SOME INVERSE PROBLEMS FOR CONVECTION ...

MSC 35R30, 35K57, 80A20 DOI: 10.14529/mmp140403

SOME INVERSE PROBLEMSFOR CONVECTION-DIFFUSION EQUATIONS

S.G. Pyatkov, Yugra State University, Khanty-Mansyisk, Russian Federation,[email protected],E.I. Safonov, Yugra State University, Khanty-Mansyisk, Russian Federation,[email protected]

We examine the well-posedness questions for some inverse problems in the

mathematical models of heat-and-mass transfer and convection-di�usion processes. The

coe�cients and right-hand side of the system are recovered under certain additional

overdetermination conditions, which are the integrals of a solution with weights over some

collection of domains. We prove an existence and uniqueness theorem, as well as stability

estimates. The results are local in time. The main functional spaces used are Sobolev spaces.

These results serve as the base for justifying of the convergence of numerical algorithms for

inverse problems with pointwise overdetermination, which arise, in particular, in the heat-

and-mass transfer problems on determining the source function or the parameters of a

medium.

Keywords: parabolic system; convection-di�usion; heat-and-mass transfer; inverse

problem; control problem; boundary value problem; well-posedness.

Introduction

We examine the question on recovering of the right-hand side and coe�cients in asecond order convection-di�usion system. Let G be a domain in Rn with boundary Γ ofclass C2 and let Q = G× (0, T ). This system is of the form

ut + A(t, x,D)u = fc, (t, x) ∈ Q, (1)

where A is a second order elliptic operator with matrix coe�cients of dimension h × h.The equation (1) is supplemented with the initial and boundary conditions

u|t=0 = u0, u|S = g(t, x), (2)

where S = (0, T )× Γ. The right-hand side and the operator A in (1) are of the form

fc =

r0∑i=1

bi(t, x)qi(t) + f, (3)

A(t, x,D)u = A =r∑

i=r0+1

qi(t)Ai(x, t) + Ar+1(x, t), Aiu =∑

|α|≤2

aiαDαu. (4)

The unknowns in (1), (2) are a solution u and the functions qi(t) (i = 1, 2, . . . , r)occurring into the right-hand side of (1) and the operator A. We consider the followingoverdetermination conditions:∫

Gi

uφi(x)dx = ψi(t), i = 1, 2, . . . , s, r = sh, (5)

36 Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫

ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ

where Gi ⊂ G are some domains. Consider also the overdetermination conditions

u(xi, t) = ψi(t), xi ∈ G, i = 1, 2 . . . , s, r = sh. (6)

The problem of this type arise when describing heat and mass transfer, convection-di�usion, and �ltration processes (see [1�5]). Inverse problems on recovering of thecoe�cients of an equation (1), depending on t with the overtermination conditions (5),where r = 1 and Gi = G, are exposed in [6�12]. The linear inverse problems on recoveringof the right-hand side are studied in [5,13] respectively. Similarly, both types of problemswith conditions (5) and (6) are examined in [4,14] and [15,16]. A large number of physicalstatements and numerical methods of solving of the above-mentioned inverse problems withcondition (6) is exposed in [27] and [28]. The problem on recovering of the source function(3) with the overdetermination conditions of the form (6) can be found in [29, Ch. 3],where the main attention is payed to numerical methods. In this monograph the problemof determination of the source function G(x, t), with given measurements (6) is examined.Here, the source function is replaced with its approximation of the form (3) which iscalculated numerically. Note that most articles are devoted to the case of some modelequations, where n = 1. We can note only articles [17,18], where problems (1), (2), (6) ingeneral statement are treated. We can also refer to monographs [2,6,14,19,20], where thereader can �nd statements of inverse problems for parabolic equations and systems andthe corresponding existence and uniqueness theorems as well as some numerical methodsfor inverse problems solving.

In the present article under natural conditions on the data of the problem, wedemonstrate that problem (1) � (5) is uniquely solvable and establish stability estimatesfor solutions. On one hand, problem (1) � (5) is of interest in its own right. On the otherhand, a solution to problem (1) � (4), (6) can be approximated by solutions to the problem(1) � (5) for a suitable choice of the weights φi = φi(x, ε) depending on a parameter ε > 0(actually, we can construct an approximation of the Dirac δ-function). The convergence canbe established under appropriate conditions on the data. The latter fact allows to constructnumerical algorithms for solving of problem (1) � (4), (6). We should note that most of theauthors for numerical solution of the inverse problems use methods based on minimizationof some functional (which is not convex) (see, for instance, [29] èëè [27]). Algorithmsrelying on an approximation of solutions to the problem (1) � (4), (6) by solutions tothe problem (1) � (5) allow to construct simpler methods and this fact is con�rmed bynumerical experiments. In the next section, we present some auxiliary statements andconditions on the data. In section 3 we state and prove our main results (Theorems 4, 5).

1. De�nitions and Auxiliary Statements

Let E be a Banach space. The symbol Lp(G;E) (G is a domain in Rn) stands forthe space of strongly measurable functions u(x) de�ned on G with values in E such thatthe norm ∥∥u(x)∥E∥Lp(G) is �nite [21]. We also use the spaces Ck(G) comprising functionshaving derivatives up to the order k inclusively in G continuous in G and admittingcontinuous extensions onto G. The notations of the Sobolev spaces W s

p (G;E), Wsp (Q;E)

and so on are conventional (see [21, 22]). If E = C or E = Cn then we use the notationW s

p (G) or Ck(G) rather than W s

p (G;E) or Ck(G;E). Thus, the inclusion u ∈ W s

p (G) (or

u ∈ Ck(G)) for a given vector-function u = (u1, u2, . . . , uk) means that every component

2014, òîì 7, � 4 37

S.G. Pyatkov, E.I. Safonov

of ui belongs toWsp (G) (or C

k(G)). In this case the norm of the vector-function is the sumof the norms of the coordinates. A similar convention is used for matrices as well. Givenan interval J = (0, T ), assign W s,r

p (Q) = W sp (J ;Lp(G)) ∩ Lp(J ;W

rp (G)) and, respectively,

W s,rp (S) = W s

p (J ;Lp(Γ)) ∩ Lp(J ;Wrp (Γ)). In what follows the symbol ∇u stands for the

vector (ux1 , ux2 , . . . , uxn), i. e. the gradient of u in the space variables. The conditionΓ ∈ Cα (α ≥ 1) is understood conventionally (see [25]).

The smoothness and consistency conditions. Fix p > n+ 2. We assume that

u0(x) ∈ W 2−2/pp (G), g(x, t) ∈ W 1−1/2p,2−1/p

p (S), g(x, 0) = u0(x)|∂G, (7)

ψi(t) ∈ C1([0, T ]), ψi(0) =

∫Gi

u0(x)φi(x)dx, i = 1, 2, . . . , s, (8)

aiα(t, x) ∈ L∞(Q) (|α| < 2), aiα ∈ C(Q) (|α| = 2). (9)

We assume below that the boundaries of the domains {Gj} (j = 1, 2, . . . , s) those in (5)belongs to the class C1. We employ the following conditions on the weights {φj(x)}:

supp φj ⊂ Gj, φj ∈ W 1q (Gj) (

1q+ 1

p= 1), j = 1, 2, . . . , s, (10)

supp φj ⊂ Gj, φj ∈ L1(G), j = 1, 2, . . . , s. (11)

Let G0 = ∪sj=1Gj, Q0 = G0 × (0, T ). Under the condition (10) we require that

bj, f ∈ C([0, T ];Lp(G0)) (j = 1, 2, . . . , r0), aiα ∈ C([0, T ],W 1

p (G0)) for |α| = 2, (12)

aiα ∈ C([0, T ], Lp(G0)) (i = r0 + 1, r0 + 2, . . . , r + 1) for |α| ≤ 1. (13)

If we replace (10) with (11) then we need the conditions

∇bk,∇f ∈ Lp(Q0), ∇u0 ∈ W2− 2

pp (G0), ∇aiα(x, t) ∈ L∞(Q0) (|α| ≤ 2), (14)

aiα ∈ C([0, T ], L∞(G0)) (|α| ≤ 1), bk, f ∈ C([0, T ];L∞(G0)) (k = 1, 2, . . . , r0), (15)

where i = r0 + 1, r0 + 2, . . . , r. In the case of the problem (1)�(5) we de�ne the matrix Bof dimension r × r whose rows with the numbers from (k − 1)h+ 1 to kh, (k = 1, 2, ..., s)are occupied by the h× r-matrices with columns∫

G

b1(x, 0)φkdx, ...,

∫G

br0(x, 0)φkdx, −∫G

Ar0+1(x, 0)u0φkdx, . . . ,−∫G

Ar(x, 0)u0φkdx.

We require thatdet B ̸= 0 ∀t ∈ [0, T ]. (16)

Assign Gδ,i = {x ∈ Gi : ρ(x, ∂Gi) > δ}, Qγδ,i = Gδ,i × (0, γ), Gδ = ∪s

i=1Gδ,i, and Qδ =Gδ× (0, T ), Qγ

δ = Gδ× (0, γ) (δ ≥ 0), Qγ = G× (0, γ). Let A =∑

|α|≤2 aαDα, where aα are

h× h-matrices. This expression can be also rewritten in the form A = −∑n

i,j=1 aij∂xixj+∑n

i=1 ai∂xi+ a0. We say that A is elliptic whenever

Ren∑

i,j=1

(aij(x, t)ξi, ξj) ≥ δ0

n∑i=1

|ξi|2, ∀ξ1, ξ2, . . . , ξn ∈ Rh,∀(x, t) ∈ Q. (17)



Here δ0 is some positive constant. The following theorems are valid.

Theorem 1. Assume that G is a bounded domain with boundary of class C2, theconditions (7) hold, the coe�cients of the operator A =

∑|α|≤2 aαD

α satisfy the condition

aα(t, x) ∈ L∞(Q) (|α| < 2), aα ∈ C(Q) (|α| = 2) (18)

and the ellipticity condition (17) holds. If f ∈ Lp(Q) then there exists a unique solutionu ∈ W 1,2

p (Q) to the problem

ut + A(t, x,Dx)u = f, u|t=0 = u0(x), u|S = g, (19)

satisfying the estimate

∥u∥W 1,2p (Q) ≤ c

[∥f∥Lp(Q) + ∥g∥

W1−1/2p,2−1/pp (S)

+ ∥u0∥W 2−2/pp (G)

], (20)

where c is a constant independent of f, u0 and a solution u. If we additionally assume that

∇u0 ∈ W2− 2

pp (G0), ∇aα(x, t) ∈ W 1

∞(Q0), |α| ≤ 2, (21)

and ∇f ∈ Lp(Q0) then a solution u possesses the property ∇u ∈ W 1,2p (Qδ) for all δ > 0

and, for a �xed δ > 0, we have the estimate

∥∇u∥W 1,2p (Qδ)

+ ∥u∥W 1,2p (Q) ≤ c

[∥f∥Lp(Q) + ∥∇f∥Lp(Q0)+

∥g∥W

1−1/2p,2−1/pp (S)

+ ∥u0∥W 2−2/pp (G)

+ ∥∇u0∥W 2−2/pp (G0)

].

(22)

Proof. The �rst claim follows from Theorem 10.4 in [25]. The second claim is justi�edconventionally with the use of the �nite di�erence method and Lemma 4.6 of Ch. 2 in [26].

2Theorem 2. Assume that G is a bounded domain with boundary of the class C2, thecoe�cients of A =

∑|α|≤2 aαD

α satisfy (18), the ellipticity condition (17) holds, and

f ∈ Lp(Qγ) (γ ∈ (0, T ]). Then there exists a unique solution u ∈ W 1,2

p (Qγ) to the problem

ut + A(t, x,Dx)u = g, u|t=0 = 0, u|S = 0, (23)

satisfying the estimate∥u∥W 1,2

p (Qγ) ≤ c∥g∥Lp(Qγ), (24)

where c is a constant independent of γ.

Proof. A solution sought agrees on [0, γ] with a solution uγ to the problem (19) with the

right-hand side gγ =

{g, t ∈ [0, γ]0, t ∈ (γ, T ]

∈ Lp(Q) and the homogeneous initial and boundary

conditions. Theorem 1 yields the estimate ∥uγ∥W 2,1p (Q) ≤ c∥gγ∥Lp(Q) ≤ c∥g∥Lp(Qγ) which

validates (24).2

Theorem 3. Assume that the conditions of Theorem 2 and condition (21) on aα hold.Then a solution u to problem (23) for a �xed δ1 > 0 satis�es the estimate

∥u∥W 1,2p (Qγ) + ∥∇u∥W 1,2

p (Qγδ1

) ≤ c(∥g∥Lp(Qγ) + ∥∇g∥Lp(Qγ0 )), (25)

where c is independent of γ.

Proof. The proof is in line with that of Theorem 2.2

2014, òîì 7, � 4 39


2. The Basic Results

De�ne the constants q0i (i = 1, 2, . . . , r) as solutions to the system

ψjt(0) +r∑

i=r0+1

q0i∫G

Aiu0φj dx+∫G

Ar+1u0φj dx =r0∑i=1

q0i∫G

bi(x, 0)φj dx+∫G

fφj dx, (26)

where j = 1, 2, . . . , s. In the following theorem we assume that

the operator A0 =r∑

i=r0+1

q0iAi + Ar+1 is elliptic. (27)

Theorem 4. Let the conditions (7) � (8), (9), (10), (12) � (13), (16), (27) hold. Then thereexists γ0 > 0 such that, for t ∈ [0, γ0], there exists a unique solution (u, q⃗) (q⃗ = (q1, .., qr))to the problem (1) � (5) such that u ∈ W 1,2

p (Qγ0), qi(t) ∈ C([0, γ0]), i = 1, 2, ..., r. Asolution depends continuously on the data of the problem, i. e., for every two solutions(ui, qi1, . . . , q

ir) (i = 1, 2) to the problem (1) � (5) from the class

ui ∈ W 1,2p (Qγ0), qij ∈ C([0, γ0]), (i = 1, 2, j = 1, 2, . . . , r),

corresponding two di�erent collections of the data f i, ψij, u

i0, g

i (j = 1, 2, . . . , s, i = 1, 2)satisfying (7) � (8), (9) � (10), (12) � (13) and such that the condition (16) holds and

∥f i∥Lp(Q) +s∑

j=1

∥ψij∥C1([0,T ]) + ∥f i∥C([0,T ];Lp(G0))+

+∥ui0∥W 2−2/pp (G)

+ ∥gi∥W

1−1/2p,2−1/pp (S)

≤ R, (i = 1, 2, )

there exists γ1 = γ1(R) such that the stability estimate

∥u1 − u2∥W 1,2p (Qγ1 ) +

r∑j=1

∥q1j − q2j∥C([0,γ1]) ≤ C(∥f 1 − f 2∥Lp(Qγ1) +r∑

j=1

∥ψ1j − ψ2

j∥C1([0,γ1])+

+∥f 1 − f 2∥C([0,T ];Lp(G0)) + ∥u10 − u20∥W 2−2/pp (G)

+ ∥g1 − g2∥W

1−1/2p,2−1/pp (Sγ1 )

),

is valid, where C is a constant independent of R and γ1 and Sγ1 = ∂G× (0, γ1).

Proof. We construct a function Φ as a solution to the problem

Φt + (r∑

i=r0+1

q0iAi + Ar+1)Φ = f +

r0∑i=1

q0i bi(x, t), Φ|t=0 = u0(x), Φ|S = g. (28)

A solution to this problem exists and possesses the properties those of Theorem 1. Makethe change V = u − Φ and q⃗ = q⃗1 + q⃗0 (q⃗i = (qi1, . . . , q

ir), i = 0, 1). In this case V is a

solution to the problem

Vt + A0V +r∑

i=r0+1

q1iAiV =r0∑i=1

biq1i −

r∑i=r0+1

q1iAiΦ, V |t=0 = 0, V |S = 0. (29)

Theorem 2 implies that for every f ∈ Lp(Qτ ) the problem

Vt + A0V = f, V |t=0 = 0, V |S = 0 (30)



is uniquely solvable and∥V ∥W 1,2

p (Qτ ) ≤ c∥g∥Lp(Qτ ), (31)

where the constant c is independent of τ ∈ [0, T ]. Examine an auxiliary problem

Vt + A0V +r∑

i=r0+1

q1iAiV = g(x, t), V |t=0 = 0, V |S = 0. (32)

Let q⃗1 ∈ Bµ,τ = {q⃗1 ∈ C([0, τ ]) : ∥q⃗1∥C([0,τ ]) ≤ µ} (τ ≤ T ). By Theorem 2, we have

V + (∂t + A0)−1

r∑i=r0+1

q1iAiV = (∂t + A0)−1g(x, t). (33)

The estimate (31) yields

∥(∂t + A0)−1r∑

i=r0+1

q1iAiV ∥W 1,2p (Qτ ) ≤ c∥

r∑i=r0+1

q1iAiV ∥Lp(Qτ ) ≤

∥q⃗1∥C([0,τ ])cc1∥V ∥W 1,2p (Qτ ) ≤ µcc1∥V ∥W 1,2

p (Qτ ).(34)

In what follows, we assume that µ ≤ µ0 =1

2cc1. In this case we infer

∥(∂t + A0)−1(r∑

i=r0+1

q1iAiV )∥W 1,2p (Qτ ) ≤

1

2∥V ∥W 1,2

p (Qτ )

and thereby (33) is uniquely solvable and a solution V ∈ W 1,2p (Q) satis�es the estimate

∥V ∥W 1,2p (Qτ ) ≤ 2∥(∂t + A0)−1g∥W 1,2

p (Qτ ) ≤ 2c∥g∥Lp(Qτ ), ∀q⃗1 ∈ Bµ,τ . (35)

Integrating equation (29) with the weight φi over G, we obtain∫G

Vtφi dx =∂

∂t

∫G

V φi dx = ψ̃it(t), ψ̃i(t) = ψi −∫G

Φφi dx,

ψ̃it +∫G

A0V φidx+r∑

j=r0+1

q1j∫G

AjV φidx =r0∑j=1

q1j∫G

bjφidx−r∑

j=r0+1

q1j∫G

AjΦφidx, (36)

where i = 1, 2, . . . , s. Examine the matrix B(t) of dimension r × r whose rows with thenumbers from (k − 1)h+ 1 to kh, (k = 1, 2, ..., s) are occupied by the h× r-matrices withthe columns ∫

G

b1φkdx, ...,

∫G

br0φkdx,−∫G

Ar0+1Φφkdx, . . . ,−∫G

ArΦφkdx.

Note that B(0) = B with the matrix B that was de�ned before (16). Demonstrate that theentries of B(t) are continuous in t. The continuity of the entries from the �rst r0 columnsresults from (9). Consider the last r − r0 columns. Since Φ ∈ W 1,2

p (Q), the embeddingtheorems yield

Φ ∈ C([0, T ];Wαp (G)), α ≤ 2(1− 1/p). (37)

2014, òîì 7, � 4 41


The latter results, for example, from Theorem 1.8.3 in [21], where the Banach spaces A0, A1

are replaced with W 2p (G) and Lp(G). In view of our conditions on p, 2(1− 1/p) > n/p+1

and, in particular, W2(1−1/p)p (G) ⊂ Cβ(G) with β ≤ 2− 2/p− n/p [21]. Thus,

W 1,2p (Q) ⊂ C([0, T ];W

2(1− 1p)

p (G)) ⊂ C([0, T ];Cβ(G)) β ≤ 2− 2

p− n

p. (38)

In particular, changing Φ on a set of a zero measure if necessary, we can assume that thefunctions Dα

xΦ for |α| ≤ 1 are continuous. We have AjΦ = Aj0Φ + Aj1Φ (j > r0, whereAj0Φ =

∑|α|=2

ajαDαΦ and Aj1Φ =

∑|α|<2

ajαDαΦ. In view of (13) and the above-pointed

properties of Φ, for |α| ≤ 1 the functions∫GajαD

αΦφi dx are continuous in t. Consider thecase of |α| = 2. We conclude that∫

G

ajα(x, t)DαΦφidxdt =

∫Γi

aα(x, t)Dα′ΦφinkdΓ−

∫Gi

(aαφi)xkDα′

Φ dx,

where DαΦ = ∂∂xk

Dα′Φ, nk is a coordinate of the unit outward normal to Γi = ∂Gi. By

the same reasons, the obtained integrals are continuous in t.In view of continuity of the entries of B(t), there exists τ0 > 0 and a constant δ̃ > 0

such that |detB(t)| > δ̃, ∀t ∈ [0, τ0]. The equalities (36) can be rewritten as

B(t)q⃗1 =

ψ̃1t + (A0V, φ1) +

r∑i=r0+1

q1i (AiV, φ1)

...

ψ̃st + (A0V, φs) +r∑

i=r0+1

q1i (AiV, φs)

, (u, v) =∫G

uv̄dG.

Thus,

q⃗1 = g0 +R(q⃗1), (39)

where g0 = B(t)−1

ψ̃1t...

ψ̃st

, R(q⃗1) = B(t)−1

(A0V, φ1) +

r∑i=r0+1

q1i (AiV, φ1)

...

(A0V, φs) +r∑

i=r0+1

q1i (AiV, φs)

and V is

a solution to the problem (29), which exists whenever q⃗1 ∈ Bµ,τ , with µ ≤ µ0 è τ ≤ T .Consider the vector

g0 = B(t)−1

ψ̃1t...

ψ̃st

.Let τ ≤ τ0. We have ∥g0∥C([0,τ ]) ≤ c max

1≤i≤s∥ψ̃it∥C([0,τ ]). In view of (26), ψ̃it(0) = 0. By

continuity of ψ̃it, ∃τ1 ≤ τ0: ∥g0∥C([0,τ ]) ≤ µ0

2, ∀τ0 ≤ τ1. Show that there exists τ2 ≤ τ1 such

that the equation (39) is uniquely solvable in the ball Bµ0,τ2 = {q⃗1 : ∥q⃗1∥C([0,τ2]) ≤ µ0}.Derive estimates for the operator R(q⃗1). Let q⃗1, q⃗2 ∈ Bµ0,τ , (τ ≤ τ1), and V 1, V 2 are



solutions to problem (29) (with q⃗1 and q⃗2 on the right-hand side) satisfying the initial andboundary conditions V j|t=0 = 0, V j|S = 0 (j = 1, 2). We have

V jt + A0V j +

r∑j=r0+1

qjiAiVj =

r0∑i=1

qji bi −r∑

i=r0+1

qjiAiΦ, Vj|t=0 = 0, V j|S = 0, (40)

where i = 1, 2, . . . ,m, j = 1, 2. Estimate ∥R(q⃗1)−R(q⃗2)∥C([0,τ ]). We obtain

∥R(q⃗1)−R(q⃗2)∥C([0,τ ]) ≤ c max1≤j≤s

∥(A0(V 1 − V 2), φj)+r∑

i=r0+1

(q1i (AiV1, φj)− q2i (AiV

2, φj))∥C([0,τ ]).(41)

Subtracting the equations (40) for j = 1, 2, we arrive at the equality

(V 1 − V 2)t + A0(V 1 − V 2) +r∑

i=r0+1

q2i (AiV1 − AiV

2) =

r0∑i=1

(q1i − q2i )bi −r∑

i=r0+1

(q1i − q2i )AiV1 −

r0∑i=1

(q1i − q2i )AiΦ.(42)

Note that in view of (35) the functions V j meet the estimate

∥V j∥W 1,2p (Qτ ) ≤ 2C∥

r0∑i=1

qji bi −r∑

i=r0+1

qjiAiΦ∥Lp(Qτ ) ≤ C1∥q⃗j∥C([0,τ ]) ≤ C1µ0, (43)

where C1 is independent of x and τ . Estimate (35) and equalities (42) yield

∥V 1 − V 2∥W 1,2p (Qτ ) ≤ C2∥q⃗1 − q⃗2∥C([0,τ ]) +max

i∥AiV

1∥Lp(Qτ )C3∥q⃗1 − q⃗2∥C([0,τ ]) ≤

∥q⃗1 − q⃗2∥C([0,τ ])(C2 + C3∥V 1∥W 1,2p (Qτ )) ≤ ∥q⃗1 − q⃗2∥C([0,τ ])(C2 + C3C1µ0).

(44)

Estimate the right-hand side of (41) using the inclusion φj ∈ W 1q (Gj). Consider |(A0v, φj)|

for v = V 1 − V 2. We have A0v = A0v + A1v, where A0v =∑

|α|=2

aαDαv and A1v =∑

|α|<2

aαDαv. For |α| ≤ 1, we infer |

∫GaαD

αvφidx| ≤ c∥Dαv∥L∞(G0) ≤ c∥v∥W 1∞(G0). Thus,

|∫G

A1vφidx| ≤ c1∥v∥W 1∞(G). (45)

Consider the expression∫GA0vφidx. It contains the summands∫

G

aα(x, t)Dαvφidxdt =

∫Γi

aα(x, t)Dα′vφinkdΓ−

∫Gi

(aαφi)xkDα′

vdx,

where Dαv = ∂∂xk

Dα′v, nk is the k-th coordinate of the unit normal to Γi = ∂Gi. The

second integral is estimated as∫Gi

|aα∥Dα′v| · |φixk

|dx+∫G0

|aαxk| · |Dα′

v| · |φi|dx ≤

2014, òîì 7, � 4 43


≤M∥Dα′v∥L∞(G0)

(∫Gi

|φixk|qdx

) 1q+M1∥Dα′

v∥L∞(G0)

(∫Gi

φqidx

) 1q.

Therefore, the second integral is estimates by c1∥v∥W 1∞(G). Estimate the �rst integral with

the use of the trace theorems (see, for instance, [21,25,26]). In view of (38) we can assumethat v ∈ C([0, T ];C1(G)), and thus aα ∈ C(Q0) and

|∫Γi

aαDα′vφinkdΓ| ≤ ∥v∥W 1

∞(G)∥aα∥L∞(G0)∥φi∥Lq(Γi) ≤ c2∥v∥W 1∞(G).

We involved the fact that φi ∈ W1−1/qq (Γi) due to embedding theorems. These two

inequalities ensure that there exists c3 such that

|∫G

A0vφidx| ≤ c3∥v∥W 1∞(G). (46)

By the embedding theorems and the interpolation inequalities (see (38), [21]), we infer

∥v∥W 1∞(G) ≤ c4∥v∥

W1+n

pp (G)

≤ c5∥v∥θW

2− 2p

p (G)

∥v∥1−θLp(G), θ(2− 2

p) = 1 + n

p. (47)

In this case relations (45), (46), (47) imply that

max1≤j≤s

∥(A0(V 1 − V 2), φj)∥C([0,τ ]) ≤ C6∥v∥θC([0,τ ];W

2− 2p

p (G))

∥v∥1−θC([0,τ ];Lp(G)). (48)

As it was noted, v ∈ C([0, τ ];W2−2/pp (G)) after a possible modi�cation on a set of zero

measure. Next, v =∫ t

0vτ (x, τ)dτ and thereby

∥v∥C([0,τ ];Lp(G)) ≤ τ1q ∥vt∥Lp([0,τ ];Lp(G)), (1/p+ 1/q = 1). (49)

Therefore, ∥(Av, φj)∥C([0,τ ]) ≤ Cτ1−θq ∥v∥W 1,2

p (Qτ ). In this case we have

∥(A0(V 1 − V 2), φj)∥C([0,τ ]) ≤ c7τ1−θq ∥V 1 − V 2∥W 1,2

p (Qτ ), (50)

where C1 is independent of τ . Using (44), we obtain

∥(A0(V 1 − V 2), φj)∥C([0,τ ]) ≤ c8τ1−θq ∥q⃗1 − q⃗2∥C([0,τ ]), (51)

Consider the second summand in (41). We have

r∑i=r0+1

(q1i (AiV1, φj)− q2i (AiV

2, φj)) =

r∑i=r0+1

(q1i − q2i )(AiV1, φj) +

r∑i=r0+1

q2i (AiV1 − AiV

2, φj).

The summands (AiV1, φj), (Ai(V

1 − V 2), φj) are estimated as the expression (A0(V 1 −V 2), φj) (see (50)). We have

∥r∑

i=r0+1

(q1i − q2i )(AiV1, φj)∥C[0,τ ] ≤ ∥q⃗1 − q⃗2∥C[0,τ ]τ

1−θq c9∥V 1∥W 1,2

p (Qτ ) ≤

τ1−θq C(µ0)∥q⃗1 − q⃗2∥C([0,τ ]).

(52)



Similarly, in view of (44), we infer

∥r∑

i=r0+1

q2i (Ai(V1 − V 2), φj)∥C[0,τ ] ≤ c10τ

1−θq ∥q⃗1 − q⃗2∥C([0,τ ]). (53)

Relations (51) � (53) imply that

∥R(q⃗1)−R(q⃗2)∥C[0,τ ] ≤ c11τ1−θq ∥q⃗1 − q⃗2∥C([0,τ ]), ∀qi ∈ Bµ0,τ .

Find τ2 ≤ τ1 such that c11τ1−θq

2 ≤ 12. In this case

∥R(q⃗1)−R(q⃗2)∥C([0,τ2]) ≤1

2∥q⃗1 − q⃗2∥C([0,τ2]). (54)

Since R(q⃗2) = 0 for q⃗2 = 0, we have

∥R(q⃗1)∥C([0,τ2]) ≤1

2∥q⃗1∥C([0,τ2]). (55)

In this case the operator R0(q⃗1) = g0 + R(q⃗1) is contractive and takes the ball Bµ0,τ2 intoitself. Indeed,

∥R0(q)∥C([0,τ2]) ≤ ∥g0∥C([0,τ2]) + ∥R(q⃗1)∥C([0,τ2]) ≤µ0

2+µ0

2= µ0.

The �xed point theorem implies the existence of a solution q⃗1 to equation (39) in this ball.By construction, V is a solution to problem (29). Assign γ0 = τ2. Demonstrate that Vsatis�es the conditions

∫G

V φjdx = ψ̃i(t). Integrate equation (29) over G with the weight

φi. We have

∂∂t

∫G

φiV dx+∫G

A0V φidx+r∑

j=r0+1

q1j∫G

AjV φidx =r0∑j=1

q1j∫G

bjφidx−r∑

j=r0+1

q1j∫G

AjΦφidx.

The function qj satis�es (36); subtracting the i-th equation from the previous equality, we

obtain that(∫G

φivdx − ψ̃i

)t= 0 or

∫Gφivdx − ψ̃i = (

∫G

φivdx − ψ̃i)|t=0 = 0 in view of

the consistency conditions. Thus, v is a solution to our problem. The stability estimatefrom the statement of the theorem was actually obtained in the proof and we omit thearguments.

2Theorem 5. Let conditions (7) � (9), (11), (14) � (16), (27) hold. Then there exists γ0 > 0such that on the segment [0, γ0] there exists a unique solution (u, q1, .., qr) to problem (1)� (5) such that

u ∈ W 1,2p (Qγ0), qi(t) ∈ C([0, γ0]), i = 1, 2, ..., r.

A solution continuously depends on the data of the problem, i. e., for a �xed δ1 > 0 andevery two solutions (ui, qi1, . . . , q

ir) (i = 1, 2) to the problem (1)�(5) of the class

ui ∈ W 1,2p (Qγ0), qij ∈ C([0, γ0]), (i = 1, 2, j = 1, 2, . . . , r),

2014, òîì 7, � 4 45


corresponding two di�erent collections of the data f i, ψij, u

i0, g

i (j = 1, 2, . . . , s, i = 1, 2)such that conditions (7) � (9), (11), (14) � (16), (27), (16) hold and

∥f i∥Lp(Q) +s∑

j=1

∥ψij∥C1([0,T ]) + ∥f i∥C([0,T ];L∞(G0)) + ∥∇f i∥Lp(Q0)

∥ui0∥W 2−2/pp (G)

+ ∥∇ui0∥W 2−2/pp (G)

+ ∥gi∥W

1−1/2p,2−1/pp (S)

≤ R, i = 1, 2,

there exists a constant γ1 = γ1(R) such that the stability estimate

∥u1 − u2∥W 1,2p (Qγ1 ) + ∥∇(u1 − u2)∥W 1,2

p (Qγ1δ1

) +r∑

j=1

∥q1j − q2j∥C([0,γ1]) ≤

C(∥f 1 − f 2∥Lp(Qγ1 ) +r∑

j=1

∥ψ1j − ψ2

j∥C1([0,γ1]) + ∥f 1 − f 2∥C([0,T ];L∞(G0))+

∥∇(f 1 − f 2)∥Lp(Q0) + ∥u10 − u20∥W 2−2/pp (G)

+

∥∇(u10 − u20)∥W 2−2/pp (G0)

+ ∥g1 − g2∥W

1−1/2p,2−1/pp (Sγ1 )

,

(56)

holds, where Sγ1 = ∂G× (0, γ1) and the constant C is independent of R and γ1.

Proof. The proof relies on the same scheme as that of the proof of the previous theorem.As before, the claim is reduced to the solvability of equation (39).

2

The authors were supported by the Russian Foundation for Basic Research (Grant 12-01-00260a).

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Received September 9, 2014

ÓÄÊ 517.95 DOI: 10.14529/mmp140403

ÍÅÊÎÒÎÐÛÅ ÎÁÐÀÒÍÛÅ ÇÀÄÀ×È ÄËß ÑÈÑÒÅÌÓÐÀÂÍÅÍÈÉ ÊÎÍÂÅÊÖÈÈ-ÄÈÔÔÓÇÈÈ

Ñ.Ã. Ïÿòêîâ, Å.È. Ñàôîíîâ

Â íàñòîÿùåé ñòàòüå ìû ðàññìîòðèì âîïðîñû êîððåêòíîñòè íåêîòîðûõ îáðàò-

íûõ çàäà÷ äëÿ ìàòåìàòè÷åñêèõ ìîäåëåé ïðîöåññîâ òåïëîìàññîïåðåíîñà è êîíâåêöèè-

äèôôóçèè. Êîýôôèöèåíòû è ïðàâàÿ ÷àñòü ñèñòåìû âîññòàíàâëèâàþòñÿ ïðè âûïîëíå-

íèè íåêîòîðûõ äîïîëíèòåëüíûõ óñëîâèé ïåðåîïðåäåëåíèÿ. Ýòè óñëîâèÿ åñòü çíà÷åíèÿ

èíòåãðàëîâ ðåøåíèÿ ñ âåñàìè ïî íåêîòîðîé ñîâîêóïíîñòüþ îáëàñòåé. Äîêàçàíû òåîðå-

ìû ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè è óñòàíîâëåíû îöåíêè óñòîé÷èâîñòè. Ïîëó÷åííûå

ðåçóëüòàòû ÿâëÿþòñÿ ëîêàëüíûìè ïî âðåìåíè. Â êà÷åñòâå îñíîâíûõ ôóíêöèîíàëüíûõ

ïðîñòðàíñòâ èñïîëüçóþòñÿ ïðîñòðàíñòâà Ñîáîëåâà. Ðåçóëüòàòû ñëóæàò îñíîâîé ïðè

îáîñíîâàíèè ñõîäèìîñòè ÷èñëåííûõ àëãîðèòìîâ ðåøåíèÿ îáðàòíûõ çàäà÷ ñ òî÷å÷íûìè

óñëîâèÿìè ïåðåîïðåäåëåíèÿ, êîòîðûå âîçíèêàþò, â ÷àñòíîñòè, â çàäà÷àõ òåïëîìàññî-

ïåðåíîñà îá îïðåäåëåíèè ôóíêöèè èñòî÷íèêîâ è ïàðàìåòðîâ ñðåäû.

Êëþ÷åâûå ñëîâà: ïàðàáîëè÷åñêèå ñèñòåìû; êîíâåêöèÿ-äèôôóçèÿ; òåïëîìàññîïåðå-

íîñ; îáðàòíàÿ çàäà÷à; çàäà÷à óïðàâëåíèÿ; êðàåâàÿ çàäà÷à; êîððåêòíîñòü.

Ëèòåðàòóðà

1. Àëåêñååâ, Ã.Â. Îïòèìèçàöèÿ â ñòàöèîíàðíûõ çàäà÷àõ òåïëîìàññîïåðåíîñà è ìàã-íèòíîé ãèäðîäèíàìèêè / Ã.Â. Àëåêñååâ. � Ì.: Íàó÷. ìèð, 2010.

2. Belov, Ya.Ya. Inverse Problems for Parabolic Equations / Ya.Ya. Belov. � Utrecht:VSP, 2002.



3. Babeshko, O.M. On taking into account the types of sources and settling zonesof pollutants / O.M. Babeshko, O.V. Evdokimova, S.M. Evdokimov // DokladyMathematics. � 2000. � V. 61, � 2. � P. 283�285.

4. Êàëèíèíà, Å.À. ×èñëåííîå èññëåäîâàíèå îáðàòíîé çàäà÷è âîññòàíîâëåíèÿ ïëîò-íîñòè èñòî÷íèêà äâóìåðíîãî íåñòàöèîíàðíîãî óðàâíåíèÿ êîíâåêöèè-äèôôóçèè /Å.À. Êàëèíèíà // Äàëüíåâîñòî÷íûé ìàòåìàòè÷åñêèé æóðíàë. � 2004. � Ò. 5, � 1.� Ñ. 89�99.

5. Îáðàòíàÿ çàäà÷à âîññòàíîâëåíèÿ ïëîòíîñòè èñòî÷íèêà äëÿ óðàâíåíèÿ êîíâåêöèè-äèôôóçèè / Þ.À. Êðèêñèí, Ñ.Í. Ïëþùåâ, Å.À. Ñàìàðñêàÿ, Â.Ô. Òèøêèí // Ìà-òåìàòè÷åñêîå ìîäåëèðîâàíèå. � 1995. � Ò. 7, � 11. � Ñ. 95�108.

6. Iskenderov, A.D. Inverse Problem for a Linear System of Parabolic Equations /A.D. Iskenderov, A.Ya. Akhundov // Doklady Mathematics. � 2009. � Ò. 79, � 1.� P. 73�75.

7. Ismailov, M.I. Inverse Problem of Finding the Time-Dependent Coe�cient of HeatEquation from Integral Overdetermination Condition Data / M.I. Ismailov, F. Kanca// Inverse Problems In Science and Engineering. � 2012. � V. 20, � 24. � P. 463�476.

8. Ivanchov, M.I. Inverse Problem of Simulataneous Determination of Two Coe�cientsin a Parabolic Equation / M.I. Ivanchov // Ukrainian Mathematical Journal. � 2000.� V. 52, � 3. � P. 379�387.

9. Jing, Li. An Inverse Coe�cient Problem with Nonlinear Parabolic Equation / JingLi, Youjun Xu // Journal of Applied Mathematics and Computing. � 2010. � V. 34,� 1-2. � P. 195�206.

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11. Kerimov, N.B. An Inverse Coe�cient Problem for the Heat Equation in the Caseof Nonlocal Boundary Conditions / N.B. Kerimov, M.I. Ismailov // Journal ofMathematical Analysis and Applications. � 2012. �V. 396, issue 2. � P. 546�554.

12. Êîæàíîâ, À.È. Ïàðàáîëè÷åñêèå óðàâíåíèÿ ñ íåèçâåñòíûì êîýôôèöèåíòîì, çàâè-ñÿùèì îò âðåìåíè / À.È. Êîæàíîâ // Æóðíàë âû÷èñëèòåëüíîé ìàòåìàòèêè èìàòåìàòè÷åñêîé ôèçèêè. � 2005. � Ò. 45, � 12. � C. 2168�2184.

13. Vasin, I.A. On the Asymptotic Behavior of Solutions to Inverse Problems for ParabolicEquations / I.A. Vasin, V.L. Kamynin // Siberian Mathematical Journal. � 1997. �V. 38, � 4. � P. 647�662.

14. Prilepko, A.I. Methods for solving inverse problems in Mathematical Physics /A.I. Prilepko, D.G. Orlovsky, I.A. Vasin. � New-York: Marcel Dekker, Inc. 1999.

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2014, òîì 7, � 4 49


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Ðàáîòà ïðîâîäèëàñü ïðè ôèíàíñîâîé ïîääåðæêå Ðîññèéñêîãî ôîíäà ôóíäàìåí-òàëüíûõ èññëåäîâàíèé. Ãðàíò �12-01-00260à.

Ñåðãåé Ãðèãîðüåâè÷ Ïÿòêîâ, äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,êàôåäðà ≪Âûñøàÿ ìàòåìàòèêà≫, Þãîðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò (ã. Õàíòû-Ìàíñèéñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected].

Åãîð Èâàíîâè÷ Ñàôîíîâ, àñïèðàíò, êàôåäðà ≪Âûñøàÿ ìàòåìàòèêà≫, Þãîð-ñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò (ã. Õàíòû-Ìàíñèéñê, Ðîññèéñêàÿ Ôåäåðàöèÿ),[email protected].

Ïîñòóïèëà â ðåäàêöèþ 9 ñåíòÿáðÿ 2014 ã.


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