A representation of solution of the identification problem of the coefficients at second order operator in the multi-dimensional parabolic equations system Текст научной статьи по специальности «Математика»

УДК 517.9

A Representation of Solution of the Identification Problem

of the Coefficients at Second Order Operator

in the Multi-Dimensional Parabolic Equations System

Galina V. Romanenko*

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny 79, Krasnoyarsk, 660041

Russia

Received 10.10.2013, received in revised form 19.11.2013, accepted 22.12.2013 An identification problem of the coefficients at differential operator of second order and sum of lowest terms in system of multidimensional parabolic equations with Cauchy data was studied in this article. The theorems of existence and uniqueness of the solution for direct and inverse problems have been proved. The method of weak approximation is used to the proof existence of solutions.

Keywords: inverse problem, identification problem, method of weak approximation, system of equations in partial derivatives, existence and uniqueness of the solution.

Introduction

In the investigation of coefficient inverse problems for partial differential equations, using some additional information on the solution, the original problem is reduced to a certain auxiliary direct problem. As a rule integrodifferential or non-classical "loaded" equation is obtained [3,5]. The following investigation method was proposed in [1]: initial inverse problem is reduced to two auxiliary direct problems, one of which contains an expression for the unknown coefficient. This approach was used to reduce inverse problem to auxiliary direct problems in [6], in which were proved theorems of existence and uniqueness of assigned problems. At international conference "Inverse and Ill-Posed Problems of Mathematical Physics" was submitted the result of two two-dimensional parabolic equation system of a similar type. Theses are publicized in [4].

The case of system of m multidimensional parabolic equations (m > 2 — any finite number) with Cauchy data was obtained in this article. To prove existence of solutions of given problems method of weak approximation [2,7] was used which firstly proposed in the works of N. N. Yanenko and A. A. Samarskiy.

1. Formulation and reduction of the problem to the direct problems

Consider in the domain r^T] = {(t,x, z) | x e Rn, z e R, 0 ^ t ^ T} the Cauchy problem for a system of parabolic equations (i = 1, m)

ut = ai(t)uizz(t,x,z) + b(t)Axui(t, x, z) + Ai(t,z)^Biz(ui) + ^ gk(t)ufcj , (1)

* [email protected] © Siberian Federal University. All rights reserved

where Bz(ui) = c\(t)ulzz(t, x, z) + c\(t)ulz(t, x, z) + (t)ui(t, x, z), with initial data

ui(0,x,z) = u0(x,z). (2)

Let the continuous functions al(t),b{t),cil(t),gk{t), (l = 1, 2, 3, i,k = l,m) be bounded on [0, T] and al(t) > a0 >0, b(t) > b0 >0, c\(t) > c0 >0. Let the functions ul0(x,z) be defined as the real-valued and be defined on Rn+1. The functions Xl(t,z) are to be determined simultaneously with the solution ul(t,x,z) of problem (1), (2).

Suppose the overdetermination conditions are given

ul(t, 0, z) = 4>l(t,z), (3)

and consistency conditions are u0(0, z) = (0, z). Assume that the following conditions are fulfilled

m

Bz (v^+E gk (t)vk k = 1

> ^ > 0, ^ are const. (4)

Theorem 1.1. If there are solutions tp(t,x) and fl(t,z) of the following Cauchy problems

= b(t)AxV>,

ip(0, x) = w0(x), (5)

f = af+m,z) -al<m,(y-f(W,0) (B,(/t) + £gkmk\

" Bi (# ) + £ gk (t)vk \ k=1 !

k= 1

f (0, z) = v0(z), (6)

the functions u1 (t, x, z) and X1 (t, z) defined by

u1 (t, x, z) = tp(t, x)f (t, z),

Xi (t,z)

vi (t, z) - a (t)Vzz (t, z) - f (t, z)yt(t, o)

m ,

Bz(Vi) + Z gk (t)vk k= 1

are the solution of the inverse problem (1)-(3), in the assumption that

u0(x,z) = wo(x)v0(z). (7)

Proof. We verify the theorem by direct substitution in the equation of (1), (2) expressions for the unknown functions.

We substitute in equations of systems (1) the expressions ul(t,x,z) = ^(t,x)fi(t, z),

X% z) = #(t,z) - ai(t)*''(t$ - f t(t>z)^(t>0) and obtain

BZ(Vi) + E gfmk k = 1

+ f-* =ai (t)f+b(t)fiAx^+

+ mz) - aW"^ - f %z)Mt>0) U f) + £ gk (t)f k) .

Bim+Y, gkk(t)^k v k=i J

k=1

The following is true by reason of operator's linearity B*

m

+ *(M) - a(,)*„(M) -/• (MfeM) B*/ + £rf(,)fM

Bi(*')+E 9?(t)*? V ?-1 '

k=1

We group relative to /® and y,

(J - b(i)Ax^ /" = ( / - ai(t)/i* - / + E 9? (t)/? ) x

?=1

\

*"(t, z) - a"(t)*i *(t, z) - /"(t, z)y,(t, 0)

m

B"(**)+£ 9?(t)*?

?=1 /

If y(t, x) is solution of problem (5) and /®(t, z) is solution of system (6), this be identical

Vi = 1, m.

If conditions (7) are valid, the functions u®(t, x, z) = y(t, x)/®(t, z) satisfy to initial data (2)

u®(0, x, z) = y(0, x)/®(0, z) = wo(x)v0(z) = Uo(x, z).

We test execution overdetermination conditions (3). Let A®(t, z) = u®(t, 0, z) — z). We can proof A®(t, z) = 0. Consider the system of equations (1) in x = (0,0,..., 0). Here and further we understand that x = 0 such as n-dimensional vector x = (0,0,..., 0).

i(t, 0, z) = ai(i)u**(t, 0, z) + b(i)AxUi(i, 0, z) + Ai(i, z) ( B"(A"(t, z)) + 9?(t)A?(t, z) ) +

fc=i

+ *(t,z) - a(t)*Z^ - f ^^ 0) B"(*") + E 9?(t)*? .

B i(*)+£ 9? (t)*? V k-1 )

?=1

u,(t, 0, z) = ai(t)(uZz(t, 0, z) - *(t, z)) + b(t)Axui(t, 0, z)+

+ z) - fi(t, z)A^(t, 0)b(t) + Ai(t, z) ^(Ai(t, z)) + ]=gk(t)Ak(t, z)J .

We derive Cauchy problem for a system of parabolic equations with homogeneous initial data

At(t,z)= ai(t)AZZ(t,z)+ A®(t, z) J Bj(Ai(t,z)) + £gk(t)Afc(t, z) ) ,

fc=i

A4(0,z) =0.

The unique solution of this problem is A®(t, z) = 0. Therefore w®(t, 0, z) = z), because A®(t, z) = u®(t, 0, z) — z). Thus overdetermination conditions are attested.

The theorem is proved. □

x

u

2. Proof the solvability of the direct problem (6)

To prove existence of the solution of direct problem (6) we consider in domain G[0,T] {(t, z) | z G R, 0 < t < T} auxiliary problem

dfi

/ \

ß*(t,z) - /i(t,z)^i(t, 0)

BZ+ E gkW

V k=1

BZ (/gk (t)/M ,

/

k=1

/ ^0,z) = vo (z)

(8)

here ^®(t, z) = ^£(t, z) — a®(t)^Zz(t, z) are known functions, and the patch function ($), Vi = 1,m is defined in R. Patch function is an arbitrary continuously differentiable function with properties:

öi

5%

â, â > 2 '

S^(â) > - > 0, â G R and (â) = ^ 2 d"T (S«i(â)) < 2, (l = 1,..., 4).

—, â < —, 3 ' ^ 3 '

The functions f ®(t, z) are determined. The functions vj (z) are defined in the real-valued and be defined on R.

To prove the existence of a solution supporting the direct problem (8) we use the weak

approximation method. We fix constant t > 0, tJ = T. The problem is broken up into three

t

fractional steps and is linearized displacement on variable t upon —.

/f =3 ai(t)/ZZ(t,z), jr < t < (j + i)t

/iT = 3 (c1(t)/ZZ(t, z) + c2(t)/ZT(t, z))Sgi(AiT(t, z)), ( j + ^t < t < (j + 3 )t,

3

1 \

2 \

37

/ ml

/tT = 3 I (c3(t) + ^(t))/iT(t,z)+ gk(t)/kT (t - 3,z) I S,i(AiT(t,z)),

k=1,k=i

j + 3/ t <t < (j + 1) t ,

/iT(0, z) = vo(z), j =0,1, 2,..., ( J - 1), Jt = T,

(9) (10)

(11)

(12)

. ^(t,z)-/iT(t-3,z) ^(t,0) here AiT (t,z) =-m^-.

BZ (^)+E gk (t)^fc k=1

Concerning the input data v0(z),^®(t, z) we suggest that they are sufficiently smooth, have all continuous derivatives occurring in the next lower relations and satisfy them for all i = 1, m

dzlv0 (z)

+

d 3l1

dt dzl1

^i(t, z)

+

dl2

dzl2

^i(t, z)

^ C,

11 =0,1,..., 4, 12 =0,1,..., 6, Vi = 1, m. (13)

It was proved that fixed constant t* : 0 < t* < T exists, which depends on constant limiting input data (13), constant ao, b0, c0 and ^ from (4) such as in domain G^f-, = {(t, z)|0 < t < t*, |z| < M}, uniformly on t estimates are hold.

E

dzl/ iT (t,z)

< C, l = 0,1,..., 4, (t, z) G G

[o,t*],

]T|/r(t,z)| < C, (t,z) G G

[0,t*]-

The equations of problem (9)-(12) are differentiated on variable z ones or twice. Uniformly on t estimates are hold

£l/£ (t,z)| + E I/tzz (t, z)| < C, (t,z) G G[0,tB.], i =1,i

i=i

i=i

with estimates

E

i=i

dZl/ iT (t'z)

< C, l = 3, 4, (t, z) G G

[o,t*],

fulfillment of conditions of Arzela's theorem about compactness is guaranteed.

By Arzela's theorem some subsequence fiTk (t,z) of sequence fiT (t,z) (Vi = 1,m) of problem solutions (9)-(12) converges with derivatives on z till second order to functions f®(t, z) G Ctz (G[0 f*i ). By the convergence theorem of the weak approximation method fi (t, z) are solution

of problem (8) and /i(t, z) G C,'z2 (G[0 t~„] ), here

CiZ2(G[o,¿i]) = \ f (t,z) /it(t,z), dZr/i(t,z) G C(G[o,r.]),l = 0,1, 2, i = 1,

dl

dzl

and

E

dl .

a? / '(t'z>

< C, l = 0,1, 2.

(14)

Let the following conditions satisfy with t G [0, t*]

ß(t,z) - vio(z)^t(t, 0)

m

BZ(^)+ E gk(t)^fc k=1

> Vi = 1, m.

(15)

To show that the solution of problem (8) equals to solution of direct problem (6), we can prove fulfilment with t G [0, t*]

ßi(t,z) - /i(t,z)^t(t, 0K *

BZ (^)+ E gk (t)^fc k=1

> —, V i = 1, m. / 2 , ,

The system of problem (8) is integrated on temporary variable in the range from 0 to t:

/i(t, z)= v0(z) + z)dn, i = 1, m,

here

^(t,z) = ai(t)/izz +

/ \

ßi(t,z) - /i(t,z)^t(t, 0)

BZ (^)+E gk (t)^k V k=1

bz (/'O + E gik (t)/k

/

k=1

m

As conditions (4) are complied, therefore equalities are true £®(t, z) - f ®(t, z Vt(t, 0) _ £®(t, z) - 0)) 0) ¡0

V i = 1,

Bz + E gk w Bz + E gk w Bz + E gk w

k= 1 k = 1 k=1

t G

The conditions (13), (15) are hold and (14) is valid, therefore equalities will be true with

0,

2A4 (¿4)

ß4(t,z) - f4(t,z)^t(t, 0)

> - , Vi = 1,

BZ(^)+ E gk(t)V>

k (iW.fc 2

m.

k=1

Here A® (J®) are some positive constants, which estimate the input data and depend on J®, constant C from (13), and are also constant limiting coefficients a®(t). By definition of patch function Sgi (0) we have

J®

SSi(A®(t, z)) = A®(t, z), with t e [0,t*], here t* = min ^t*, ' Vi = 1 m'

Thus we prove the existence solution f®(t, z) of problem (6) in class Ct1'z2(G[0it*]). The unique solution of the problem is obtained by the instrumentality of proof that the difference of two putative solutions comes to nought. The proof isn't adduce in this paper.

Theorem 2.1 (Unique existence). Let the conditions (4), (13), (14) are hold. Fixed constant t* : 0 < t* ^ T exists, which depends on constants limiting input data (13), constant a0, b0, c0 and fa from (4). Then in the class

el,z2(G[0,t.]) = {f®(t, z) |f®t(t, z), ddJf®(t,z) e C(G[0,t.]), l = 0,1, 2, i = J ,

there exists a unique solution /®(t, z) (Vi = 1,m) of problem (6), which satisfies the following relation

2 m d'

EE

i=0 k = 1

dzlf k (t,z)

< C, (16)

3. Proof existence of the solution of the inverse problem

We consider the problem (5) in domain n[0/T] = {(t,x)|0 ^ t ^ T,x G Rn}. The unique existence solution conditions are formulated in following theorem.

Theorem 3.1 (Unique existence). Let w0 G C(Rn) be bound.Then in the class

C/JCn^T]) = Mt,x)|^(t,x),D^(t,x) G C(n[o,T]), M < 2},

there exists a unique solution y(t, x) of problem (6) and the following relation is valid.

£ |D^^t(t,x)| < c.

m.

The solution of inverse problem (1)-(3) are considered in domain r[o,T] = n[o,T] U G[0,T]. In view of the fact that the functions ul(t,x,z), Xl(t,z) are expressed in terms of known functions notably

U(t, x, z) = <(t, x)f l(t, z),

Xi(t, z)

rt(t, z) - ai(t)^zz(t, z) - fi(t, z)pt(t, 0)

m

Bi m + z gk w

k = 1

where <p(t,x) G Cl£(n^Tj), fl(t,z) G Cl'J:(Got*]) are solutions of problems (5), (6), the estimate is valid

EE E

i=l l=0 |a|<2

dl

dzl x v 7

+EE

i=1 l=0

dl . d? x'(t-z)

< C.

(17)

On the account of theorems 1.1-3.1, the following theorem is true Theorem 3.2 (Existence). Let the conditions of theorems 1.1-3.1 are valid. Then in the class

Z(t*) = {ui(t,x,z),Xi(t,z) | u'(t,x,z) G cWpot.]), \%z) G C^Go^i = 1m} ,

there exists a solution ui(t,x,z), Xi(t,z) of inverse problem (1)-(3) and the relation (17) are defined.

4. Proof uniqueness of the solution of the inverse problem

Let us the conditions of theorem 3.2 are true. We use the proof by contradiction. Let u\(t,x, z), X\(t, z) h u2(t,x, z),X\(t, z), (i = l,m) are two classical solution of inverse problem (1), (2). Here the functions u\(t,x,z), X\(t,z) are the solution, which defined theorem 1.1 and satisfied the condition (3), and the functions ul2(t,x, z), Xl2(t, z) are an another solution of problem (1), (2), which satisfied the condition (17). Then the relations are valid

ult — a(t)ulzz(t, x, z) + b(t)Axui(t, x, z) + X\(t, zW Ez(ul) + £ gk(t)

\ k = 1 m

(t)u2zz(t, x, z) + b(t)Axu2(t, x, z) + X2(t, z) I Bi(u2) + £ gk(t)

u2t — a (t)u2zz\~t, x

kk

k=l

u\(0, x, z) = u0(x, z), ul2(0, x, z) = u0(x, z),

u1(t, 0,z) = ^(t,z), u2(t, 0,z) = ^(t,z).

The differences u\(t, x, z) — u\(t, x, z) = u%(t, x, z), X\(t, z) — X2(t, z) = X1 (t, z) are the solution of the problem

i(t)u\z (t,x,z)+ b(t)Axui (t,x,z)+ Xl(t,z)i Ez (ui gk (t)uk) +

k=1

+ Xi(t,z)[ Ez (u2) + Y, gk (t)ukk) , (18)

k=1

2

ui(0,x, z)=0, ui(t, 0, z) = 0, i = 1, m.

(19)

Assuming that x = 0 in system (18), we express coefficients A®(t, z) through (19) and substitution in (18).

ut = ai(t)MZz(t, x, z) + b(t)Axui(t, x, z) + AÎ (t, z) ( B;(u4) + g? (i)uM +

V ?=1

+ b(')A-"Ü'' 0'z) (b: („2) + f g? (t)u2

Bz+ E g?(t)^fc V k = 1

k = 1

u®(0, x, z) = 0, i = 1, m. We consider nonnegative never-decreasing function on segment [0, t*]

(20) (21)

gj (t) = sup

r[0,t]

^Da„i(e,x,z)

|a| < 2.

We will consider first equation of system (20) as parabolic equation relative to function u1, second one is relative to function u2 etc, m-th is relative to function um with initial data (21).

The principle of the maximum was employed to each equation, then received estimates were added

E„i(^,x,z)

< eC«C(g2(t) + go(t))e, (e,x,z) G G[0,t], 0 < t < t*,

whence we obtain estimate as nonnegative functions gk(t)

go(t) < Ct(g2(t) + go(i)) < G(g2(t) + Si(t) + So(t))t, 0 < t < t*. We differentiate system (20), (21) on variable x ones or twice

D>t = ai(t)Da„i(t, x, z)z: + 6(t)Da(Ax„i(t, x, z)) + Al(t, z)Bi +

+j (gk(t)Da„?) + b(t)Ax0,z) b: (Da„2) + j (g?wd^),

k=1

b: (^ ) + e g?(t)^? ?=1

?=1

Daui(0,x,z) = 0, a = 1, 2, i =1,m, and we receive similar estimates

gq(t) < Ct(g2(t)+ go(t)) < C(g2(t) + gi(t) + go(t))t, q = 1, 2 0 < t < t*. All of them are added

go(t, z) + gi(t, z) + g2(t, z) < C(g2(t, z) + gi(t, z) + go(t, z))t, 0 < t < t*.

Hereof equality g0(t, z) + g1(t, z) + g2(t, z) = 0 is true with t G [0, Z], where Z < ^, therefore,

C

u" (t, x, z) = 0, (t, x, z) G r[o,z], i = 1, m.

2

Arguments having replicated for t G [0,2Z], we receive

w®(t, x, z) = 0, (t, x, z) G r[o,2Z], i =1,m. In finite number of steps we obtain estimate

w®(t,x, z) = 0, (t, x, z) G r[ojt*], i = 1,

m.

In consideration of uj(t, x, z) = u|(t,x, z), (t, x, z) G T^i*], (i = 1,m), from (18), we receive that for A®(t, z) = Aj(t, z) — A|(t, z), (i = 1, m) correlations exist

A4(t,z) BZW + E skW* =0, i = 1, m.

r.kf+\J,k

\ k=1 Whence taking into account (4), it follows that

A®(t, z) = Aj(t, z) — A2(t, z) = 0, t G [0, t*], i = 1, m.

Theorem 4.1 (Uniqueness). Let the conditions of theorem 3.2 are valid. Then in the class Z(t*) there exists the unique solution u®(t, x, z), A®(t, z) of inverse problem (1)—(3) and the relation (17) is true.

5. The example of initial data, for which the theorems conditions are valid

We examine the following Cauchy problem for system of parabolic equations in the capacity of example.

Consider in domain r[0,0.5] = {(t, x, z) | x G R, z G R, 0 < t < 0.5} the set of equations

ui = ^(tKz (t,x, z) + 6(t)uXx(t,x,z)+ A4(t,z) ^ BZ (ui) + ff gk (t)ufcj , (22)

where Bz(u) = uzz(t, x, z) + uz(t, x, z) + u(t, x, z), with initial data

l(0, x, z) = u0(x, z) = (sin(z) + (i + 2))(sin(x) + 1), i =1,m. (23)

The continuous functions a®(t) = 6®(t) = cl(t) = c2(t) = c3(t) = gk(t) = 1, k = 1,m, Vi = 1, m, are bounded on [0, T]. The functions w0(x, z) are defined as the real-valued and be defined on R2. The functions A®(t, z) are to be determined simultaneously with the solution U(t,x,z) of problem (22), (23).

The overdetermination conditions are given

U(t, 0, z) = ^(t, z) = (t + 1)(sin(z) + (i + 2)), i = 1, m, and consistency conditions are valid

u0(O, z) = ^(0, z) = sin(z) + (i + 2). The fulfillment of the following conditions is required

m

ibz w + E gk(t)^fc i =

k = 1

m

ci(t)^Zz (t, z) + 4(t)V>Z (t, z) + 4(t)^(t,z) + E gk(t)^fc

k = 1

M — const.

> M > 0,

u

We can easily verify that these conditions are true

Bi (*") + E 9? (t)*?

?=1

= (t +1)

m • sin(z) + cos(z) + (i + 2) + ^(k + 2)

?=1

> M > 0,

here the choice depends on the amount of equation m and the number i. These inequations are fulfilled, for instance, with = 1. Consider the Cauchy problem

"dt = ^xx, ^(0, x) = wo(x) = sin(x) + 1.

The solution if this problem is the function <^(t, x) = e-t sin(x) + 1. This is easily seen by substituting the function y(t, x) in equation

-e( ' sin(x) = -e( ' sin(x), ^(0, x) = w0(x) = sin(x) + 1.

The function w0(x) = (sin(x) + 1) G C(Rn) is bounded. For following problem

f = /* * + (Bi (/") + E 9?(t)f ?

(i + 2) + t sin(z) + 2 sin(z)

?=1

(t + 1)(m • sin(z) + cos(z) + (i + 2) + £ (k + 2))

?=1

(24)

f(0,z) =v0 (z) = sin(z) + (i + 2), the following functions /®(t, z) = (t + 1)(sin(z) + (i + 2)) are solutions of problem (24) , with

B"(/") + E 9?(t)f ? = (t + 1) ( m • sin(z) + cos(z) + (i + 2) + ^(k + 2) ) , ?=1

?=1

A" (t,z) = ^

*"(t, z) - a"(t)*i*(t, z) - /"(t, z)^(t, 0)

B"(*")+£ 9?(t)*? ?=1

(i + 2) + t sin(z) + 2 sin(z)

(t + 1)(m • sin(z) + cos(z) + (i + 2) + £ (k + 2))

?=1

The given solution is inserted in the system of problem (24) sin(z) + (i + 2) = -(t + 1) sin(z)+

+ (t + 1) ^m • sin(z) + cos(z) + (i + 2) + + 2)^ x

(i + 2) + t sin(z) + 2sin(z)

(t + 1)(m • sin(z) + cos(z) + (i + 2) + £ (k + 2))

?=1

sin(z) + (i + 2) = -(t +1) sin(z) + (i + 2) + t sin(z) + 2sin(z), so we obtain correct identity.

The accomplishment of the following conditions is required for existence of the solution of problem (24)

d^ (z) +

d 3l1 ■

V (t,z)

dt dzli

+

dl2 • V*(t,z)

dzl2

< C,

/1 =0,1,..., 4, /2 =0,1,..., 6, Vi = 1,m.

The given condition is true on account of the limitations of all derivatives of functions (z) = sin(z) + (i + 2) and ^(t, z) = (t + 1)(sin(z) + (i + 2)). By theorem 1.1 the functions w®(t, x, z) are represented in form

ui(t, x, z) = y>(t, x)fi(t, z) = (t + 1)(e-t sin(x) + 1)(sin(z) + (i + 2)).

Let's test whether the functions u®(t, x, z) satisfy to the system of equations. The functions

= (sin(z) + (i + 2)) (sin(x)(e-i - (t + 1)e-t) + 1) = (sin(z) + (i + 2)) (sin(x)e-i(-t) + 1) = —(sin(z) + (i + 2))(t + 1) sin(x)e-t, = —(sin(x)e-t + 1)(t + 1) sin(z),

(i + 2) +1 sin(z) + 2sin(z)

A*(t, z)

(t + 1)(m • sin(z) + cos(z) + (i + 2) + £ (k + 2))

k=1

m m

BZ(u*) + 53(t)uk = (t + 1)(sin(x)e-t + 1)(m • sin(z) + cos(z) + (i + 2) + ^(k + 2))

k=1 k=1

are substituted in system (22)

((sin(z) + (i + 2)) (sin(x)e-i(—t) + 1) = — (sin(x)e-t + 1)(t + 1) sin(z) —

- (sin(z) + (i + 2))(t + 1)sin(x)e-t + (t + 1)(sin(x)e-t + 1)x

x m • sin(z) + cos(z) + (i + 2) + ^(k + 2) I x

k=1

(i + 2) +1 sin(z) + 2 sin(z)

(t + 1)(m • sin(z) + cos(z) + (i + 2) + £ (k + 2))

k = 1

The elementary transformations are reduced to

((sin(z) + (i + 2)) (sin(x)e-i(-t) + 1) =

= — (sin(x)e-t + 1)(t + 1) sin(z) - (sin(z) + (i + 2))(t + 1) sin(x)e-t+

+ (sin(x)e-t + 1)((i + 2) +1 sin(z) + 2sin(z))

After cancellation we obtained identity Vi = 1, m. The functions u0(x, z) are

u0(x, z) = wo(x)v0(z) = (sin(z) + (i + 2))(sin(x) + 1).

The conditions for existence of the solution of the problem (24) are

^(M) - vi0(z)^i(t, 0) (i + 2)+ t sin(z) + 2sin(z)

BZ(V*) + E (t)Vk (t + 1)(m • sin(z) + cos(z) + (i + 2) + £ (k + 2)) k=1 k=1

x

here the choice 5* depends on the amount of equation m and the number i. Due to the limited functions in relation above and also true to the fact that the m and i are final numbers, we can choose 5* : 0 < 5* < 1.

This example is showed, that the solution set of problem (1)-(3) is nonempty.

The research was supported by the Russian Foundation for Basic Research (No. 12-01-31033).

References

[1] Yu.E.Anikonov, On the methods of study of multidimensional inverse problems for evolutional equations, Dokl. Math., 331(1993), no. 3, 409-410.

[2] Yu. Ya. Belov, Inverse Problems for Partial Differential Equations, Utrecht, VSP, 2002.

[3] Yu.Ya.Belov, I.V.Frolenkov, Coefficient Identification Problems for Semilinear Parabolic Equations, Dokl. Math., 72(2005), no.2, 737-739.

[4] G.V.Romanenko, I.V.Frolenkov, On the representation of the solution of the inverse problem for a system of twodimensional parabolic equations, International Conference "Inverse and Ill-Posed Problems of Mathematical Physics ", dedicated to the 80th anniversary of the birthday of Academician M. M. Lavrent'ev, Siberian Mathematical Publishers, Novosibirsk, 2012, 103-104.

[5] I.V.Frolenkov, E.N.Kriger, An existence of the solution for identification problem of coefficient in special form at source function, Vestnik, Quart. J. of Novosibirsk State Univ., Series: Math., mech. and informatics, 13(2013) no. 1 , 120-134.

[6] I.V.Frolenkov, G.V.Romanenko, On the solution of the inverse problem for multidimensional parabolic equations, Sib. J. Math. XV, 50(2012), no. 2, 139-146.

[7] N.N.Yanenko, Fractional Steps for Solving Multidimensional Problems of Mathematical Physics, Nauka, Novosibirsk, 1967 (in Russian).

О представлении решения задачи идентификации коэффициентов при дифференциальном операторе второго порядка в системе многомерных параболических уравнений

Галина В. Романенко

Исследована обратная задача с данными Коши для системы многомерных параболических уравнений, содержащих неизвестные коэффициенты перед дифференциальным оператором второго порядка по выделенной переменной и суммой младших членов. Начальные данные имеют специальный вид и заданы в виде произведения двух функций, зависящих от разных переменных. Получены достаточные условия существования и единственности решения вспомогательной прямой и исходной обратной задач. Для доказательства используется метод слабой аппроксимации.

Ключевые слова: обратная задача, задача идентификации, коэффициентные обратные задачи, метод слабой аппроксимации, системы уравнений в частных производных.