Nonstationary Radiative–Conductive Heat Transfer Problem in a Semitransparent Body with Absolutely Black Inclusions

: The paper is devoted to a nonstationary initial–boundary value problem governing complex heat exchange in a convex semitransparent body containing several absolutely black inclusions. The existence and uniqueness of a weak solution to this problem are proven herein. In addition, the stability of solutions with respect to data, a comparison theorem and the results of improving the properties of solutions with an increase in the summability of the data were established. All results are global in terms of time and data.


Introduction
Complex heat transfer problems, in which it is necessary to simultaneously take into account the transfer of energy by thermal radiation and thermal conductivity, arise in various fields of science and industry. The discussion on the properties of complex heat transfer problems and the methods for solving them constitutes an extensive physical literature (see, for example, [1][2][3][4]).
Mathematical problems of radiative-conductive heat transfer are nonstandard, interesting and rather complicated. Heat radiation is nonlinearly dependent on temperature, and integro-differential equations or nonlocal boundary conditions are used to describe radiation heat transfer. Various nonlinear nonlocal boundary and initial-boundary value problems arise in this field.
The first mathematical results in this direction were obtained by A.N. Tikhonov [5,6] in the late 1930s. The construction of the mathematical theory of radiative-conductive heat transfer problems was continued for roughly forty years [7][8][9][10][11][12][13][14]. In the early 1990s, many mathematicians were paying attention to such problems. As a result, over the past 30 years, a large number of papers have been devoted to the solvability of complex heat transfer problems (cf. ). Naturally, the above list is not exhaustive.
To date, the solvability of various statements of complex (radiative-conductive) heat exchange problems in systems consisting either only of radiation-opaque bodies or only of radiation-semitransparent bodies has been studied in sufficient detail. At the same time, the problems of radiative-conductive heat exchange in systems consisting of both radiation-opaque and of radiation-semitransparent bodies remain to date unexplored. This specific area of study, to the best of the author's knowledge, has only been the subject of the following articles: [64][65][66].
In this paper, the existence and uniqueness of a weak solution to a nonstationary boundary value problem governing radiative-conductive heat transfer in a semitransparent body containing several absolutely black inclusions were proven. All results are global in terms of time and data. The unknown functions u and I physically represent the absolute temperature and radiation intensity. The problem was considered in a gray approximation.
The technique used was developed in [41,53,57]. In the stationary version, this problem was studied in [65].
The paper is organized as follows. In Section 2, the physical sense of the problem is explained. Section 3 is devoted to notations. In Section 4, the boundary value problem for the radiative transfer equation is considered. Section 5 contains the formulation of the main results of the paper. In addition, in this section, the problem is reduced to the equivalent initial-boundary value problem for a nonlinear operator-differential equation (Problem P) with only one unknown function u. In Section 6, a number of important auxiliary assertions are proven. In Section 7, an auxiliary problem P [n] is introduced and its solvabilty is proven. In Section 8, a priori estimates for weak solutions to Problems P and P [n] are established. Section 9 establishes the stability of weak solutions to Problem P with respect to the data. A comparison theorem, which, in particular, implies the uniqueness of a weak solution, is also proven. Section 10 contains the proof of the existence of a weak solution to the problem P. Finally, Section 11 establishes the validity of the main results of the article.

Physical Statement the Problem
Let G be a bounded convex domain in R 3 and {G b,j } m j=1 be a system of strictly internal subdomains of the domain G. Assume that G b,i ∩ G b,j = ∅ for all i = j and We assume that each of the domains G b,j , 1 ≤ j ≤ m is an absolutely black body and the domain G s is occupied by a semitransparent optically homogeneous material with a constant absorption coefficient κ > 0 and a scattering coefficient s ≥ 0.
The unknown functions u(x, t) and I(ω, x, t) physically represent the absolute temperature at point x ∈ G at moment t ∈ (0, T) and the intensity of the radiation propagating at point x ∈ G s in direction ω ∈ Ω = {ω ∈ R 3 | |ω| = 1}, respectively. The function u is defined on the set Q T = G × (0, T). Its restrictions to the set Q s,T = G s × (0, T) and to the set Q b,T = G b × (0, T) will be denoted by u s and u b , respectively. The function I is defined on the set D s × (0, T), where D s = Ω × G s .
Here, c p is the heat capacity coefficient, λ(x, u) is the thermal conductivity coefficient, and f is the density of heat sources. The function h(u) = σ 0 |u| 3 u for u > 0 corresponds to the hemispherical radiation density of an absolutely black body according to the Stefan-Boltzmann law, where σ 0 > 0 is the Stefan-Boltzmann constant. Equation (1) describes the heat transfer process in the gray semitransparent medium G s . The terms 4κ h(u s ) and κ Ω I dω in it correspond to the densities of the energies emitted and absorbed in G s , respectively. Equation (2) describes the heat transfer process in opaque inclusions G s . Equation (3) describes the transfer of radiation in a radiating, absorbing and scattering medium G s . The term ω · ∇I = (3) denotes the derivative of I along the direction ω. We denote by S the scattering operator: with the scattering indicatrix possessing the following properties: We regard R 3 as the Euclidean space of elements x i y i . Assume that the domain G s is Lipschitz.
Thus, the domains G b,j , 1 ≤ j ≤ m are also Lipschitz. We denote by dω and dσ(x) the measures induced by Lebesgue measure in R 3 on Ω and ∂G s , respectively. We also assume that the boundary ∂G s is piecewise smooth in the following sense. There exists a closed subset G ⊂ ∂G s such that meas (G; dσ) = 0; moreover, for each point x ∈ ∂ G s = ∂G s \ G, there exists a neighborhood of it, in which the boundary ∂G s is continuously differentiable. Note It is clear that the outward normal n to the boundary ∂G s is defined and continuous on ∂ G s and the outward normal to the boundary of the set G b coincides with −n(x) for x ∈ ∂ G b .
We introduce the sets: Denote by I| Γ ± the values (traces) of the function I on Γ ± , where I| Γ − and I| Γ + are interpreted as the values of the intensity of radiations entering into G s and coming out of G s .
Endow the system (1)-(3) with the boundary conditions: and the initial condition: Here, ∂ Q T = ∂ G × (0, T) and ∂Q b,T = ∂G b × (0, T) are the lateral surfaces of cylinders Q T = G × (0, T) and Q b,T = G b × (0, T). By tru s and tru b , we denote the values (traces) of u s and u b on ∂Q b,T . It is assumed that the body G is surrounded in a vacuum. Therefore, on the boundary of the body, the boundary condition (2) is set, which means the absence of heat flux. On the boundary ∂G b , separating the semitransparent material G s and absolutely black inclusions G b , we set two boundary conditions. They account for incoming and outgoing energy flows using a heat transfer mechanism. In addition, it is taken into account that absolutely black inclusions emit energy and absorb the incident radiation on them. Here, γ is the heat transfer coefficient and: represents the flux of radiation coming out of G s and absorbed at ∂G b . The condition (7) means that on the boundary ∂G b , the intensity of radiation entering into G s is equal to the intensity of radiation leaving the set G b . In (8), J * denotes the intensity of external radiation incident on ∂ G.
Let u be a real number or a real-valued function. We put Let S be a set where the measure dµ is given. We denote by L p (S; dµ) the Lebesgue space of functions f defined on Z that are measurable with respect to the measure dµ and have the finite norm:

Spaces of Functions on G, G s and ∂G b
We set: Let functions f , g defined on G or G s are such that f g ∈ L 1 (G) or f g ∈ L 1 (G s ). In these cases, we use the notations: Let functions f , g defined on ∂G b are such that f g ∈ L 1 (∂G b ). In this case, we use the notation: Let u be a function defined on G. We denote by u s , u b and u b,j , the restrictions of u to G s , G b and G b,j , 1 ≤ j ≤ m, respectively.
By W 1,2 (G), we understand the space: (where W 1,2 (G s ) and W 1,2 (G b,j ) are the classical Sobolev spaces) equipped with the norm: If u ∈ W 1,2 (G), then by tr u s and tru b we denote the traces of the restrictions u s and We remind the important multiplicative inequalities: which hold for all p ∈ [2, 6], q ∈ [2, 4].

Spaces of Functions on Q T and ∂Q b,T
We set: ). The inequalities (10)-(12) imply the estimates: tr u s L r 2 (0,T;L q 2 (∂G b )) ≤ C 2,r 2 ,q 2 (G, tr u b L r 2 (0,T;L q 2 (∂G b )) ≤ C 2,r 2 ,q 2 (G, which hold for all exponents r 1 , q 1 , r 2 , q 2 , such that: From the estimates (13), (15), it follows that if u, |u| 1/2 u ∈ V 2 (Q T ), then u ∈ L 5 (Q T ), tr u b ∈ L 4 (∂Q b,T ) and the following estimates hold: We also draw attention to the following multiplicative inequality, which follows from (11), (12): This inequality, in particular, implies that if u ∈ V 2 (Q T ), the sequence {u k } ∞ k=1 ⊂ V 2 (Q T ) is bounded in V 2 (Q T ) and u k → u in L 2 (Q T ) as k → ∞, then tr u k s → tr u s , tr u k b → tr u b in L 2 (∂Q b,T ).

Spaces of Functions on D s and Γ
Remind that: We set: L p (D s ) = L p (D s ; dωdx). We introduce the following measures on Γ and Γ ± : We set: . By the weak derivative in direction ω of a function f ∈ L 1 (D s ), we understand a function z ∈ L 1 (D s ), denoted by z = ω · ∇ f and satisfying the integral identity: We denote by W p (D s ) the Banach space of functions f ∈ L p (D s ) possessing the weak derivative ω · ∇ f ∈ L p (D s ) and equipped with the norm: We will denote by f | Γ − and f | Γ + the traces of the function f ∈ W p (D s ) on Γ − and Γ + , respectively. It is known that f | Γ ± ∈ L p loc (Γ ± ). Moreover, if f ∈ W p (D s ) and f | Γ − ∈ L p (Γ − ), then f | Γ + ∈ L p (Γ + ).
We refer to [67][68][69] for more detailed information about the properties of functions f ∈ W p (D s ) and their traces f | Γ ± .

Boundary Value Problem for Radiative Transfer Equation
For almost all t ∈ (0, T), the unknown function I(t), involved in the problem (1)- (9), is a solution to the following subproblem: where F = 1 π h(tr u s (t)), J b = 1 π h(tr u b (t)). We formulate some results on the properties of the problem (20)- (22) which follow from [67,69].
. By a solution to the problem (20)-(22), we mean a function I ∈ W 1 (D s ) that satisfies Equation (20) almost everywhere on D s , condition (21) almost everywhere on Γ − b , and condition (22) almost everywhere on Γ − .
then a solution to the problem (20)- (22) exists and is unique. Moreover, I ∈ W p (D s ) and for 1 ≤ p < ∞ the following estimates hold: and for p = ∞, the following estimates hold: In addition: and as a consequence, if We denote by A the resolving operator for the problem: . This operator is linear and continuous.
We denote by A the resolving operator for the problem: This operator is also linear and continuous.
We introduce the operators A Ω : These operators are linear and continuous. Their continuity follows from the estimates (23), (25) and the estimate: We introduce the characteristic functions 1 D s , 1 G s , 1 ∂G b and 1 − of sets D s , G s , ∂G b and Γ − . Note that I = 1 D s is the solution to the problem (20)- (22) where It follows from (25) that: We also introduce the operators C : It follows from (29) that: We draw attention to the following equality proven in: [65] 4κ

Main Results: Reducing the Problem (1)-(9) to Problem P 5.1. Formulation of the Main Results
In what follows, it is assumed that the following conditions on the data are satisfied.
(A 1 ) The function λ(x, u) is defined on G × R, and for any u ∈ R, it is measurable with respect to x. Furthermore: where λ min and λ max are constants. In addition, the following holder condition holds: where L is a constant.
where c p and c p are constants.
We introduce the spaces: By a weak solution to the problem (1)-(9), we mean a pair of functions (u, I) ∈ V (Q T ) × L 1 (0, T; W 1 (D s )) such that: (1) The following identity holds: where: Here and below, by (2) For almost all t ∈ (0, T), the function I(t) satisfies Equation (3) almost everywhere on D s and the conditions (7), (8) almost everywhere on Γ − b , Γ − , respectively. meaning that: where for almost all t ∈ (0, T).

Remark 1.
The fulfillment of the identity (34) is equivalent to the fact that (c p u, v) G ∈ W 1,1 (0, T) for all v ∈ V; moreover:

Remark 2.
It follows from Theorem 1 that I * ∈ L r * (0, T; W q * (D s )) and: In addition, for almost all t ∈ (0, T), the following estimates hold: In what follows, the following notations are used: The main results of this paper are the following theorems.

Theorem 2.
A weak solution to the problem (1)-(9) exists and is unique.
Consider that u and I are interpreted as the absolute temperature and the radiation intensity. Therefore, it is important to show that u and I are nonnegative under some natural assumptions on the data. It is clear that (u, I) = (0, 0) is a solution to the problem (1)-(9) with u 0 = 0, f = 0 and J * = 0. Thus, Theorem 4 implies the following result. Corollary 1. Let (u, I) be a weak solution to the problem (1)- (9). If u 0 ≥ 0, f ≥ 0 and J * ≥ 0, then u ≥ 0 and I ≥ 0.
The following three theorems show that an increase in summability exponents of f and g leads to improved properties of a weak solution.

Reducing the Problem
We put: It follows from (37), (38) It follows from (35) that: Using these formulas, we exclude the function I from the problem (1)- (9) and arrive at the problem: in which only one function u is unknown. This problem will be called Problem P.
Remind that V (Q T ) = {u ∈ V 1,0 2 (Q T ) ∩ L 4 (Q s,T ) | tru b ∈ L 4 (∂Q b,T )}. Therefore, it follows from u ∈ V (Q T ) and the boundedness of the operators C : By a weak solution to Problem P, we mean a function u ∈ V (Q T ) satisfying the identity: where:

Remark 3.
Due to the equality (31) instead of the formula (55), it is possible to use an alterna- Remark 4. It is easy to see that if (u, I) ∈ V (Q T ) × L 1 (0, T; W 1 (D s ) is a weak solution to the problem (1)-(9), then u is a weak solution to Problem P. On the other hand, if u ∈ V (Q T ) is a weak solution to Problem P, then defining I by the formula (35) for almost all t ∈ (0, T), we obtain the pair (u, I) ∈ V (Q T ) × L 1 (0, T; W 1 (D s )) that is a weak solution to the problem (1)- (9). The fact that I ∈ L 1 (0, T; W 1 (D s )) follows from the continuity of the operator A : L 1 (G s , ∂G b ) → W 1 (D s ) and properties h(u s ) ∈ L 1 (Q s,T ), h(tr u b ) ∈ L 1 (∂Q b,T ) that the function u ∈ V (Q T ) possesses.

and Some of Their Properties
We set: It follows from (28) that: The following three statements are proven in [65]. .
We denote by 1 si and 1 b the characteristic functions of sets E si and S b , respectively. We set: Lemma 3. Let: be simple functions defined on G s , ∂G b , respectively, and let: be other simple functions defined on G s , ∂G b , respectively. Then: (u, v) and Some of Their Properties We also set: where h [n] = min{max{h(u), −n}, n}. Note that: Lemma 4. For all n ≥ 1, the following inequality holds: Proof of Lemma 4. We construct sequences of simple functions {u N I } ∞ N=1 and {u N I I } ∞ N=1 of the forms (58) such that: It follows from Lemma 3 and the monotonicity of the function h that: It is clear that (h [n] Passing in the inequality (61) to the limit as N → ∞, we arrive at the inequality (60).

Lemma 5.
Assume that w ∈ C(R), w be a non-decreasing function such that w(0) = 0. Then, for all n ≥ 1, M > 0, the following inequalities hold: Proof of Lemma 5. Let {u N I } ∞ N=1 and {u N I I } ∞ N=1 be the same sequences of simple functions as in the proof of the previous lemma.
It follows from Lemma 3 and the monotonicity of the function h that: Passing in (64) to the limit as N → ∞, we arrive at the inequality (62). The proof of the inequality (63) is quite the same.

An Auxiliary Problem P [N] and Its Solvability
Consider an auxiliary Problem P [n] , which differs from Problem P only in that in its formulation, the function h(u) is replaced by h [n] = min{max{h(u), −n}, n}, where 1 ≤ n is a natural parameter.
By a weak solution to Problem P [n] , we mean a function u ∈ V 1,0 2 (Q T ) satisfying the identity: where is given by the formula (59).
We set V k = span {e 1 , . . . , e k }, k ≥ 1 and will seek an approximate solution to Prob- Note that u 0,k → u 0 in L 2 (G) as k → ∞, moreover c 1/2 p u 0,k L 2 (G) ≤ c 1/2 p u 0 L 2 (G) . The Caratheodory theorem implies the existence of a time-local solution u (k) . It is defined on the whole interval (0, T) by virtue of the global to time a priori estimate: To obtain this estimate, we substitute v = u (k) (t) in (66) and use the inequalities: which follow from the condition (32) and Lemma 4, and arrive at the inequality: Integrating it, we deduce the inequality: Applying the inequality (13) with r , q and r * , q * instead of r 1 , q 1 , the inequality (14) with r * , q * instead of r 2 , q 2 and using the inequality (53), we arrive at the estimate (67).
By virtue of the Riesz precompactness criterion for L 2 (Q T ), the estimates (67) and (68) allow us to select a subsequence, such that u (k ) → u strongly in L 2 (Q T ) and almost everywhere on Q T .
It is clear that λ(·, u (k ) )∇u (k ) → λ(·, u)∇u weakly in L 2 (Q T ) and therefore From the estimate (19) applied to u (k ) − u, it follows that tr u (k ) s → tr u s and tr u It is also easy to see that h [n] (u (k ) T], and integrating the result over t from 0 to T, we have: Passing to the limit at k = k → ∞, we establish the validity of the identity (65) for an V k is dense everywhere in W 1,2 (G), then the identity (65) holds for all v ∈ W 1,2 (G).

Estimates for Weak Solutions to Problems P and P [N]
We need the following statement, following from [41], in Lemma 4.4.

Lemma 6.
Assume that a function u ∈ V 1,0 2 (Q T ) satisfies the identity: where: Assume also that w ∈ C 1 (R), w ≥ 0, w(0) = 0 and W (M) (u) = Then: Lemma 7. Let u be a weak solution to Problem P or to Problem P [n] .
Proof of Lemma 7. A weak solution to Problem P satisfies the identity (69) with: Using Lemma 6, we arrive at the equality: Note that: Using these inequalities, the inequality (62) and the estimates: we arrive from (73) at (72). The inequality (72) for the weak solution to Problem P [n] is established in the same way. The only difference is that the inequality (63) is used instead of the inequality (62). Theorem 9. Let u be a weak solution to Problem P or to Problem P [n] . If the assumptions (41)- (43) are satisfied, then |u| γ−1 u ∈ V 2 (Q T ) for all γ ∈ [1, p/2]; moreover, the estimate (44) holds. In addition, u ∈ C([0, T]; L s (G)) for all s ∈ [1, p).

Theorem 10.
Let u be a weak solution to Problem P or to Problem P [n] . If the assumptions (49), (50) are satisfied, then u ∈ L ∞ (Q T ) and the estimate (51) holds.
Setting γ = γ k = (1 + δ) k , k ≥ 0, we obtain the inequality: It is easy to check that r, q and r * , q * satisfy (16) in the role of r 1 , q 1 and r * , q * satisfy (16) in the role of r 2 , q 2 . Using (13) and (15), we arrive at the inequality: which implies the inequality: Iterating these inequalities, we find: Thus: The limit passage as k → ∞ leads to the estimate: which implies that u ∈ L ∞ (Q T ) and: Taking into account the estimate (44) with γ = 1, we obtain the estimate (51).

Stability and Uniqueness of Weak Solutions to Problem P: Comparison Theorem
The proof given in this section uses some ideas of the method [70] proposed for proving comparison theorems for quasilinear elliptic equations. Special modifications of this method for some nonstationary radiative-conductive heat transfer problems were used in [41,53,57].
The following theorem concerns the stability of weak solutions to Problem P with respect to data. Theorem 12. Let u 1 and u 2 be two weak solutions to Problem P with (u 0,1 , f 1 , J 1 * ) and (u 0,2 , f 2 , J 2 * ) instead of (u 0 , f , J * ). Then, the estimates (39), (40) hold.

Proof of Theorem 12. We put
δ is a parameter. We introduce the sets: We introduce the function v δ = δ −1 (∆u) [0,δ] = min{δ −1 [∆u] + , 1}. It is clear that Subtracting from each other the identities (34) corresponding to the definitions of the weak solutions u 1 and u 2 leads to the identity: where ds and taking into account that: (the inequality (78) follows from (27)), we have: Using the fact that ∇v δ = δ −1 ∇(u 1 − u 2 ) almost everywhere on Q (0,δ) T and ∇v δ = 0 almost everywhere on Q − T ∪ Q δ T and taking into account assumptions (32), (33), we find that: We note also that: We set ∆h(u s ) = h(u 1 for (x, t) ∈ ∂Q b,T and introduce the sets: From the formula (56), it follows that: Noticing that: we find that: It follows from the inequality (79) and the estimates (80)-(82) that: We pass to the limit as δ → 0 in this inequality. Since: then: The first three terms on the right hand side of (83) tend to zero as δ → 0, since: Thus, (83) implies the inequality: The following inequality can be established in the same way: Adding (84) and (85), we obtain the inequality: The inequalities (84), (86) imply the estimates (39), (40).
Corollary 3 (Uniqueness theorem). If a weak solution to Problem P exists, then it is unique.
10. Solvability of Problem P Theorem 13. A weak solution to Problem P exists and is unique.
Proof of Theorem 13. Firstly, we suppose that assumptions (49), (50) hold. By Theorems 8 and 10, for all n > 0, there exists a function u ∈ V 1,0 , which is a weak solution to Problem P [n] and satisfies the estimate: where M ∞ does not depend on n. By this estimate, h [n] (u) = h(u) for n > h(M ∞ ). Therefore, a weak solution to Problem P [n] with n > h(M ∞ ) is simultaneously a weak solution to Problem P. Now, we prove the existence of a solution without additional assumptions (49), (50).
Thus, the function u (−N) is a weak solution to Problem P corresponding to the data in the role of u 0 , f and J * . Since: then by virtue of Corollary 2, the sequence {u (−N) } ∞ N=1 is non-increasing. Therefore, from the estimates (93), (94) it follows that there exists a function u ∈ V 2 (Q T ) ∩ L 5 (Q T ) such that u (−N) → u weakly in L 2 (0, T; W 1,2 (G)), weakly stars in L ∞ (0, T; L 3 (G)), strongly in L 5 (Q T ) and almost everywhere on Q T as N → ∞. In addition, tr u (−N) s → tr u s in L 2 (∂Q b,T ) and tr u (−N) b → tr u b in L 4 (∂Q b,T ). As a consequence, h(u (−N) ) → h(u) in Therefore, a(u (−N) , v) → a(u, v) weakly in L 1 (0, T) and b(u (−N) , v) → b(u, v) in L 1 (0, T) for all v ∈ V. Passage to the limit as N → ∞ in the identity (96) gives the identity (34).
We proved the existence of a weak solution to Problem P. Its uniqueness follows from Corollary 3.

Justification of the Main Results
Proof of Theorem 2. Note that (see Remark 4) the pair (u, I) ∈ V (Q T )×L 1 (0, T; W 1 (D)) is a weak solution to the problem (1)- (9) if and only if u ∈ V (Q T ) is a weak solution to problem P and I is expressed by the formula: Therefore, the existence and uniqueness of a weak solution to the problem (1)-(9) follow directly from Theorem 13.
Proof of Theorem 4. Assume that the conditions of Theorem 4 be satisfied.
By Corallary 2 we have u 1 ≤ u 2 . Thus, h(u 1 s ) ≤ h(u 2 s ), h(tr u 1 b ) ≤ h(tr u 2 b ) and Proof of Theorem 5. Assume that the conditions of Theorem 5 are satisfied.
Proof of Theorem 7. Assume that the conditions of Theorem 7 are satisfied.

Conclusions
In this paper, the author continues to construct a mathematical theory of complex heat transfer problems.
A nonstationary initial-boundary value problem governing a radiative-conductive heat transfer in a convex semitransparent body with an absolutely black inclusions was considered. To describe the process, a system consisting of two heat equations and the integro-differential radiative transfer equation was used. This system is supplied by boundary conditions, which describe the energy exchange between semitransparent body, external media and opaque inclusions.
The unique solvability of this problem was proven. In addition, the stability of solutions with respect to the data was proven, which established a comparison theorem. Besides, results on improving the properties of solutions with an increase in the summability of the data were established. All results are global in terms of time and data. The considered mathematical model of radiative-conductive heat transfer contains a number of simplifying assumptions. One should consider the process of heat transfer in a system of bodies, and not in one convex body. In a more complex model, it should be taken into account that the properties of the semitransparent medium and the radiation intensity depend on the radiation frequency. In addition, inclusions may not be completely black, but gray or even "colored". The author expects to study the more complex corresponding models in the near future.

Funding:
The results were obtained within the framework of the state assignments of the Russian Ministry of Education and Science (project FSWF-2020-0022).