On the Approximation of a parabolic inverse problem by pseudoparabolic one Текст научной статьи по специальности «Математика»

УДК 517.9

On the Approximation of a Parabolic Inverse Problem by Pseudoparabolic One

Anna Sh. Lyubanova*

Institute of Space and Information Technologies, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 29.10.2011, received in revised form 23.12.2011, accepted 10.02.2012 The properties of the solution to the inverse problem on the identification of the leading coefficient of the multi-dimensional pseudoparabolic equation are studied. It is proved that the inverse problem for the pseudoparabolic equation approximates the appropriate inverse problem for the parabolic equation of filtration. The existence and uniqueness of the solution to the parabolic inverse problem is established.

Keywords: filtration, inverse problems for PDE, pseudoparabolic equation, parabolic equation, existence and uniqueness theorems.

Introduction

An inverse problem for the pseudoparabolic equation

(u + L1u)t + L2u = f (0.1)

with the differential operators Li and L2 of the second order in spacial variables is discussed in this paper. We are interested in finding the leading coefficients of L2 in (0.1) from the additional boundary data. Applications of this problem deal with the recovery of unknown parameters indicating physical properties of a natural stratum which should be determined on the basis of the investigation of its behaviour under the natural non-steady-state conditions (see [1] for details). This leads to the interest in studying the inverse problems for (0.1) and its analogue.

The investigation of inverse problems for pseudoparabolic equations goes back into 1980s. The first result obtained by Rundell in [2] is concerned with the inverse problems of the identification of an unknown source f in the (0.1) with linear elliptic operators Li and L2, Li = L2. Rundell proved the global existence and uniqueness theorems in the case that f depends only on x or t. Another kind of inverse problems is considered in [3,4]. These works are devoted to problems of reconstructing the kernels in integral term of (0.1) with the integro-differential operator L2. As for the determination of unknown coefficients in (0.1) we mention the results of Mamayusupov [5], Lubanova and Tani [6]. Mamayusupov proved the uniqueness theorem and found an algorithm for solving the inverse problem with respect to u(t,x), functions b(y), c(y) and a constant a for the equation

ut — Aut = aAu + b(y)uy + c(y) + S(t, x, y), for (x, y) £ R2, t > 0

provided that u(t, x, 0), uy(t, x, 0) and u(0,x,y) are given. Here S(t,x,y) is the Dirac delta function.

In [6] an inverse problem of identification of an unknown leading coefficient in the operator L2 for (0.1) was discussed (see Problem 1 below). The existence, uniqueness and regularity of

* lubanova@mail.ru © Siberian Federal University. All rights reserved

the solution to the inverse problem were established there. The statement of the inverse problem was motivated in [1].

A main goal of this paper is to investigate the behavior of the solution to the inverse problem considered in [6] as n ^ 0. It is well known [7] that when passing to the limit n ^ 0 equation (0.1) formally tends to coincide with the standard linear equation of filtration in a porous medium

ut + ¿2 u = f, (0.2)

The direct initial boundary value problem for pseudoparabolic equation in a bounded domain Q C Rn approximates the appropriate problem for parabolic equation [8]. In particular, under certain assumptions the solution un of equation (0.1) with the initial data un(0, x) = u0(x) tends to the solution u of (0.2) with the same initial condition in the L2-norm for all t > 0 as n ^ 0. It was established in [1] that the inverse problem for the pseudoparabolic equation also approximates weakly the appropriate inverse problem for the parabolic one in the case when L2 = nd2/dx2, L2 = k(t)d2/dx2, n is a positive real number and k(t) is an unknown coefficient. In the present paper this result will be extended to the inverse problems for (0.1) and (0.2) with any number of space variables. Such an investigation is also of an interest in studying the inverse problems for evolution equations whose principal terms contain unknown coefficients. The considerable results in this sphere are obtained for parabolic equations (see [9-12] and references given there).

The paper is organized as follows. In Section 1 for the convenience of the reader we repeat the formulation of the inverse problem for (0.1) and the relevant material from [6] without proofs and comments, thus making our exposition self-contained. In Section 2 we discuss the behaivior of the solution to the inverse problem as n ^ 0 and prove the existence and uniqueness theorem for the relevant parabolic inverse problem. Section 3 contains the conclusions and comments to the main results of the paper.

1. Preliminaries

Let Q be a domain in Rn with a boundary dQ e C2, T an arbitrary real number and QT = Q x (0,T). Throughout this paper we use the notation: || • || and (•, •) are the norm and the inner product of L2(Q), respectively;

o

|| • ||j and (-, •) are the norm of Wj(Q) and the duality relation between Wj (Q) and W2-j(Q), respectively (j = 1,2); as usual W20(Q) = L2(Q).

Let M : W2(Q) ^ (W^Q))* be a linear differential operator of the form

Mv = -div(M(x)Vv) + m(x)v, (1.1)

where M(x) = (mj(x)) is a matrix of functions mj(x), i, j = 1, 2,... ,n. We assume that the following conditions are fulfilled.

I. mj(x), dmj/dxi , i, j, l = 1, 2,..., n, and m(x) are bounded in Q. M is an operator of

o

elliptic type, that is, there exist positive constants mi and m2 such that for any v GWj (Q)

mi||v||2 < (Mv,v)1 < m2|M|i. (1.2)

II. There exists a positive constant m3 such that for any v e W|(Q)

||Mv|| < m3||v||2. (1.3)

III. mjj(x) = mjj(x) for i, j = 1, 2,..., n and m(x) ^ 0 for x e Q. We proceed to study the following inverse problem [6].

Problem 1. For a given constant n and functions f(t,x), g(t, x), p(t, x), Uo(x), w(t, x), ^i(t), b2(t) find the pair of functions (u(t, x), k(t)) satisfying the equation

ut + nMut + k(t) Mu + g(t,x)u = f (t,x), (t,x) £ QT, (1.4)

and the conditions

(u + nMu)|t=0 = Uo(x), x £ Q, (1.5)

u|dQ = p(t,x), t £ [0,T], (1.6)

nddut + k(t)|u}^(t,x) dS + Bi(t)k(t) = ^(t), t £ [0,T]. (1.7)

d

Here — = (n, M(x)V) and n is the unit outward normal to dQ. dv

We use functions a(t,x), hn(t,x) and b(t,x) as the solutions of the Dirichlet problems

Ma = 0 in Q, a|dQ = P(t,x); (1.8) Mb = 0 in Q, b|dQ = w(t,x),

hn + nMhn =0 in Q, hn|dQ = w(t,x), (1.9)

(Mvi,v^i M = (M(x)Vvi, Vv2) + (m(x)vi,v2), v,,v2 £ W2i(Q);

^(t) = (Ma, b) iM, F(t, x) = at — f (t, x) + g(t,x)a, (1.10)

(t) = ^(t) — 2 (Mat, hn>i,M + (f (t,x) — at, hn),

max (Ma, hn>, M, b, = max wi(t), = max (t).

te[o , t] 1' M te[o , T] te[o , t]

By a solution {u, k} of Problem 1 we mean that

(1) k(t) is continuous for 0 < t < T;

(2) u £ C1 ([0, T]; W2(Q));

(3) the equation (1.4) and the conditions (1.5)—(1.7) are satisfied.

The existence and uniqueness of the solution to Problem 1 is established by the following theorem [6].

Theorem 1.1. Let the assumptions I-III be fulfilled and n be a positive constant. Assume that

(i) f £ C([0, T]; L2(Q)), p £ C1 ([0, T]; W23/2(dQ)), Uo £ L2(Q), g £ C(Qt),

w £ C 1([0, T]; W^2(dQ)), £ C 1([0,T]), B2 £ C([0,T]);

(ii) f, Uo, p, w, bi are nonnegative and

/ hn dx > ho = constant > 0, t £ [0, T]; (1.11)

JQ

(iii) there exist constants ai; i = 0,1, 2, such that 0 ^ ao, a, ^ 1, ao + a, < 2,

(1 — ao) (t) + (1 — a,) ^(t) > a2 = constant > 0, t £ [0, T],

x(0) + a(0, x) — Uo(x) ^ 0 for almost all x £ Q, g(t, x)x(t) + x'(t) + F(t, x) ^ 0 for almost all (t, x) £ Qt,

where x(t) = n (ao^i(t) + «^(t)) [fQ hn dx] 1 ; (iv) for any t £ [0, T]

(t) > = constant > 0 holds and g(t, x) satisfies the inequality

$

Qt n

max g(t, x) < — + ^ + 2n 1 max(a, hn)

[0,T ]

-1

ko n '

Then Problem 1 has a unique solution (u, k) £ C 1([0,T]; W22 (Q)) x C([0, T]). Moreover, u and k satisfies the estimates

0 ^ u(t, x) ^ x(t) + a(t, x) for almost all (t, x) £ Qt, (1-12)

l|u(t)||2 + |k(t)||2 + n (||u(t)||2 + ||ut(t)H2) < C, t £ [0,T], (1.13)

k0 < k(t) < k1 (1.14)

with positive constants C and k1 = a-1 maxte[0 {$n(t) + (|g|(a + x(t)), hn)} .

2. Approximation of Parabolic Inverse Problem

As mentioned above, when passing to the limit n ^ 0 equation (0.2) formally tends to coincise with the linear parabolic equation and Problem 1 transforms to the following parabolic inverse problem.

Problem 2. Given f(t,x), g(t,x), ^(t, x), u0(x), ^1(t), (t); find the pair of functions (u(t, x), k(t)) satisfying the equation

ut + k(t) Mu + g(t,x)u = f (t,x), (t,x) £ QT, (2.1)

and the conditions

u|t_0 = u0(x), x £ Q, (2.2)

u|dn = £(t,x), t £ [0,T], (2.3)

r f)

k(t) / ds + ¿1(t) k(t) = &(t), t £ (0,T). (2.4) Jan dv

Hereafter, by the solution of Problem 2 we mean a pair (u(t, x), k(t)) such that

a) u £ V = {v| v £ LTO(0,T; W22(Q)), vt £ LTO(0,T; L2(Q))}, k(t) £ LTO(0,T);

b) system (2.1)-(2.4) is satisfied.

We shall denote the solutions of Problem 1 with the initial data

(un + nMun) = u0 + nMu0 = U (2.5)

t_0

and Problem 2 by (un,kn) and (u, k), respectively. In this section we make use of the inequality

dv dv

< C^ ||v|a ||v||1-a + ||v|0 (2.6)

Lq(dn)

n n_1

valid for any v G W| (O) where a = ^--, q G

2(n-1) 2(n _ 1)

n ' n _ 2

for n ^ 3 and q G [1, to]

for n = 2. (2.6) is easily derived from the multiplicative inequality [13]. The constant C2 depends on n, q, mesQ, m2 and m3. We also use the property of the function hn established by the following lemma.

Lemma 2.1. Let w £ C([0, T]; W3/2(Q)). Then the solution of the problem (1.9) satisfies the estimate

||hn||2 + nl|hn||? < nC3 (2.7)

where a positive constant C3 depends on m?, m2, mesQ, ||b|| and does not depend on n.

Proof. To obtain the estimate (2.7) we multiply the equation (1.9) by hn in terms of L2(Q) and integrate by parts in the left-hand side. This gives

r dhn

||hn||2 + 1 = v — w ds.

Jan dv

By Holder's inequality for n > 2

dv

w ds

ten

< dv

LP(dn)

lw|Lp/(p-i)(ôn)

(2.8)

(2.9)

where p = 2(n — 1)/n. From (2.6) and the embedding theorem [13] it follows that for any v £ W22(Q)

dv

^ C4 ||v||i, |v|iP/(P-1)(dQ) < C51| v 12. (2.10)

dV LP(dQ) K '

Here constants C4 and C5 depend on m2, m3, n and mesQ. Applying (2.10) to (2.9) yields

f dhn

'an

dv

w ds

< C4C5||hn|i|6|2 = |i.

Estimating the right-hand side of (2.8) with the help of this inequality, one can obtain the estimate (2.7). The lemma is proved. □

The main result of this section is formulated in the next theorem.

Theorem 2.2. Let n G (0,no], n ^ 2, the condition (ii) of Theorem 1.1 and the assumptions I-III are fulfilled. Let

(i'') f G L2(0, T; Wl(fi)) n C(QT), p G C 1([0, T]; W^2(ôfi)), uo G Wf(n), g G C(Qr), w G C 1([0, T]; W^2^)), G C1 ([0,T]), ^2 G C([0,T]);

(iii'') u0 and p obey the compatibility condition wo(x)|ôn = P(0, x),

a(0, x) _ uo(x) _ noMuo ^ 0, x G O, F (t, x) > 0, (t, x) G , ^(t) + ^(t) > a2 = const > 0, t G [0,T].

(iv'') there exist positive constants , such that

< &(t) < ^2, t G [0,T],

(2.11) (2.12)

(2.13)

Then

un ^ u *- weakly in LTO(0,T; W22(Q)),

u( ^ ut *- weakly in LTO(0,T; L2(Q)) and weakly in L2(0,T; W^Q)),

kn ^ k *- weakly in LTO(0,T)

as n ^ 0. Moreover,

0 < r(n) < kn(t) < a-1 + tni0£lXT](ga'hn)) = k2 (2.14)

where r(n) is a continuous function of n on [0, n0] and r(0) > 0.

Proof. Without loss of generality we can assume no to be chosen so that no ^ 1,

0 < = ^2 - max yMat||iM&M + |M|||hn°||| < (t) < (2.15)

- t£[0,T] I. 2 J

-1

maxg(t,x) < + +2C3n01/2 max ||a

Q

T

t£[0,T ]

because of (2.7). Therefore the hypotheses of the theorem imply that all assumptions of Theorem 1.1 are fulfilled with a0 = a0 = 0. This shows that Problem 1 has a unique solution (un(t,x), kn(t)) G C1 ([0, T]; W22(Q)) x C([0,T]) and the estimates (1.12)-(1.14) hold for any n, 0 < n ^ no- Our next step is to get a uniform lower bound (2.14) for kn and then uniform estimates for the derivatives of un. Let us set

wn(t,x)= a(t,x) - un(t,x). (2.16)

The function w(t,x) satisfies the equation

w} + nMwn + kn(t) Mwn + g(t,x)wn = F(t,x), (t,x) G QT, (2.17)

and the conditions

(wn + nMwn)|t=0 = a(0,x) - U0(x), x G Q, (2.18)

wn|dfi = 0, t G [0,T], (2.19)

r ( dwn dwn 1

j\n-Q^ + k^^^dS = (<1 + *)kn + n(Ma„hn) 1M - <2, t G [0,T]. (2.20)

As was shown in [6], multiplying (2.17) by hn(t, x) in terms of L2(Q), the integration by parts in the left side and substituting (2.20) into the resulting equation leads to the equation

kn(t)(<?1(t) + *(t) + ^(wn,hn)) = $n(t) - (g(t, x)(a - wn),hn) (2.21)

by virtue of (1.8), (1.9), (1.10), (1.11).

According to Theorem 1.1 the pair (wn,kn) G C 1([0,T]; W22(Q)) for every 0 < T < Since the problems (2.17)-(2.20) and (2.17)-(2.19),(2.21) are equivalent, the pair (w, k) also solutions the problem (2.17)-(2.19),(2.21). Let us set

kn = min kn(t). (2.22)

0 t£[0,T]

We multiply (2.17) by Mwn in terms of the inner product of L2(Q) and integrate by parts in the following way:

1 d

1 d |M| 2,m + ||Mwn ||2 + k00 ||Mw0 ||2

2 dt" M1'M 2 dt

—gw0,Mw0)i,m — ^ F — ds + <F,Mw0)^. (2.23)

By (1.2),(1.3),(2.6) and the Young inequality,

f F^ ds

/ao

< Cr((^^ + 1) ||F||2 + ||w0||2,m) + ||Mw0||2 (2.24)

where C7 = const > 0 depends on C2, mesQ, m,, i = 1,2,3. Then (1.1),(2.23), (2.24) give

||w||2,m + n||Mw0||2 + kn £ ||Mw0||2dr < + C9 ^ |M|?,mdr. (2.25)

The positive constants C8 and C9 depends on C7, ||g||Ci(QT), m;, i = 1, 2, 3. In accordance with Gronwall's lemma, it follows from (2.25) that

|?,m + n||Mwn||2 + k0n J ||Mw0||2dr < . (2.26)

Here Cio =const > 0 depends on C8, C9 and does not depend on n and k0.

Let us come back to the equation (2.21). We first note that the numerator of (2.21) is bounded below by a positive constant independent of k0 when no is small enough. Indeed, by (2.7) and (2.26),

| (gw0,h0)| < Cnn1/2 (2.27)

The constant C?? > 0 depends on C2, Cio and does not depend on n and k0. Thus, (2.13), (2.15) and (2.27) give

^2 — n(Mat,h0)1M — (F, h0) + (gw0, h0) > ^ — C?2n?/2. (2.28)

Here the positive constant C12 depends on C??, ||g|C(QT), maxte[o,T] ||a|| and does not depend on n and k0. If we choose no < (^2C121)2, then — C12n1/2 > 0. Furthermore, by (2.10),(2.26),

1 (w0 ,h0) ^| L ^ w ds|+|(Mw0 ,h0 )| < Mpw (2.29)

where C13 = const > 0 depends on C4, C5, Cio, no, m,, i = 1, 2,3, and does not depend on n and k0. Thus, by (2.12), (2.21), (2.22), (2.28), (2.29), we have

k0 > C14 (k0)1/4[a2(k0)1/4 + Cis]"1,

whence

a2k0 + Cis(k0)s/4 — C14 > 0. (2.30)

Here C14 = — C12n1/2, Since there exists a unique positive real root yo of the equation

G(y) = a2y4 + Cisy3 — C14 = 0,

w

(2.30) implies (kg )3/4 > y0 > 0. From the obvious inequality

f («2 + C13)y3 - C14, 0 < y < 1,

G(y) < 4

I («2 + C13)y4 - C14, 1 < y <

we conclude that y0 > y* if

y = C14 otherwise, y0 > (y*)3/4. Thus we get

* = ^(«2 + C13)-1/3 < 1;

kn

> min{y*, (y*)4/3} = r(n) > 0. (2.31)

It is clear that r(n) is continuous function of n on [0,n0] and r(0) > 0.

Now the uniform estimates of un and Mun become evident. By (2.26) and (2.31),

||un+ n||Mun||2 + r(n) /i|Mun||2dT < C15(r(n))-1/2 + C16. (2.32)

0

The constants C15, C16 depends on C10, |M|C([o,T];Wi(Q)) and does not depend on n. The uniform estimate of ^ can be derived from (2.17)-(2.19), (2.21), (2.31) and (2.32). Multiplying (2.17) by Mwg in terms of the inner product of L2(Q) and integrating by parts in the resulting equation we obtain

1 |'wn||?,m + ^ |Mwg||2 + 1 -dt||Mwn||2

kn (t) 11 t'l1'M kn (t) 11 tM 2 dt1

1 C dwn 1

ds -7^(F + gwn,Mwn\ . (2.33)

kn (t^an dv kn (t^ ' '

By the smoothness of f, the embedding theorem and (2.10)?

/ F^ ds

/an dv

< M||fHc(n) + ||w2(n)) |wn|1 (2.34)

The constant C17 depends on C4, n and mesQ. Therefore, taking into account (2.31), (2.32), (2.34) we can readily derive the estimate

J ||w?|2,m dr + n J ||Mw?||2 dr + ||Mwn||2 < (rC8^ (2.35)

from (2.33). Here the constant C18 depends on k1, c, ||Mu0||, ||F||c(qt) and does not depend on n. (2.17), (2.19) and (2.35) lead to the estimate

||wt||2 + nMwtM0,м < + C20 (2.36)

where the constants C19 and C20 depend on C18, k2, ||F||c([0,T];L2(Q)), ||g||c(QT) and does not depend on n. Thus, from (2.14), (2.32), (2.35), (2.36) it follows that there exists a subsequence (uni, kn 1) of (un, kn) and a pair of functions (u, k) such that

uni ^ u *- weakly in LTO(0,T; W22(Q)), (2.37)

uni ^ ut *- weakly in LTO(0,T; L2(Q)) and weakly in L2(0,T; W2°(Q)), (2.38)

kn 1 ^ k *- weakly in LTO(0,T) (2.39)

as n ^ 0. By the compactness theorem [14], (2.37)-(2.39) implies

uni ^ u in L4(0,T; W^Q)), (2.40)

kni ^ k weakly in L4(0,T) as n ^ 0. (2.41)

We are now in a position to show that the pair (u, k) is a solution of Problem 2. In fact, the pair (uni, kni) satisfies the identity

f {(u^ + guni,v) + n^Mu^,v)1 + kni(Muni,v)1 }dt = f (f,v)dt (2.42) ./o Jo

o

for every v G L2(0,T; W^ (Q)). In view of (2.37)-(2.41) we can pass to the limit in (2.42). Since

n; / (Mu?1 ,v)1 dt ^ 0 o1

as n; ^ 0 (because of (2.36)), we have

J |(ut,v) + k(t) (Mu,v)1 + (gu,v)j dt = J (f, v) dt (2.43)

o

for every v G L2(0,T; W^ (Q)). Moreover, by (1.12), (2.14), (2.16), (2.31), (2.32), (2.35) and (2.36), the estimates

r(0) < k(t) < ^2a—1, (2.44)

0 < u(t,x) < a(t,x), (2.45)

£ |k||2 dT + ||MuyL^(o,T;L2(n)) < m1(rC(108))3/2 + £ IM? dт, (2.46)

llui||L~(0,T;L2(Q)) < C(r(0))1/2 + C20 + |ai|L~(0,T;L2(fi))) (2.47)

are valid. From (2.43)-(2.47) it follows that the pair (u, k) satisfies equation (2.1) for almost all (t,x) G Qt. Furthermore, by (1.6), (2.5), (2.37), (2.38) u(t,x) obeys (2.2), (2.3).

It remains to prove that the condition (2.4) is also fulfilled. Let v(t, x) = v(t,x)h(t) where v(t, x) and h(t) are arbitrary functions of classes LTO(0, T; W21(Q)) and L2(0, T), respectively, = Then the identity

jj {K1 + guni, v) + n^Mu?1 ,v)1M + kni (t)((Muni ,v)1M + ^(t))} h dt

= fT ((f,v) + ^2)h dt (2.48)

Jo

holds because of (1.7). A passage to the limit in (2.48) similar to the above yields

J {(ut + gu,«)0 + k(i)((Mu,v)1iM + ^1)}hdt = J ((f,v) + ^2)hdt. (2.49)

By virtue of (2.1), integrating by parts in the second term of the left-hand side of (2.49) gives

Jo |k(t) J ds + ¿1(t)k(t) - ^(t)J h(t) dt = 0

for any h(t) G L2(0,T), which implies that the pair (u, k) satisfies (2.4) for almost all t G (0,T). The theorem is proved. □

Under the hypotheses of Theorem 2.2 the solution to Problem 2 is unique in the class V x

LTO(0, T).

Theorem 2.3. Let the conditions of Theorem 2.2 be fulfilled. Then Problem 2 has a unique solution (u(t, x), k(t)). The pair (u(t, x), k(t)) satisfies the estimates (2.44)-(2.47) and ut G L2(0,T; W(Q)).

Proof. The existence of the solution to Problem 2 and the estimates (2.44)-2.47) were proved in Theorem 2.2. It remains to establish the uniqueness.

Let (u1(t,x), k1(t)) and (u2(t, x), k2(t)) be two solutions of Problem 2. Then the pair (w(t,x), p(t)) = (u1 - u2, k1(t) - k2(t)) solutions the problem

wt - k1(t)Mw = -p(t) Mu2, (t, x) G QT, (2.50)

w|t=0 = w|dQ = 0, (2.51)

Mt) / ds = - ^ p(t), t G [0, T]. (2.52)

■/an dv k2 (t)

Multiplying (2.50) by Mw in terms of the inner product of L2 (Q) and integrating by parts, we can easily obtain

2 -tMwM0,м + k1(t)||Mw||2 < |p(t)| |Mu2| ||Mw||. (2.53)

From (2.6) with q = 2, (2.14), (2.52) and the Young inequality it follows that |p(t)| ||Mu2| ||Mw|| < C21 ||Mu2| ||w||1/M ||Mw||3/2

< ||Mw|2 + 2C2°) ||Mu2||4 ||wM2,м (2.54)

where C21 = const > 0 depends on C2, ^2, r(0), a3, m,, i = 1, 2, 3. Since u2 G LTO(0,T; W22(Q), according to Gronwall's lemma, (2.51),(2.53) and (2.54) implies that w = 0 for almost all (t, x) G QT and p = 0 for almost all t G (0, T). The theorem is proved. □

Conclusions

In this paper we discussed the behaivior of the solution to the Problem 1 as n ^ 0. It was established that Problem 1 for the pseudoparabolic equation approximates weakly Problem 2 for the parabolic one under the hypotheses of Theorem 2.2 when n ^ 0. Theorems 1.1 and 2.2 remains true if w G C([0,T]; W23/2(dQ)) and <1 G C([0,T]).

In general Problem 1 does not approximate Problem 2. As was shown in [1], if the initial and boundary data do not satisfy (2.11), then Problem 1 may be unsolvable.

Theorem 2.2 implies that Problem 2 for the relevant parabolic equation is solved relying on the results on Problem 1. The uniqueness of the solution to Problem 2 is provided by Theorem 2.3.

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Об аппроксимации параболической обратной задачи псевдопараболической задачей

Анна Ш. Любанова

Исследуется обратная задача идентификации одного из старших коэффициентов псевдопараболического уравнения. Доказывается, что обратная задача для псевдопараболического уравнения аппроксимирует соответствующую обратную задачу для параболического уравнения. Устанавливается также существование и единственность решения параболической обратной задачи.

Ключевые слова: фильтрация, обратные задачи для уравнений в частных производных, псевдопараболическое уравнение, параболическое уравнение, теоремы существования и единственности.