г e Cs can be found in [22, Chapter 1]. In what follows we assume that the parameter p > n + 2 is fixed. Let Eg(xi) be a the ball of radius S centered at xi (see (3)). The parameter S > 0 will be referred to as admissible if Eg(xi) c G for interior points xi e G, Eg(xi) П Eg(xj) = 0 for xi = Xj, i,j = 1, 2,... ,r, and, for every point xi e Г, there exists a neighborhood U (the coordinate neighborhood) about this point and a coordinate system y (local coordinate system) obtained by rotation and translation of the origin from the initial one such that the yn-axis is directed as the interior normal to Г at xi and the equation of the boundary UnГ is of the form yn = ш(у'), w(0) = 0, \y'\ < S0, y' = (yi,..., yn-i); moreover, we have ш e C3(E'S(0)) (Eg(0) = {y' : \y'\ < S}) end G n U = {y : \y'\ < S, 0 < yn — ш(у') < Si}, (Rn \ G) n U = {y : \y'\ < S, — Si <yn — u(y') < 0}. The numbers S, Si for a given domain G are fixed and without loss of generality we can assume that Si > (M + 1)S, with M the Lipschitz constant of the function ш. Assume that QT = (0, т)x G, Gg = Ui(Eg(xxi) nG), Qg = (0, T) x Gg, QT = (0, т) x Gg, Sg = (0, T) x Ui(Eg(xi) n Г).
Consider the parabolic system
Lu = ut + A(t, x, D)u = f (t, x), (t, x) e Q = (0, T) x G, G c Rn, (4)
where
nn
A(t,x,D)u = —^2 aij(t,x)uXjXj +22 ai(t,x)uXi + ao(t,x)u, i,j = i i=i
aij, ai are matrices of dimension h x h, and u is a vector of length h. The system (4) is supplemented with the initial and boundary conditions (2). We assume that there exists an admissible number S > 0 such that
aij e C (Q), ak e Lp(Q), Ik e C 1/2,1 (S), aij e L^(0,T; W^ (Gg)); (5)
ak e LP(0,T; Wp(Gg)), i,j = 1, 2,...,n,k = 0,1,...,n. (6)
The operator L is considered to be parabolic and the Lopatiskii condition holds. State these
n
conditions. Introduce the matrix A0(t,x,£) = — У aij(t,x)^i^j (£ e Rn), and assume that
i,j = i
there exists a constant Si > 0 such that the roots p of the polynomial
det (Ao (t,x,ip) + pE) = 0 (E is the identity matrix) meet the condition
Rep < —Si\e\2 V£ e Rn y(x,t) e Q. (7)
The Lopatinskii condition can be stated as follows: for every point (to, xo) e S and the operators
n
A0(x,t,D) and E0(x,t,D) = У Yi(t,x)dXi, written in the local coordinate system y at this
i=i
point (the axis yn is directed as the normal to S and the axes yi,... ,yn-i lie in the tangent plane at (x0,t0)), the system
(XE + A0(x0, t0, ig, dyn))v(z)
0, E0(x0,t0,ig',dyn )v(0)
hj,
(8)
where g' = (gi,..., gn-i), yn e R+, has a unique solution C(R+) decreasing at infinity for all g' e Rn-i, \ arg X\ < n/2, and hj e C such that \g'\ + \X\ = 0.
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We also assume that there exists a constant e1 > 0 such that
Re (-Ao(t,x,^)n,n) > £i|£|2Ini2 G R", П G Ch,
(9)
where the brackets (■, ■) denote the inner product in Ch (see [22, Definition 7, Sec. 8, Ch. 7]). Let
det £ YiVij I > £o > 0,
£ i=i '
where v is the outward unit normal to Г, e0 is a positive constant, and
uo(x) G W2-2/p(G), g & Wp0’2k0 (S), B(x, 0)uo(x)|r = g(x, 0) Ух G Г,
(10)
(11)
where k0 = 1/2 — 1/2p. Fix an admissible S > 0. Construct functions фi(x) G Q0°(R") such that
r
Ф^) = 1 in BS/2(xi) and фi(x) = 0 in R" \ B3S/4(xi) and denote ф(x) = У ф^). Additionally
i=1
it is assumed that
ф^)ио(x) G Wp-2/p(G), фд G Wkl2kl (S) (ki = 1 — 1/2p), (12)
Г G C2, Yk G C 1’2(S~s) (k = 0,1, 2,... ,n). (13)
The proof of the following theorem can be found in [18].
Theorem 1. Assume that the conditions (5)-(13) hold for some sufficiently small admissible
S > 0 and the function ф, f G Lp(QT), f ф G Lp(0,T; Wp,(G)), and т G (0,T]. Then there exists
a unique solution и G W^’2(Qt) to the problem (4), (2). Moreover, фщ G Lp(0,T; W^(G)) and фи G Lp (0, т; Wp (G)). If g = 0 and u0 = 0 then we have the estimates
IMlwp’2(qt) ^ cWf Wlp(qt),
\\U\\wf2(QT) + WWut WLp(0,T ;W1 (G)) + №uWLp(0,t ;W3(G)) < c [Wf WLP(QT) + ||фf WLp(0,t ;W1(G))\
(14)
where the constant c is independent of f, a solution u, and т G (0,T].
2. Main results
Consider the problem (1)-(3), where
Г " "
A L0 ^ ^ qk(t)Lk, Lku ^ ^ aij ^, x)uXjXj + ^ ^ ai (t,x')uXi + a0 ^, x')u,
k=m+1 i,j = 1 i=1
and k = 0,m + 1,m + 2,...,r. The unknowns qi are sought in the class C([0,T]). Construct a matrix B(t) of dimension r x r with the rows
< f 1 (t, xj ),ej >,..., < fm (t, xj ), ej >,< Lm+1u0 (t, xj ), ej >,...,< Lr u0 {p, xj ), ej > .
We suppose that
fj G C 1([0, T ]), <u0(xj ),ej >= fj (0) (j = 1, 2,... ,r), y G C l/2,1(S) П C 1’2(Sf), (15)
akj G C (Q) П L°(0,T; W° (Gs)), a<k G Lp (Q) П L°(0,T; Wlp(Gs)) (i,j = 1,...,n), (16)
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fi € Lp(Q) П LTO(0, T; Wp(G$)) (i — 0, l,..., m), (17)
foe some admissible 5 > 0, p > n + 2, and к — 0, m + l,... ,r, l — 0, l,.. .n;
(t,xi),fi(t,xi) € C([0,T]) (18)
for all possible values of i, к, l. We also need the condition (C) there exists a number 5o > 0 such that
|det B(t) \ fi 5o a.e. on (0, T).
Note that the entries of the matrix B belong to the class C([0,T]). Consider the system
fjt(0) + < Louo(0,xj),ej > — < fo(0,Xj),ej >—
m mi
V qok < fk (0,xj ),ej > + ^2 qok < Lk uo(0,xj ),ej >, j — l,...,r, (19)
k = 1 k=m+1
where the vector qo — (qo1,qo2,... ,qor) is unknown. Under the condition (C), this system is
r
uniquely solvable. Let A1 — Lo — У qokLk. Now we can state our main result.
k=m+1
Theorem 2. Let the conditions (9)—(13), (C), (15)—(18) hold. Moreover, we assume that the conditions (7), (8) are fulfilled for the operator dt + A1. Then there exists a number тo € (0, T] such that, on the interval (0, тo), there exists a unique solution (u, q1,q2,..., qr) to the problem (1)-(3) such that u € Lp(0, тo; Wp(G)), ut € Lp(QT ), qi (t) € C ([0, тo ]), i — l,... ,r. Moreover, Фи € Lp(0, тo; W°3(GS)), put € Lp (0,тo; W^)).
Proof. First, we find a solution to the problem
m
ф + А1ф — fo + qoifi ((x,t) € Q), Ф1 t=o — u0(x), ВФ|й — g. (20)
k=1
By Theorem 1, Ф € Wp;’2(Q), p^t € Lp(0,T; Wp,(G)), фФ € Lp(0,T; W^(G)). As a consequence of Theorem III 4.10.2 in [24] and embedding theorems [20, Theorems 4.6.1,4.6.2.], we infer фФ € C([0,T]; Wp3-2/p(G)) c C([0,T]; C3-2/p-n/p(G)). Hence, фФ € C([0,T]; C2(G)) after a possible change on a set of zero measure. The equations (20) and (18) imply that Фt(t,xj) € C([0,T]). Note that this function is defined, since every summand in (20) with the weight ф belongs to Lp(0,T; Wp,(G)) c Ca(G; Lp(0,T)) (a fi l — n/p) (see the embedding theorems in [25] and the arguments below). Multiply the equation (20) scalarly by ej and take x — xj. We obtain the equality
< Фt(0,Xj),ej > + < Louo(0,xj),ej > — < fo(0,xj),ej >—
mr
V qok < fk (0,xj ),ej > + ^2 qok < Lk uo(0,xj ),ej >, j — l,...,r. (21)
k = 1 k=m+1
The relations (19) and (21) imply that < Фt(0,Xj),ej >— fijt(0). After the change of variables q — qo + q1 and u — w + Ф in (1), we arrive at the problem
r m r
Lw — wt + Arw — ^2 q1kLkW — ^2 fiq1i +$3 qliLiФ — F, w\t=o — 0, Bw\s — 0, (22)
k=m+1 i=1 i=m+1
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< w(t,Xj),ej >= 'Фз (t) = Фз (t)- < Ф(t,Xj),ej >£ C 1([0, T]), фз (0) = Vjt(0) = 0. (23)
Fixing the vector q1 = (q11,... ,q1r) £ C([0, r]) and determining a solution w to the problem (22) on (0, r), we construct a mapping w = w(q1) = L-1 F. Demonstrate that there exists R0 > 0 such that, for q1 £ BRo, the problem
Lv = g, v\t=o =0, Bv\s = 0 (24)
for every g £ HT и r £ (0,T] has a unique solution in the class v £ Wp),2(QT), фvt £ Lp(0,r; W1(G)), fv £ Lp(0,r; W^(G)) satisfying the estimate
IbllwyCQT) + Wvvt\\bp(0,T ;Wp (G)) + \\фЛьр(0,т ;W3(G)) < cllgl|HT (25)
where the constant c is independent of r and the vector q1 £ BRo and the space HT is endowed with the norm
\\/\\ht = II/\\lp (qt) + ||ф/\\lp (0,T W(Q)).
In accord with Theorem 1, the problem
L01v = vt + A1 v = g, v\t=0 = 0, Bv\s = 0
for every g £ HT has a unique solution such that v £ W12(QT), фvt £ Lp(0,r; W^(G)), фv £ Lp(0, r; W3(G)) and
\\v\\w1’2(QT)
+ \Wvt lb
1 (G))
+ \ \ Pv IIL p (0,t ;W3(G)) Ф c1\\g\\HT
P
(26)
where the constant c1 is independent of r. In this case the question of solvability of the problem (24) is reduced to the same question for the equation
r
/ -53 quLiL-11/ = g, (27)
i=m+1
where / = L01 v. We have the estimate
-53 qnLiv
i=m+1
<
Ht
с11йНса0,т ])(IMI
w!’2(Qt )
+ \\Fvt\\
Lp (0,T
Wp (G)) + |fv|Lp(0,T;W3(G))) , (28)
where the constant c depends on the coefficients of the operators Lk in Q and is independent of r and q1. Indeed, the following estimate is obvious
r
-53 q1k Lkv
k=m+1
Ht
r
Ф |qi\c([0,T]) £ llLk*'IIHt .
k=m+1
(29)
Estimate the quantity \\Lkv\\Ht . To this aim, we estimate the norms of every of the summands in this quantity. For example, estimate the norm
\aij vXiXj\\Ht Ф cd\aij vXiXj\\Lp (Qt ) +53 11ф ( aij vXiXj )xi \\Lp (Qt 3 ^
l=1
Ф + ( 11v \ Lp (0,T ;W2(G)) + \ \ Pv \ I Lp (0,т ;W^(G))) + 53aijxi vXi Xj II Lp (Qt ) , (30)
l=1
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where the constant c1 depends on the norms \\akj\\Lmq). The last summand here is estimated as follows:
vXiXj\\lp(Qt ) < C2(\\<fv\\Lp(0,T ;W^ (g)) + \\v\\lp(0,t ;W^ (О))) Ф
l = 1
Ф c3(\\<fv\\Lp(0,T ;Wp(G)) + \v\Lp(0,T ;wp (О))) , (31)
where the constant c2 depends on the norms \\Vakj\\Lp(0,Tl^Gs))- Thus, we infer
\\aij vXiXj\\ht Ф c4{\v\Lp(0,T ;W2(G)) + \\Vv\\Lp(0,T ;W3(G))) ,
where the constant c4 is independent of т. Similarly, we derive that
(32)
n
\\ai vXi\\Ht Ф c0<
\ak vx
Lp(QT)
+ V \\‘fi(aivxi )xt\\Lp(QT))
<
l=1
< V\\^v\\l^(Qt ) + \\Vv\\Lp(0,t;W2 (о))) , (33)
where the constant c1 depends on the norms of ak, akxi in Lp(Q) and the norms of ak in L^(Qg). However (see Lemma 3.3 in [22]), we have
\\^v\\l^(Qt) < ci\v\w1-2(qt),
where the embedding constant is independent of т. Summing the estimates obtained we justify (28). Using (28) and the estimate of Theorem 1, we conclude that
'y ( q1iLiL-1 f
i=m+1
Ht
Ф c2\\q1\\c([0,T])\\f \\ht ,
(34)
where c2 is independent of т and q1 e BRo. Let R0 = 1/2c2. In this case c2\q1 \C([0,T]) ^ 1/2
and thereby the equation (27) has a unique solution satisfying the estimate \\f \\Ht ф 2||д||Нт, which along with Theorem 1 ensures (25).
Assume that w is a solution to the problem (22), (23). Take x = xj in (22) and multiply the equation scalarly by ej. The traces of all function occurring into the equation exist. First, our conditions for coefficients and embedding theorems yield фw e C([0,T]; C2(G)) (see the above arguments for the function Ф). Second, as we have indicated above, every of the summands in (22) with the weight ф belongs to Lp(0,T; W^(G)) C Ca(G; Lp(0,T)) (a < 1 — n/p) (see embedding theorems in [25]). We arrive at the system
</pjt,ej > + <Arw(t,Xj),ej > — ^ qu <Liw(t,Xj),ej >=
i=m+1
m r
= Y^ < fi(t, Xj ),ej >q1i(t)+ qu <Li&(t,Xj ),ej > (j = 1, 2,...,r), (35)
i=1 i=m+1
which can be rewritten in the form
Bq1 = Ф + R(q1),
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where coordinates of the vectors ф and R(qi) agree with the functions < ijt,ej > and < A0w(t,Xj),ej > — У[=m+i 4u < Liw(t,xj),ej > (w = w(q1)); respectively, j-th row of the matrix B(t) of dimension r x r is written as
< f 1 (t, xj ), ej >,..., < fm(t, xj ), ej >, < Lm+1 Ф(С xj ), ej >,...,< Lr Ф(С xj ) , ej >,
where j = 1,... ,r. This matrix differs from B by the entries < Li^(t,xj ),ej >. It is easy to prove that this matrix is nondegenerate as well on some segment [0, to]. Indeed, the embedding theorems (see Lemma 3.3 of Chapter 1 in [22]) imply that УФ, §XiXj G Ce/2,e(Qs/2) for в < 1 — (n + 2)/p and all i,j and, therefore,
| <Lk^(t,xj) Lk u0(t,xj ),ej > \ T ^ ' sup \\aik(t,xj m^Xk Xi (t,xj) u0xk Xi (xj )| +
i,k=1 te[0’T ]
n
+ V sup \\ak(t,xj)\\^Xi(t,xj) — u0Xi(xj)| + sup ||ak(t,xj)\\^(t,xj) — uo(xj)| T cte/2,
i=1 te[0,T] te[0,T]
on [0, T], where, by the norm of a matrix (for example, \\ak(t,xj)||), we mean the norm of the corresponding linear operator ak (t,xj) : Ch ^ Ch. Taking the condition (C) into account, we can say that there exists to > 0 such that
|detB(t)| > S0/2 Vt T to. (36)
We thus obtain the integral equation
q1 = B 1гФ + Ro(ql), ЯоШ = B 1R(q1), (37)
where the operator R0(q1) : C([0, t]) ^ C([0, t]) (t T to) is bounded. Check the conditions of the fixed point theorem. Denote R0t = 2||B_1Ц|с([0,т]). Let q01, q02 be two vectors of length r with coordinates qj (i = 1, 2,...,r, j = 1, 2) lying in the ball BRo = {q : ||<q||c([0,r]) T R0}. The functions w1 = w(q01),w2 = w(q02) are solutions to the equation (22) satisfying homogeneous initial and boundary conditions. Let v = w1 — w2. We infer
r m r
Lv = vt + A1v — ^2 q2iLiV = ^Z fiq — q2) + 53 q — q2)Liwb v = w1 — w2. (38)
i=m+1 i=1 i=m+1
In view of (23) and the definition of R0r, R0r ^ 0 as t ^ 0. Hence, there exists a parameter t1 T to such that, for t T t1, R0r T Ro. Let R = R0ri. We now derive that there exists a parameter t0 T t1 such that the equation (37) has a unique solution in the ball BR of the space C([0, to]). Take t T t1. Let q01, q02 G BR. We have
IIR0(q01) — R0(q02)||c([0,r]) T c1|R(q01) — R(q02)\c([0,r]) T
rr
T c2 xj )|C([0,r]) + \\q2iLiv(t,xj )\c([0,r])) T
j = 1 i=m+1
rr
T C3 ^2(\\^(ф xj ) II C([0,r ]) + ^ j \\Li v(t,xj )|C([0,r])), (39)
j=1 i=m+1
where v is a solution to the problem (38). Note that
\\Lk v(t,xj ) || C([0,r]) T CT^ (y^Vv\^i,2(gT) + llv\ Wj1,2(Qt )),
(40)
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where the constant c is independent of т € (0, T] and в > 0. Validate this inequality. In view of the conditions on the coefficients akl, akl(t,xj) € C([0,T]). Fix an arbitrary s € (n/p, 1 — 2/p). The embedding Wp;(GS/2) C C(GS/2) [20, Theorems 4.6.1,4.6.2.] yields
\\ail (t,xj )vXiXi (t,xj )Ус([0,т]} ^ cllvXiXj (t,xj )Ус([0,т]} ^ c1 llvxfcXj (t, x)\\L^(0,t ;Wg(Gs/2)) ^
^ C2\\Vv(t,x)\\L^(0,T;Wp+s(Gs/2}}' (41)
Next, we employ the interpolation inequality (see [20])
\\v\\w;° (G} < CMewp (G}\v\ — (Gy s1 < s0 < s2> 0s1 + (1 — 0)s2 = s0 and the inequality
\9\l^(0,t;E} < т(p-1}/p\\gt\\Lp(0,t;E}, Vg € W1(0,t; E), g(0) = 0,
(42)
(43)
<
resulting from the Newton-Leibnitz formula. Here E is a Banach space. We obtain that
HVv(t,x)llL^(0,T ;Wl+s(G,/2}} ^ C\\Vv(t,x')WL^(0,T ;W2-2/p (Gs/2}}\\Vv(t, x')\\L,x,(0,t ;Lp (Gs/2}}
< C1T(1-e}(p-1}/p(yVv\\wi,2{Q} + \\v\\w}.2(Q}), (2 — 2/p)e = 1 + s. (44)
Here we have used the inequality
\\Vv(t,x')\lLm(0,T;Wp2-2/p(Gs/2}} ^ C\\Vv(t,x)\\wp’2(G
S/2}},
(45)
where the constant c is independent of т (in the class of functions vanishing at t = 0). Estimate the lower-order summands of the form akvXi(t,xj), aftv(t,xj) in Liu(t,xj). We conclude that (s € (n/p, 1 — 2/p), (2 — 2/p)01 = 1 + s)
<
\ai vXi (t,xj )|C([0,t]} ^ C\\vXi (t,xj ) || C([0,t]} ^ C1 IIv(t, x) IIL^ (0,t ;wp+s(Gs
< \\v(t,x)\\91 2 2/p IlvU^M1-9' L (g }} < C2T(1-9l}(p-1}/pM\w 1,2(QT}. (46)
11 V ’ /UL^(0,t ;wp-2/p (Gs/2}}" V ’ ;"L^(0,t ;Lp(gs/2}} ^ 2 11 11wp (qt } v >
We have used the estimate (45) applied to v rather than Vv. The second summand is estimated similarly. The estimates (39)-(46) ensure that
ll-^gbO — Й0(сТ02)\с([0,т]} < C4т в (\\pVv(t,x)\\wp-2(QT} + \v(t,x)\wp-2(QT ^ (47)
where the constant c4 is independent of т and в = min(1 — 0, (1 — 01)(p — 1)/p). Since v is a solution to the problem (38) and т < т1, we can employ (25) and obtain that
m r
\pVv(t,x)\w1'2(QT} + \v(t,x)\w1’2(QT} < C\\Y1 fi(4i — 9л)+ (q1 — qi)LiW1 H , (48)
„ .... HT
i=1
i=m+1
where the constant c is independent of т. Every of the functions w1, w2 is a solution to the problem (22), where the right-hand side contains the components of the vector q01 or q02. The estimate (25) yields
\\<fVwj (t,x)\wp-2(QT} + \\wj (t,x)\wP’2(QT } < C\\Y1 fiqj + Щ qi LiФ
i=1 i=m+1
Ht
(49)
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Sergey G. Pyatkov, Vladislav A. Baranchuk
On some Inverse Parabolic Problems...
The estimate (48), (49) and the conditions on the coefficients imply that
W^Vwj(t,x)\\wi,2{QT) + \\wj(t,x)\\wi,2{QT) p c1(R). (50)
\\<pVv(t,x)\\wi,2{QT) + \\v{t,x)\\wi,2{QT) < c2\cToi - cf02||c([0,T]), (51)
where the constant ci are independent of т. In turn, these estimates and those in (47) validate the estimate
||Ro(g0l) - R0(^02) || С([0,т]) ^ c5Tв 11^01 - ?02||с([0,т]) (52)
with a constant c5 independent of т. Choose a parameter т0 < т1 such that c5 (т0)e < 1/2. The
fixed point theorem ensures solvability of the equation (37) in the ball BR.
Show that w satisfies the overdetermination conditions (23). Multiply the equation (22) scalarly by ej and take x = xj in the equation. We obtain the equality
r
< w(t,xj),ej >t + < L0w(t,xj),ej > — qi < Liw(t,xj),ej >=
i=m+1
m r
= ^2 < fi(t,xj),ej >qi(t)+ qi <Li^(t,xj),ej >> j = 1,2,...,r, (53)
i=1 i=m+1
Subtracting this equality from (35), we obtain that pj — < w(t,xj),ej >t= 0. Integrating this equality from 0 to t, we derive that pj(t)— < w(t, xj), ej >= 0, since the agreement conditions imply that pj(0) =0, < w(0,xj),ej >= 0. Thus, we infer pj(t) =< w(t,xj),ej > and the equality (23) holds. □
In the case of the unknown lower-order coefficients, the results can be reformulated in a more convenient form. In this case the operator A is assumed to be representable in the form
A L0 ^ ^ qi(t)li, L0U ^ ^ aij (t,x)uXjXj I ^ ^ ai(t,xPUXi + ao(t,x)u,
i=m+1 i,j = 1 i=1
n
liU = ^2 bij(t,x)uxj + hi0(t,x)u. (54)
j=1
Moreover, the rows of the matrix B(t) of dimension r x r are as follows:
< f1(t: xi), ei >:...: < fm(t,xi),ei >: < lm+1u0(t: xi), ei >: . . . : < lr u0(t: xi) ,ei > .
We suppose that
Pj e wp(°,T), < u0(xj),ej >= pj(0), j = 1,2,...,r,
(55)
fi,bkj e Lx,(0, T; W1(GS)) n L^(0,T; Lp(G)), f0 e LP(Q) П Lp(0,T; W^)), (56)
for some admissible S > 0, where i = 1,... ,m, j =0,1,... ,n, k = m + 1,... ,r. The remaining coefficients satisfy the conditions
aij e C(Q), ak e Lp(Q), lk e C1/2’1(S) П C1’2(Ss), aij e L^(0,T; W^ (Gs)); (57)
ak e Lp(Q) П Lp(0,T; W^(GS)), i,j = 1, 2,... ,n, k = 0,1,...,n. (58)
The corresponding theorem is stated in the following form.
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Sergey G. Pyatkov, Vladislav A. Baranchuk
On some Inverse Parabolic Problems...
Theorem 3. Assume that the parabolicity condition and the Lopatinskii condition (7), (8) for the operator dt + L0, the conditions (9)-(13), (55)—(58), (С) for some admissible 6 > 0 and p > n + 2 hold. Then, for some yo G (0,T], on the interval (0, y0), there exists a unique solution (u, qi, q2,. .., qr) to the problem (1)-(3) such that u G Lp(0,Y0; Wp(Gf), ut G Lp(QY0), фи G Lp(0,Y0; W3(G)), фщ G Lp(0,Yo; W1(G)), eji(t) G Lp(0,Yo), i = l,...,r.
The proof is omitted, since it is quite similar to that of the previous theorem.
References
[1] G.I.Marchuk, Mathematical Models in Environmental Problems, Studies in Mathematics and its Applications, Vol. 16, Elsevier Science Publishers, Amsterdam, 1986.
[2] M.N.Ozisik, H.R.B.Orlande, Inverse Heat Transfer, Taylor & Francis, New York, 2000.
[3] O.M.Alifanov, E.A.Artyukhov, A.V.Nenarokom, Inverse problems of complex heat transfer, Yanus-K, Moscow, 2009 (in Russian).
[4] A.I.Prilepko, V V.Solov’ev, Solvability theorems and Rothe’s method for inverse problems for a parabolic equation. I, Differ. Equations, 23(1987), no. 10, 1230-1237.
[5] V.V.Solov’ev, Global existence of a solution to the inverse problem of determining the source term in a quasilinear equation of parabolic type, Differ. Equations, 32(1996), no. 4, 538-547.
[6] A.I.Prilepko, V.V.Solov’ev, Solvability of the inverse boundary-value problem of finding a coefficient of a lower-order derivative in a parabolic equation, Differ. Equations, 23(1987), no. 1, 101-107.
[7] A.I.Prilepko, D.G.Orlovsky, I.A.Vasin, Methods for solving inverse problems in Mathematical Physics, Marcel Dekker, Inc., New-York, 1999.
[8] M.A.Kuliev, Multidimensional inverse problem for a multidimensional parabolic equation in a bounded domain, Nonlinear boundary value problem, 14(2004), 138-145 (in Russian).
[9] O.A.Afinogenova, Yu.Ya.Belov,I.V.Frolenkov, Stabilization of the solution to the identification problem of the source function for a one-dimensional parabolic equation, Doklady Mathematics, 79(2009), no. 1, 70-72.
[10] Yu.Ya.Belov, K.V.Korshun, An identification problem of source function in the Burgers-type equation, J. Sib. Fed. Univ., Math. Phys., 5(2012), no. 4, 497-506 (In Russian).
[11] M.S.Hussein, D.Lesnic, M.I.Ivanchov, Simultaneous determination of time-dependent coefficients in the heat equation, Computers and Mathematics with Applications, 67(2014), 1065-1091.
[12] N.I. Ivanchov, N.V.Pabyrivska, On Determination of Two Time-Dependent Coefficients in a Parabolic Equation, Siberian Mathematical Journal, 43(2002), no. 2, 323-329.
[13] D.Lesnic, M.Ivanchov, Determination of the time-dependent perfusion coefficient in the bioheat equation, Applied mathematics letters, 39(2015), 96-100.
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On some Inverse Parabolic Problems...
[14] M.S.Hussein, D.Lesnic, Identification of time-dependent conductivity of an diffusive material, Applied mathematics and computations, 269(2015), 35-58.
[15] M.Ivanchov, Inverse Problems for Equation of Parabolic Type, Math. Studies. Monograph Series, Vol. 10, WNTL Publishers, Lviv, 2003.
[16] Ya.Ya.Belov, Inverse Problems for Parabolic Equations, VSP, Utrecht, 2002.
[17] S.G.Pyatkov, V.V.Rotko, On some parabolic inverse problems with the pointwise overdetermination, AIP Conference Proceedings, 1907(2017), 020008.
[18] V.A.Baranchuk, S.G.Pyatkov, On some clases of inverse problems with pointwise overdetermination for mathematical models of heat and nass transfer, Bulletin of Yugra State University, 3(2020), 38-48 (in Russian).
[19] A.V.Mamonov, Y-H.R.Tsai, Point source identification in nonlinear advection-diffusion-reaction systems, Inverse Problems, 29(2013), no. 3, 035009.
DOI: 10.1088/0266-5611/29/3/035009
[20] H.Tribel, Interpolation Theory. Function Spaces. Differential Operators, Johann Ambrosius Barth, Heidelberg, 1972.
[21] R.Denk, M.Hieber, J.Pruss, Optimal Lp — Lq-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z, 257(2007), no. 1. 193-224.
[22] O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, American Math. Society, Providence, R.I., 1968.
[23] O.A.Ladyzhenskaya, V.Ya.Rivkind, N. N.Ural’tseva, Classical solvability of diffraction problems in the case of elliptic and parabolic equations, Sov. Math., Dokl., 5(1965), 1249-1252.
[24] H.Amann, Linear and Quasilinear Parabolic Problems. I, Monographs in Mathematics, Vol. 89, Birkhauser Verlag, Basel, etc., 1995.
[25] H.Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glasnik Mat., Ser.III, 35(55)(2000), no. 1, 161-177.
О некоторых классах параболических обратных задач с точечным переопределением
Сергей Г. Пятков Владислав А. Баранчук
Югорский государственный университет Ханты-Мансийск, Российская Федерация
Аннотация. В работе рассматривается вопрос о корректности в пространствах Соболева обратных задач о восстановлении коэффициентов параболической системы, зависящих от времени. В качестве условий переопределения рассматриваются значения решения в некотором наборе точек области, лежащих как внутри области, так и на ее границе. Приведены условия, гарантирующие существование и единственность решений задачи в классах Соболева.
Ключевые слова: параболическая система, обратная задача, конвекция-диффузия, точечное переопределение.
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