On some classes of inverse parabolic problems of recovering the thermophysical parameters

n                           nri

A i = Z a ki ( t , x ) ^d x _kx l ⁺ Z a k ( t , ^x )^d x _k ⁺ a 0 , c = c o ( t , x ) - Z ^fi ( t , ^x ) q i ( t ) •

• k, l=1                        k=1

The equation (1) is furnished with the initial and boundary conditions ut=0 = Uo, Bu\S = g (t, x ), S = ( 0, T )хГ,(2)

,      „           n    du , where Bu = u or Bu =--+ au an

n

= Z aijuxi Vj + au i,j=1

( V = ( V 1 ,V 2

.,v_n ) is the outward unit normal to Г ,

and the overdetermination conditions

u ( t,b j ) = ^ j ( t ) , b j е G, j = 1,2, ^ , 5 . (3)

The unknowns in (1)-(3) are a solution u and the functions qi (t) (i = 1,2,^,5). The problem (1)(3) arise in describing the heat and mass transfer, diffusion, and filtration processes, ecology, and many other fields. The problem of determining thermophysical and mass transfer characteristics with the use of inverse modelling is studied in [1], where the results are used for describing the temperature regimes of the soils of the northern territories. We can refer to the monograph [2] devoted to inverse parabolic probems and to [3–6], where the main statements of inverse problems and some applications can be found. The number of theoretical results devoted to the problems (1)–(3) is sufficiently small. We should refer to the articles [7-10], where in the case of n = 1 the thermal conductivity depending on time is defined and existence and uniqueness theorems are established with the additional data being the values of a solution at some points lying in the domain or on its boundary. The thermal conductivity independent of one of the spacial variable and some other coefficients are identified in [11, 12] with the use of the Cauchy data on the lateral boundary of the cylinder and integral data. Existence of a solution is proven and stability estimates are exposed. The monograph [3] (see also the results in [13]) contains the existence theory for inverse problems of recovering the coefficients in the leading part of the equation independent of some part of variables with the overdetermination data given on sections of the spacial domain by planes. In view of the method, all coefficients also are independent of some spacial variables. More complete results for the problems (1)–(3) can be found in [14–17], where the well-posedness of the inverse problems in question is established for the case of the additional data are the values of a solution on some spacial manifolds or at some collection of points. However, in these articles c (t, x) = 1 except for the article [15], where c(t,x) = c(t) in the case of the pointwise ovedetermination. The existence and uniqueness theorems in the case of the unknown heat capacity and the integral overdetermination data are exposed in [18-20], where c(t,x) = c(t) or c(t,x) = const. Note that inverse problems with pointwise data have been studied by A.I. Prilepko and his followers and a number of classical results is presented in [2]. Similar results under different conditions on the data and in some other spaces can be found in [21, 22]. Our results are close to those in [23]. In contrast to this article, the heat capacity here is unknown. The main results of the article are exposed in Sect. 2.

The definition of the Sobolev spaces W ps ( G;E ) , W ps ( Q;E ) ( E is a Banach space) can be found in [24]. If E = R or E = Rⁿ then we omit the notation E and write W ps ( Q ) . The definitions of the Holder spaces C^a,e ( Q ) ,C^a,e ( 5 ) can be found in [25]. By the norm of a vector, we mean the sum of the norms of its coordinates. Given the interval J = ( 0, T ) , put W s ( Q ) = W pS ( J;L_p ( G ) ) n L_p ( JW p ( G ) ) , W p,’^r ( S ) = W p ( J;L_p ( Г ) ) n L_p ( J;W p ( Г ) ) . All function spaces as well as the coefficients of the equation (1) are assumed to be real. In what follows, we suppose that p >  n + 2. The definition of the boundary of class C^s , s >  1, can be found in Ch. 1 in [25]. Denote by B₅ ( b ) the ball of radius § centered at

• b . Fix a parameter 5 >  0 such that B § ( b ) n B § ( b j ) = 0 for i * j and B § ( b i ) n Г = 0 , i, j = 1,2, . ,.,s . Denote Q = ( 0, t ) x G and G § = u i B § ( b i ) . Construct nonnegative functions

p j g C ”(R ⁿ ) such that p j = 1 in B _{5 /2} ( b j ) , p ( x ) > 0 and p j = 0 for x ^ B _{3 5 /4} ( b j ) . Let ф = £ ф j ( x ) .

j=1

The consistency and smoothness conditions can be written as

U 0 ( x ) g W p²"^2/p ( G ± ) , B ( 0, x , D ) u 0 |_r = g ( 0, x ) , g g W p ^0,2 k ⁰ ( S ) , where k ₀ = s 1 = 1 — 1/2 p for Bu = u and k ₀ = s ₀ = 1/2 - 1/2 p otherwise;

• 3 —

p u 0 ( x ) g W p ^p ( G ) , a* kj g L _ro ( 0, T ; W p ( G 5 ) ) , a l g L p ( 0, T ; W p1 ( G 5 ) ) , where i , j = 1,2, . , n , l = 0,1, . , n , к = 0,1, . , r ;

f G C ( Q ) n L „ ( 0, TW ( G 5 ) ) , p f G L p ( 0, TW ( G ) ) , f g L p ( Q ) , where j = r + 1, . , r u l = 0, r 1 + 1, . , s . We use the inclusions of the form f g L_p ( 0, T ; W p ( G ₅ ) )

or

similar, where the set G§ consists of several connectedness components (in this case B§ (bj)). By defi- nition, this means that

^fV„ ( b j ) G L p ( o. T ; W ; ( ^B 5 ( b j ) ) )

for all j . This space is endowed with the norm

equal the sum of the norms over the corresponding connectedness components We assume that al g C(Q),al g Lp (Q),a, Jl\$ g Wp^0,2s0 (S),                           (7)

where the last inclusion is required only if Bu ^ u ,

^ j G C ¹ ( [ 0, T ] ) , p j ( 0 ) = u 0 ( b j ) , a l ( t , b j ) , f m ( t , b j ) G C ( [ 0, T ] ) ,                  (8)

where j = 1,., s, m = 0, r +1,., s, l = 0,1,., n, k = 0,1,., r. In view of (5), (8), the traces fm (t,bp),al (t,bj)      are      defined      and      fm (t,bp),al (t,bj)gLp (0,T);      moreover, fm (t,x), al (t,x) e C(G5;Lp (0,T)) (after a possible change on a set of zero measure 0) (see [26], Sect.

2,3,4, the relations (3.1)–(3.9), the corollary 4.3).

Consider the matrix B₀ of dimension s x s with rows

and assume that det B0 * 0.                                          (9)

Consider the system

B 0 q 0 = g 0 ,                                       ⁽¹⁰⁾

g 0 = ( 6 0 (0, b i ) ^ i t (0) - A >(0, b l ,D)U 0 - f ) (0, b 1 ),..., c 0 (0, bs>_я (0) - ^(0, b s , DH - f > (0, b_s )) ^T .

r

In view of (9) the system (10) has a unique solution q0 = (q01,^,q0s). Denote apl = apl + Zalplq0i i=1

and below we suppose that

n

L (0 = Z a pl ( t , x ^ ^B 0 I £ I² V _ e R ” , V ( t , x ) e Q.

• p , l = 1

^r 1

C=60- Z q0if (t,x)- ^ V( t, x )e Q, i=r+1

r where 50 is a positive constant. The operator - A0 = A0 (t, x, Dx ) + Z q0iAi( t, x, Dx) is elliptic and we i=1

can consider the problem

C ( t , x ) U t + A ⁰ ( t , x , D x ) u = f , u|_t = ₀ = u 0 ( x ) , Bu\ S = g .                      (11)

Theorem 1. Assume that the conditions (4), (7) hold and f e L_p ( Q ) . Then there exists a unique solution u e W p ’² ( Q ) to the problem (11) . If g = 0 then it satisfies the estimate

II ^u Wp -2 ( Q ^T ) ^ ⁶

II ^u 0I W _p2- ^2/ p ( G ) ⁺l l ^f L ₍ Q ^T )

where the constant c is independent of u ₀, f , t e (0, T ] . If additionally the condition (5) holds and ^ f e L_p ( 0, T ; W p ( G ₆ ) ) then p u e L_p ( 0, T ; W p3 ( G ) ) , p u t e L_p ( 0, T ; W p ( G ) ) and if g = 0 then

• ^u ll Wp ’²( Q ^T ) ⁺l ^{p u} LL _p (0, T ;W p3 (G )) ⁺l ^{p u t}tLL _p (0, r ;W p (G )) “ ^{c [}ll ^u 0l 1 ^ 2 - 2 ^p ( Q ^T ) ⁺

⁺ l ^ ^ 0 01 I w 3 ^-2/ ^p ( G ) ⁺l l ^f I Lp ( Q ^T ) ⁺l ^P^Lp _p (0, T ;W p ( G )) ^],          ⁽¹³⁾

where c is independent of u ₀, f , t e (0, T ].

Proof. The first claim results from Theorem 2.1 in [24]. The estimate (12) results from the conventional arguments (see, for instance, Theorem 2 in [22], Theorem 1 in [21]). Additional smoothness of a solution is established as in Theorem 1 in [27] (see also the proof of theorem 4, subsect. 3, sect. 2, Ch. 4 in [23]). The claim is also contained in Theorem 1 in [28] which can be applied here.

^{Denote the} left-hand ^{side of (13)} by || u||^T  ^{and the} quantity || ^f ||  _{( q} t ) ⁺|| pf||   ₍₀ , _{t ; W} 1 ₍ G )) by || f||_WT.

The corresponding Banach spaces are denoted by H ^τ and W ^τ , respectively. The space H ^τ comprises the functions u e W p ’²( Q ^r ) such that p u e L_p (0, T ; W p3 (G )), p u t e L_p (0, T ; W p ( G )), u satisfies the homogeneous initial-boundary conditions in (11).

Theorem 2. Let the conditions (4)-(9) hold. Then there exists a number T ₀ e (0, T ] such that on ( 0, T ₀ ) there exists a unique solution ( и , q₁,q ₂, - , q s ) to the problem (1)-(3) such that и e W p ’²( Q ^{T 0}), ф и e L p (O, T o ; W p (G)), ф и , e L p (0,^; w'a G )) , q j e C ( [ 0,^ ] ) , j = 1,2, - , s.

Proof. Let q = (q1,..., qs)T. Find a solution Ф to the problem (11), where we take s f=f) + E ft (t, x) q0t and the functions g, и0 are our data in (11). By Theorem 1, there exists a solu-i=ri +1

tion to the problem (11) such that Фe W p ’² ( Q ) , ф Фe L_p (0, T ; W p (G )) , ф Ф _t e L_p ( 0, T ; W p ( G ) ) . Make the change и = v + Ф . We obtain the problem

s

Lv = c (t, x) vt + S ( M) v = ( A0 - A )ф + (- c (t, x) + C (t, x ))Ф t + E ft (t,x) Mt( t),(14)

i = q+1

r where  (M) = -(A0 + A(M)),  A(M) = Ea(t)A(t-хД), c = C — c0(M,t,x), c0(M,t,x)= E f-Mi, i=1t a (t)=qt (t)- qot, M=(M1,-,ms );

vt=0 = 0 BvS = 0v(t, bj ) = Yj (t ) - Ф (t, bj ) = Yj, j = 1 -, s(15)

In view of the properties of the function Ф , D ^aфФ e W p ^C Q ) for all j and | a | <  1. The embedding theorems yield D ^a^b, ( t , x ) e C ^{1-( n + 2 ) /2 p ,2 -( n + 2 )} / ^p ( Q ) (see Sect. 6.3 and Theorem 1 (the Sect. Remarks p. 424) in [29]) and thereby D ^a ф ( t , b j ) e C ( [ 0, T ] ) . In this case the function 4 ( t , b j ) D ^a Ф ( t , b j ) , a t ( t , b j ) D ^a ф ( t , b j ) belong to C ( [ 0, T ] ) . Hence, A ^O Ф ( t , b j ) e C ( [ 0, T ] ) (after a possible change on a set of zero measure). Similarly, C ( t , b j ) e C ( [ 0, T ] ) . Consider the right-hand side in the equation for Ф . We have that f_k ( t , b j ) e C ( [ 0, T ] ) (in view of (6), (8)). From the equation we infer Ф _t ( t , b j ) e C ( [ 0, T ] ) for all j . Thus, we have reduced the problem (1)-(3) to a simpler problem ⁿ

(14H15).  Let  B^ = {реC([0,t]):|pC([0r])

r atj = E ok Ak and find the quantity R0 such that k=1

|L(f)| < ^0 ^²/2 V^ e Rⁿ,|c-C =

^r1

E Mt ft i=r+1

< ^0/2, V( t, x )e Q, Vpe B^.

In this case the operator S ( a ) is elliptic and Theorem 1 holds with S ( a ) rather than A0 . Given a vector pe BR^ , find a solution v to the problem (1)-(2) on (0,t) such that фv e Lp (0,t; Wp(G)), фvt e Lp (0, t; Wp( G)). Study the properties of this map a ^ v (A) . Theorem 1 yields r                        r1                  S v=L"1 f,f=E м1Лф( t, x)+ E Mtfxф t + E Mtft (t, x) • i=1                      t=r+1               t=r1 +1

We have the estimate iiv^T =| L-1 fLr < ciifW

The conditions on the coefficients imply the estimate

II^fV - ^c21HC([0,_r]) ’

where the constant c2 depends on the quantities ||fi||_w-  (i > r +1), ||f-|y₍₀,_T;W1₍G))  (r +1 < i< r),

||ф||_ят (we can replace - with T in these norms and thus we can take c₂ independent of - ). Take H e BR^  (i = 1,2) and consider the corresponding solutions v , v₂ to the problem (14)-(15). Let

H = (H_1;, h₂i,—, Hsi), i = 1,2. Subtracting the latter equation (1) from the former, we obtain that the difference to = v₂- v1, vi = v ( h ) meets the equation

(c + c₉ )

' ¹—²²to + 5

2       ¹

H1 ⁺H2 2

r to = E(Hj2 (1) - Hj 1 (1))Aj (1,x,D)(v1 + v2 )/ 2 + j=1

У (Hj2 (1) - Hj 1 (1))fj (1,x)(v11 + v21)/2 + E(Hj2 (1) - Hj 1 (1))Aj (1,x,D)ф j=r+1

r1   z                          X

+ E (Hj2 (1)-Hj 1 (1))fjФ1 + У fj (1,x)(Hj2 (1)-Hj 1 (1 )) = f - j=r+1                                  j=Г1+1

We have that (H1 + H₂)/2 e BR and, thus, the following estimate (see (17)) holds:

IIMH- < «Hw-•

The estimates (18), (20) ensure that

II' H- < dfU < c 2 c| |H2 - a||c([0,r]) ■(21)

^whereC2 depends ^on the norms ||(^v1 ^{+ v}2 )^/2|H , ||^f'||_w-  (i > ^r1 ⁺1) , llfi-ll^(0t;W’(G_S))  (r^{+ 1}< i< r) ,

||Ф||Ht . Let v = v(ц) be a solution to the problem (14)-(15). Taking x = bj in (1) and taking into ac count that v (1,bj) = ^j , we obtain the system

,                             r                                            r1

c(1 ■ bj^j' + 5(H)v(1 ■ bj) = У HiAi (1 ■ bj■ D)ф(1 ■ bj ) + У Hifiф 1 (1 ■ bj ) + У fi (1 ■ bj )Hj (1)- i=1                                       i=r+1                         i=r1 +1

The right-hand side can be written as B (1) ц, where the rows of the matrix B (1) are as follows:

A1 (1 ■ bj,D)ф( 1 ■ bj),—■ Ar (1,bj,D)ф( 1 ■ bj)■ fr+1ф 1 (1,bj)■—■ frф 1 (1 ■ bj)■ fr+1 (1 ■ bj)■—■ fs (1 ■ bj).

The matrix B (0) agrees with B₀ from (9) and thereby det B (0)^ 0. The functions fi (1,bj), a^lkl (1,bj), a^l_k (1,bj-) are continuous for all values of the indices. Moreover, D^a^Ф( 1,x)e C(Q) for a I< 2. Thus, the entries of B are continuous in 1 and there exists t₀< T and a constant 5₃ > 0 such that

|detB(1) | > 53 > 0V1 e[0,t₀].

In this case the system (22) is written in the form

H( 1 ) = B-¹H (H)( 1 ) = R (H) ■ H (H) =

(c (1, b1)^1' + 5 (H) v (1 ■ b1), c (1, b2)^2' + 5 (H) v (1 ■ b2),--., c (1, b_s )^_s' + 5 (H) v (1 ■ b_s))^T        (24)

The right-hand side here contains an operator taking the vector μ into the vector with the components C(1 ■ bj^j -c⁰(h, 1 ■ bj )^j + 5(h)v(1 ■ bj) (j = 1,2,—,s), where v is a solution to the problem (14),

(15). The properties of the map m ^ v(m) we have already studied. Demonstrate that there exists t1 < t0 such that the operator R(и) = B 1H(^)(t), R: C([0,t1 ])^ C([0,t1 ]) takes the ball BR^ itself and is a contraction. Consider the quantity уj (0). By construction,

C ( O, bj ^' (O) = C (O, bj )(yj' (O )-Ф' (O, bj )) = C (O, bj )yj' (O) + A (0, bj, D ) u о (bj)- s fо — Z ft(0,bj)qoi = 0,j = 1,_,л, i=г +1

since the numbers q0i are defined from (10). Let y=(c (o, bi )у1', C (o, ь2 ^V.., C (o, b )ys У.

In this case у e C([0,т]) (т< т_о ) and у(0) = 0. There exists a number t₁< т_о  such

||B-1 (t)у||с/г0 i\ < R0/2 • Note that R(0) = B-1 (t)y(t). Next, we obtain estimates assuming into

that that

Me BR^ and т < тх. In this case у e C ([0,t]) (t < t0 ) and у (O ) = 0. There exists a number t1 < то such that ||b-1 (t)y||cq0 ^ < RO /2 . Note that R(O) = B-1 (tу (t). Next, we obtain estimates assuming that д e BR and т < т1. We have sr

I^R (^{Ml) — R} ( "• )| C([o t,>  -- H*"""^Д2^{Д f}'⁽t ■ bj ^j ([0,„ ⁺

,       j=1 i=r+i ssr

Z| I AOvl ( t, bi) - Ao v2 ( t, bi )| C ([0,t]) + ЕЯ Д1 kAkvl ( t, bi ) - Д2 k Akv2 ( t, bi )| C ([0,t]) .

i=1                                                   i=1 k=1

Next, we employ the conditions on the data and the embedding W® (G) c C (G) for 0 > n / p (see Theorems 4.6.1,4.6.2 in [30]). Take вe(n/p,1 -2/p). Consider the last summand. We have

IIД1 kAkvl (t,bi ) - Д2kAkv2 (t,bi )|C([о,т]) < ||(Mlk - Д2k )(Akvl (t,bi) + Akv2 (t,bi ))||CQ0

+

( Ml k + M2 k )

⁽Ak^(v1⁽t,^bi’ ^-v2⁽t,bi”

C([0,t])

<

II^д1 k^{- д}2k Ic_([o.t]) c4 Z ||^D" ( v (', bj ) ⁺v2 ( t, b⁾⁾C([от]) +

II Mk^{+ д}2 k Ic([от]) c_5|Z ||D"v ( t. bj ) — D"v2 (I, bj )| С ([от]) < ||Mlk⁺M2kIC([0,t]) ^c7 k⁽vl ⁽t,x) ^{- v}2⁽t,x))C_([o,T|W2+в₍G)) ,

where the constant ci are independent of т . Let v e Hт . In this case фу e Lp (0,t; Wp3(G)), фvt eLp(0,t; Wp(G)) and thereby фу e C([0,t];W3^'p(G)) (see Theorem III 4.10.2 in [29] and [30]). The inequality r„                                                ^

II^фАс₍[o,T];W_p-^2/p(g)) <^c8^1 l^фvIL_p(0,t;W_p³(G)) ⁺1l^фvtlL_p(0,t;Wp(G)) j,           ⁽²⁷⁾

is valid, where the constant c₈ is independent of т e (O, T]. Indeed, consider the function w(t,x) = v(t,x) for t< т , w(t,x) = v(2т-t,x) for t e(T,2T) and w(t,x) = O for t > 2т . We use the

Pyatkov S.G.,                                        On Some Classes of Inverse Parabolic Problems

Soldatov O.A.                                           of Recovering the Thermophysical Parameters fact that v(0, x) = 0. Next, we write out (27) on [0, T] for the function w, the constant in this inequality is independent of τ . Next, we estimate the right-hand side of the inequality obtained from above with the use of the definition of w and obtain that the constant c8 in (27) is independent of т . Let 01 (3 - 2/p) = 2 + 0. We have that 01 e (0,1). Denote to = v2 - v1. Next, we use the interpolation properties of the Sobolev spaces [30] and (27). We infer

I- - ■ c([«,,],,2⁺⁰(G)) ⁵c HlC^^-^(G))   'I^Ap(G)) ⁵

^с1^т'1М¹"0^_(c))|HI^{0 5c}2^т'\\фА\ht ■ Г = ^{(1 -}1/p^{)(1 -0}1).           (28)

where we employ the obvious inequality

II4([0т];L_p (G)) ^{5 T 1P} lv*IL_p (0т;L_p (G)), v⁽⁰) = ^0-

The inequality (28) yields

II^-toС([0,т];^2⁺⁰(G)) ^{5 с}9^ l^|tolH^T ,                                  ⁽²⁹⁾

where the constant c₉ is independent of т. Аналогично получим

I-⁽v1 ⁺v2 )С₍[0,т]^_р2+0(G)) ^{- C}9^{TY |(}v1 ⁺v2 )V '

In view of the inequalities (17), (18) (written for the functions v_i ), (21), (26), and (29), we conclude that

IM1 kAkv ( t. bi)-М2 -Akv2 ( *. bi )|С ([0т]) 5 с13тЧМ1 k ” М2 k|C ([0т]).

where c12 is independent of т 5 тх (it depends on R0 ). Similar estimate holds for the second summand in (25). The first summand is estimated by

5 rf sEk m- м2 ,).f(*. М^т5 -4к- m С ([с,т])[3к;|с([о.т]>^

The final estimate is of the form (see (25))

f      5А

||^{R (М)}” ^{R (М}2 ^С([0т]) ⁵ ।^М|* ” ^М2 -^С ([0.т]) c [^Т? +SI И1с([0т_В/

Choosing т₂- т1

such that c₁₅t^y

⁺Я к/IL ([0т])

5 -^ for т 5 т₂ , we have proven that R

is a con-

traction and takes the ball B_R^ into itself for т 5 т₂ . The fixed point theorem implies the existence of a solution to the system (24). Let v = v(m) . Show that this function satisfies the overdetemination conditions in (15). Take x = b_j in (14). We obtain the system

.           .         .                             .                      r                 .                  .                 ^r1                        /          X           5

^c(*.^bj)^vt (*.^bj) ^{+ Av}(*.^bj ) = SMiAi (*.bj.D)^Ф+s fiMi^фt (t-bj )+ s fj (t.bj)Mj (t).   ⁽³³⁾

i=1                            i=r+1                       j=ri +1

Subtracting these equalities from (21), we infer v* (*,bj^-щj = 0 for all j and thereby these conditions are fulfilled. Uniqueness of a solution follows form the estimates exhibited above.

Remark 2. The corresponding stability estimate for solutions also holds.

The research was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation (theme No. FENG-2023-0004, “Analytical and numerical study of inverse problems on recovering parameters of atmosphere or water pollution sources and (or) parameters of media”).

Bulletin of the South Ural State University Series “Mathematics. Mechanics. Physics” 2023, vol. 15, no. 3, pp. 23–33