CC BY
5
DOI: 10.14529/mmph230303
ON SOME CLASSES OF INVERSE PARABOLIC PROBLEMS OF RECOVERING THE THERMOPHYSICAL PARAMETERS
S.G. Pyatkov, O.A. Soldatov
Yugra State University, Khanty-Mansiysk, Russian Federation
E-mail [email protected]
Abstract. In the article we examine the question of regular solvability in Sobolev spaces of parabolic inverse coefficient problems. A solution is sought in the class of regular solutions that has all derivatives occurring in the equation summable to some power. The overdetermination conditions are the values of a solution at some collection of points lying inside the domain. The proof is based on a priori estimates and the fixed point theorem.
Keywords: parabolic equation, inverse problem, initial-boundary value problem, existence, uniqueness.
Introduction
We consider the well-posedness questions of inverse parabolic problems. Let G be a domain in M" with boundary r e C2 . The parabolic equation is of the form
c(t,x)ut + A(t,x,D)u = f (t,x), (t,x)eQ = (0,T)xG, (1)
where the functions f,c and the elliptic operator A are as follows:
r 5
- A (t, x, D ) = Ao (t, x, Dx ) + £ q, (t ) Ai (t, x, Dx ), f = f (t, x ) + £ f (t, x ) qt (t ),
t=1 t=r+1
n n r1
A = £ a'ki (t,x+ £ak (t,x)âxk + a0,c=co (t,x)- £ ft (^x)q, (t)•
k,l=1 k=1 t=r+1
The equation (1) is furnished with the initial and boundary conditions
u,
t=0 = Щ, Bu\S = g (t, x), S = (0,T )хГ, (2)
ди "
where Bu = и or Bu =--h au = V a,,ur v.+au (v = (v,,v?,..,v„ ) is the outward unit normal to Г,
dN ч xi J ^ 1 2 n'
and the overdetermination conditions
u(t,bj ) = Wj (t),bj e G, j = 1,2,...,(3)
The unknowns in (1)-(3) are a solution u and the functions qt (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.
Prelimiraries
The definition of the Sobolev spaces Wsp (G;E), Wsp (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 Wsp (Q). The definitions of the Holder spaces Ca'/3 (q),C(S) 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 Wspr (Q) = Wsp (J;Lp (G ))o L p (J;Wrp (G)),
Wsr (S) = Wsp (J;Lp (r))oLp (J;Wrp (r)). 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 Cs, s > 1, can be found in Ch. 1 in [25]. Denote by BS (b) the ball of radius S centered at
b . Fix a parameter 8 > 0 such that BS (bi) o BS (bj) = 0 for i ^ j and BS (b) o r = 0, i, j = 1,2,...,s . Denote Qr=(0,r)xG and GS = BS(bt). Construct nonnegative functions
s
<Pj eC°°(E") such that (pt =1 in B(i 2 j. <p(x) > 0 and (pt =0 for x g B-,(i 4 j. Let (p ^(p; (x).
i=i
The consistency and smoothness conditions can be written as
Uo (x) e Wp2/p G ),B(0,x,Z))«o|r = g(0,x),g e Wp0 (S), (4)
where k0 = s1 = 1 -1/2p for Bu = u and k0 = s0 = 1/2-1/2p otherwise;
qu, (x) e Wp p (G),aj e LOT (0,T;Wp (G8)),a\ e Lp (0,T;Wp (G8)), (5)
where i, j = 1,2,n, 1 = 0,1,., n, k = 0,1,., r;
f eC (Q)o Lx (0,T;Wi(G8)) e Lp (0,T;Wlp{G)) ,ft e Lp (Q), (6)
where j = r +1,.,r1 h 1 = 0, r1 +1,.,s . We use the inclusions of the form f e Lp (0,T;Wlp (G8)) or
similar, where the set GS consists of several connectedness components (in this case
Bs (bj)). By definition, this means that f \B ^ ^ e Lp (0,T; Wlp (B8 (bj))) for all j . This space is endowed with the norm
equal the sum of the norms over the corresponding connectedness components We assume that
4 e C(Q),ak e Lp (Q),<y,clj^ eW;°,2s° (S), (7)
where the last inclusion is required only if Bu ^ u ,
eC1 ([0,T]),^. (0) = u0 (bj ), ak (t, bj ) ,fm (t, bj )eC ([0,T]), (8)
where j = 1,., s, m = 0, r +1,., s, 1 = 0,1,., n, к = 0,1,., r. In view of (5), (8), the traces
fm bp ),a1 ^bj ) are defined and fm ^bp ),a1 bj )e Lp (0,T); тот^те^
Pyatkov S.G., On Some Classes of Inverse Parabolic Problems
Soldatov O.A. of Recovering the Thermophysical Parameters
fm (t,x),ak (t,x) e C(gs;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 5 x 5 with rows
A1 , D)u0 (bj ),..., Ar (0,bj, D)u0 (bj )jr+1 (a bj 0) ..J^ ( ^f^! (o, bj ) ,...JS (a bj )•
and assume that
det B0 * 0. (9)
Consider the system
B0q0=g0, (10)
=(c0(0,bl)Vu(0)-A0(0,bl,D)u0 —f0(0,bl),...,c0(0,bs)i//st(0) — A0(0,bs,D)u0 - f0(0,bs))T.
r
In view of (9) the system (10) has a unique solution q0 = (q01,..., q0s ). Denote apl = a°pl + £aiplq0t
t=1
and below we suppose that
n
Lo (#) = £ api {t,x)ÇPSi > I # I2 e M",V(ï,x) e Q.
P,i=\
C = c0 -£ qmf (t,x)>^0 V(t,x)eQ,
t=r+1
where 80 is a positive constant. The operator -A0 = A (t,x,Dx ) + ^q^A (t, x,Dx) is elliptic and
we
;=1
can consider the problem
C (t, x) Ut + A0 (t, x, Dx )u = f, u|t=0 = U0 ( x), Bu\s = g. (11)
Theorem 1. Assume that the conditions (4), (7) hold and f e Lp (Q). Then there exists a unique solution u e Wp'2 (Q) to the problem (11). If g = 0 then it satisfies the estimate
u ,
n \w1
p
(12)
¿2(e <)*c [I lu01 p (g )+ll f\\Lp (q *)
where the constant c is independent of u0, f ,*e (0,T]. If additionally the condition (5) holds and <pf e Lp (0,T; Wj (G*)) then (pu e Lp ( 0,T ;Wp (G )), put e Lp (0,T ;Wp (G )) and if g = 0 then
Hlw^Q*) +llPull^(0,*;Wp(G)) + (0,*^)) ^ <4IW^P(Q*) +
+ |K| |w3-^ p (G) f\\Lp (Q*) +\W\lp (0,rWj,(G))], (13)
where c is independent of u0, f ,re (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\\HT and the quantity \\f\\L^(qT) +||f ^(0,r;W,(G)) by llfllWt •
The corresponding Banach spaces are denoted by HT and WT , respectively. The space HT comprises the functions u eWp'2(Qr) such that (pu e Lp (0,T;Wp3(G)), (put e Lp (0,T;Wp(G)), u satisfies the homogeneous initial-boundary conditions in (11).
Main results
Theorem 2. Let the conditions (4)-(9) hold. Then there exists a number t0 e (0,T] such that on ( 0,t0 ) there exists a unique solution (u, q1, q2,..., qs) to the problem (1)-(3) such that u e Wp'2(QT<0), (pu^Lp (0, T0;Wp(G)), qmt e Lp(0,r0; W\ (G)), qr, e C([0, r0 ]), j = 1,2,..*.
Proof. Let q = (ql,...,qs)T. Find a solution ® to the problem (11), where we take
s
f = f0 + Z f (t,x)q0i and the functions g,u0 are our data in (11). By Theorem 1, there exists a solu-
i=r +1
tion to the problem (11) such that OeWj2 (Q), qOe Lp(0,T;Wp (G)) , qOt e L p (0,T;Wp (G)) . Make the change u = v + O. We obtain the problem
Lv = c(t,x)vt+S(,u)v = (A0-A)<& + (-c(t,x) + C(t,x))®t+ £ f^x)^), (14)
where (Ju) = -(A°+A(Ju)), A(/u) = '^lul(t)4(t,x,Dx), c = C-c*(fi,t,x), c\fi,t,x)= £ f^ ,
i=l i=r+1
Pi (0 = % (0 - %i > M = (th,-■ ;Ms ) ;
vi=o = 0 ,Bvs = 0,v(t,bj ) = Wj (t) - O (t,bj ) = WjJ = 1,- • -A (15)
In view of the properties of the function O, DaqO e Wp2(Q) for all j and \ a\ < 1. The embedding theorems yield DaqOi (t,x)e C1_("+2)/2p,2_("+2)/ p(q) (see Sect. 6.3 and Theorem 1 (the Sect. Remarks p. 424) in [29]) and thereby Dao(t,bj )eC([0,T]). In this case the function akj (t,bj )Dao(t,bj ), ak (t,bj )Dao(t,bj ) belong to C([0,T]). Hence, A0o(t,bj )e C([0,T]) (after a possible change on a set of zero measure). Similarly, C(t,bj ) e C([0,T]). Consider the right-hand side in the equation for O. We have that fk (t,bj ) e C([0,T]) (in view of (6), (8)). From the equation we infer O, [t,bj j e C ([().'/']) for all j . Thus, we have reduced the problem (l)-(3) to a simpler problem
(14)—(15). Let B^ ={/2eC([0,r]):|/i|cM<^}. Consider the expression =
j=i
y^/ikiij. and find the quantity R{) suchthat
r
a, = 7 ak
k=l
rl
L
(ï)\<S0\Ç\2/2\/ÇeR",\c-C\ =
Z Hf
i=r+1
In this case the operator is elliptic and Theorem 1 holds with rather than A° . Given a vector jueBj^ , find a solution v to the problem (l)-(2) on (0,r) such that cpv eLp(0,r;Wp(G)),
cpvt e Lp (0, r; Wp (G)) . Study the properties of this map li —» v(u). Theorem 1 yields
r r s
v = L~lf, f = Zm4x)+ Z ft + Z ^f (t,x). (16)
/=1 i=r+1 /=r+l
We have the estimate
V T =
1Г11я T
L~lf ^ 4f\\ivT ' (17)
Pyatkov S.G., On Some Classes of Inverse Parabolic Problems
Soldatov O.A. of Recovering the Thermophysical Parameters
The conditions on the coefficients imply the estimate
i't'^11 ¿LM> (I«)
where the constant c2 depends on the quantities ||f ||Wt (i > r +1) , \ft (0,r;W1(G)) (r +1 <i < r1), P\\hT (we can replace t with T in these norms and thus we can take c2 independent of t ). Take ju^eBj^ (/=1,2) and consider the corresponding solutions v, . v2 to the problem (14)—(15). Let JLii = (/.iv, fi2l,.. .,//vi = 1,2. Subtracting the latter equation (1) from the former, we obtain that the difference co = v2 - v1, v, = meets the equation
M1+M2
(с, + c2) ГГ. .Г.\ K 1 2)- щ + S
2
r1
®=(o-^i (0)aj + v2)/2+
j=1
X (^72 (t)-Mj1 (t)) fj (^ x)(Vit + V2t ) /2 + Z(M2 (t)-Mj1 (t)) A (t, x,D)0
j=r+1 .7=1
r1 5
y=r+1 /=1+1
We have that ^//, + //2 j/2 e and, thus, the following estimate (see (17)) holds: The estimates (18), (20) ensure that
(21)
where c2 depends on the norms ||( V1 + V2 ) /2|| ht , \\fi IIwT (i > 1 + 1) , llf Hlm (0,r;W^(G )) (r + 1 < i < 1 ) , 1®!^ . Let v v(m) be a solution to the problem (14)—(15). Taking x = bj in (1) and taking into account that vt [t,bj j if/j , we obtain the system
r 1 S
/=1 /'=r+l 7=^+1
The right-hand side can be written as B(t)ju , where the rows of the matrix B(t) are as follows:
A1 (t, bj, D bj)^ a (t, bj, D bj) f+ (^ bj) (t, bj )jn+1 (t, b ^...f (u b).
The matrix B(0) agrees with B0 from (9) and thereby detB(0)^ 0. The functions f (t,bj),alkl (t,bj), a'k (t,bj) are continuous for all values of the indices. Moreover, Daqfl>(t,x)e C(q) for \a\< 2. Thus, the entries of B are continuous in t and there exists r0 <T and a constant S3 > 0 such that
\detB(t) \ >S3 > 0 Vt e[0,r0 ]. (23)
In this case the system (22) is written in the form
ju (t) = B~lH (//)(*) = R (//), H (//) =
(cit^)^ + S^vit^lcit,^)^ +S(jl)v(t,b2\...,c(t,bs)w;+S(iu)v(t,bs))T (24) The right-hand side here contains an operator taking the vector u into the vector with the components c{t,bj^ipj -c° (ju,t,bj^y/j + s(iu}v{t,bj^ (j =1,2,...,s), where v is a solution to the problem (14),
(15). The properties of the map // —> we have already studied. Demonstrate that there exists r, < r0 such that the operator /r///(«j(/). :( '([(),r, ]) ^( '([(),r, ]) takes the ball B,^ into
itself and is a contraction. Consider the quantity yj (0). By construction,
(°) = ^C0'^ (°> - )) = ^C0'^ (0))-
■ —
f -Z f (0,bj)q0i = 0, j = 1,.,s,
i=r+1
since the numbers q0i are defined from (10). Let
In this case feC^i]) (r<r0) and i//(0) = 0. There exists a number r, <r0 such that
B~l(t)¥
-a^D
<Rq/2. Note that /¿(0) = B (/)'//(/). Next, we obtain estimates assuming that
ju e Bo and r < r, . In this case '// e ( '([(). r]j (r < r0 ) and i//(0) = 0 . There exists a number r, < r0
JR0 such that
B~\t)¥
< Rq /2 . Note that /¿(0) = B (/ )(//(/ ). Next, we obtain estimates assuming
that /г e BR^ and x<x1. We have
£ Ik -fhiWtjWj
S r
C([0,t]) *-r
/=1t=r+1
ic([0,r])
+
31 M (i, b )- 4ov2 (t, bt
t=1
C([0,t])
+
s r
33|KAkV1 (^bt ) - K2kAkV2 (^bt
t=1 k=1
C([0,t])
(25)
Next, we employ the conditions on the data and the embedding W® (G) c C (g) for 9>n /p (see Theorems 4.6.1,4.6.2 in [30]). Take 6 e (n/p,1 -2/p). Consider the last summand. We have
blkAV1 bi ) - M2kAV2 (t,bi )C([0,r]) < ll(^1k - ^2k )(Akv1 (t,bi ) + Akv2 (t,bi )|C([0 r]) /2 +
(^1k +M2 k ),
-( Ak (Ki(t, bi ) -^(t, b ))
C([0,t])
IK -^2k|lc([0,Tl) c4 3 ö"(v1 (t'Ь/ ) + v2 (^Ь/ ))
v ' | a\ <2
ik + *2k|lc([0,t]) c5 3 ||d"v1 (^b/ )-d"v2 (t,b/ )
m<2
c([0t]) '([0,t])
+
<
IK + v2k ||c([0,t]) c7 Ip(V1 (t, X) - V2 (t, X)|C([0,T]^(G)) ,
(26)
where the constant ct are independent of t . Let v e Ht . In this case qv e Lp (0, t; Wp (G)),
qvt e Lp (0,T;Wp(G)) and thereby qv e C ([0,t]; Wp3_2/p (G)) (see Theorem III 4.10.2 in [29] and [30]).
f \
The inequality
INc([0,t];^-2/p (G))< C8 l\\\ (0tW(G) )+\Mlp (o,T;Wp(G ) )
(27)
is valid, where the constant c8 is independent of t e (0, T]. Indeed, consider the function w(t,x) = v(t,x) for t < t , w(t,x) = v(2t -t'x) for t e(T,2T) and w(t,x) = 0 for t > 2t . We use the
s
2
Pyatkov S.G., Soldatov O.A.
On Some Classes of Inverse Parabolic Problems 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 t . 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 t . Let
01 (3 -2/p) = 2 + 0 . We have that 01 e (0,1). Denote c = v2 - v1. Next, we use the interpolation properties of the Sobolev spaces [30] and (27). We infer
IMIC(M<0(G)) * clM00,,]w3-2 /p(G)) lC0]LPG) <
с1т\М1 0
Lp
0
U0,r;Z„(G))IH&" С2ТГЬ4 h*\ = (1"17 P)(1 )'
(28)
<т 14 7 P|k
\lp (o,^ (G),M (0) = °
where we employ the obvious inequality
IMIc([0,T]Lp (G))
The inequality (28) yields
INIc([°,t];^2+0(G)) < C9T\NIh
where the constant c9 is independent of т. Аналогично получим
1иm1 + m2)ic([°,t]^2+0(g)) < С9тУЫМ1 + m2)iht •
In view of the inequalities (17), (18) (written for the functions Mi), (21), (26), and (29), we conclude
(29)
(30)
that
|KAV1 ^ bi ) - M2kAV2 ^ bi !c([0,r]) < ^ W - Wk |lc([0,r]) ,
where c12 is independent of t<t1 (it depends on R0 ). Similar estimate holds for the second summand in (25). The first summand is estimated by
1 и
Z Z I
j=\i=r+\
The final estimate is of the form (see (25))
ЯИЦ
C(M) К -V2k\\c([0,t])c^r +Ё1И
(31)
(32)
Choosing т2 < t1 such that c
15
-ZIH
lc([0,r])
< — for т < t2 , we have proven that R is a con-
j=1 ■ j
traction and takes the ball into itself for r < r2 . The fixed point theorem implies the existence of a
solution to the system (24). Let v = • Show that this function satisfies the overdetemination conditions in (15). Take x = bj in (14). We obtain the system
r 1 5
c (t, bj)vt (t, bj) + Av (t, bj) = Xw4 (t, bj, D) 0 + X fW <0 fa) + Z fj (tj W (t). (33)
/=1 i=r+1 j=r1 +1
Subtracting these equalities from (21), we infer vt (t,bj) - ipj = 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 ").
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On Some Classes of Inverse Parabolic Problems of Recovering the Thermophysical Parameters
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Received May 5, 2023
Information about the authors
Pyatkov Sergey Grigorievich is Dr. Sc. (Physics and Mathematics), Professor, School of Digital Engineering, Yugra State University, Khanty-Mansiysk, Russian Federation, e-mail: [email protected].
Soldatov Oleg Al'bertovich is Post-graduate Student, Yugra State University, Khanty-Mansiisk, Russian Federation, e-mail: [email protected].
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2023, vol. 15, no. 3, pp. 23-33
УДК 517.956 DOI: 10.14529/mmph230303
О НЕКОТОРЫХ КЛАССАХ ОБРАТНЫХ ПАРАБОЛИЧЕСКИХ ЗАДАЧ ВОССТАНОВЛЕНИЯ ТЕРМОФИЗИЧЕСКИХ ПАРАМЕТРОВ
C.Г. Пятков, О.А. Солдатов
Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: [email protected]
Аннотация. В работе мы рассматриваем вопросы регулярной разрешимости в соболевских пространствах обратных коэффициентных параболических задач. Решение ищется в классе регулярных решений, которые имеют все производные, входящие в уравнение суммируемые с некоторой степенью. В качестве условия переопределения берутся значения решения в некотором наборе точек, лежащих внутри области. Доказательство основано на априорных оценках и теореме о неподвижной точке.
Ключевые слова: параболическое уравнение; обратная задача; начально-краевая задача; существование, единственность.
Литература
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- С. 92-102.
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22. Pyatkov, S.G. On some Parabolic Inverse Problems with the Pointwise Overdetermination / S.G. Pyatkov, V.V. Rotko // AIP Conference Proceedings. - 2017. - Vol. 1907. - 020008.
23. Pyatkov, S.G. Identification of Thermophysical Parameters in Mathematical Models of Heat and Mass Transfer / S.G. Pyatkov // Journal of Computational and Engineering Mathematics. - 2022. -Vol. 9, no. 2. - P. 52-66.
24. Denk, R. Optimal Lp-Lq-Estimates for Parabolic Boundary Value Problems with Inhomogene-ous Data / R. Denk, M. Hieber, J. Prüss // Mathematische Zeitschrift. - 2007. - Vol. 257, no. 1. - P. 93224.
25. Ladyzhenskaya, O.A. Linear and Quasi-linear Equations of Parabolic Type / O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva // Translations of Mathematical Monographs. - 1968.
- Vol. 23. - 648 p.
26. Amann, H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces / H. Amann // Glasnik matematicki. - 2000. - Vol. 35, no. 1. - P. 161-177.
27. Mikhailov, V P. Partial Differential Equations / V P. Mikhailov. - Moscow: Mir, 1978. - 396 p. 2 8. Nikol ' skii, S.M. Approximation of Functions of Several Variables and Imbedding Theorems /
S.M. Nikol ' skii // Grundl. Math. Wissensch., 205. - Springer-Verlag, New York, 1975. - 418 p.
Pyatkov S.G., On Some Classes of Inverse Parabolic Problems
Soldatov O.A. of Recovering the Thermophysical Parameters
29. Amann, H. Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory / H. Amann // Monographs in Mathematics. Vol. 89. - Birkhäuser Verlag Basel, 1995. - 338 p.
30. Triebel, H. Interpolation Theory, Function Spaces, Differential Operators / H. Triebel. - Berlin, VEB Deutscher Verlag der Wissenschaften, 1978. - 528 p.
Поступила в редакцию 5 мая 2023 г.
Сведения об авторах
Пятков Сергей Григорьевич - доктор физико-математических наук, профессор, Инженерная школа цифровых технологий, Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация, e-mail: [email protected].
Солдатов Олег Альбертович - аспирант, Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация, e-mail: [email protected].
