CC BY
5
Journal of Siberian Federal University. Mathematics & Physics 2023, 16(5), 661—672
EDN: TDISWG УДК 517.95
An Inverse Problem for Pseudoparabolic Equation with the Mixed Boundary Condition
Anna Sh. Lyubanova*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 05.02.2023, received in revised form 10.06.2023, accepted 20.08.2023 Abstract. In this paper we study the inverse problem on identification of the leading coefficient in the pseudoparabolic equation. The problem involves the mixed boundary condition. The unknown coefficient is recovered by additional integral boundary data. The existence and uniqueness of the strong solution are proved. The result concerns with the identification of the hydraulic properties of fissured medium.
Keywords: filtration, inverse problem, pseudoparabolic equation, existence, uniqueness.
Citation: A.Sh. Lyubanova, An Inverse Problem for Pseudoparabolic Equation with the Mixed Boundary Condition, J. Sib. Fed. Univ. Math. Phys., 2023, 16(5), 661-672. EDN: TDISWG.
Introduction
This paper is devoted to the inverse problem of identification of an unknown coefficient in the pseudoparabolic equation
(u + L1u)t + L2u = f (0.1)
with the initial data
B1u\t=0 = Uo (0.2)
and the boundary condition
B2u\an = n(t,x,k(t)) (0.3)
where Q is a bounded domain in R" with a boundary dQ, Li = nM, L2 = k(t)M, M and Bi are linear differential operators of the second order in the spacial variables, B2 is a linear operator. To find the unknown coeffcient k(t), the additional data is used in the form of the condition of overdetermination
/ B3(t)(nut + k(t)u) w(t,x) ds + k(t)vi(t) = wit) (0.4)
JSQ
where B3(t) is a linear operator in the spacial variables for every t e (0,T), u(t,x) and ^i(t), i = 1, 2, are known functions. The conditions (0.3) and (0.4) must be independent in that (0.4) may not be evident from (0.3).
* [email protected] © Siberian Federal University. All rights reserved
A main goal of this article is to investigate the correctness of the problem (0.1)-(0.4) with d
B\u = u, B2U = ^=(nut + k(t)u) + a(x)(nut + k(t)u), ¡(t, x, k(t)) = ^(t, x) — ¡i(t, x)k(t) and dN
d
B3 = I where I is the identity operator, a(x) and Hi(t,x), i = 1,2, are known functions, is
dN
the conormal derivative associated with the operator M (see Problem 1 below).
In [9,11,12], the problem (0.1)-(0.4) was considered in the case where Li = nM, L2 =
d
= k(t)M + g(t,x)I, Biu = u + Liu, B2u = u, B3 = . In [11], the existence and uniqueness of
dN
the strong solution are proved. The regularity of the solution is also investigated. The work [12] discusses the stabilization and the asymptotic behavior of the solution as t ^ It is shown in [9] that under certain conditions this solution tends to the solution of the appropriate parabolic inverse problem when n ^ 0.
Applications of the inverse problems for (0.1) with various operators Li and L2 involve the recovery of the unknown parameters indicating physical properties of a medium (the heat conductivity, the permeability of a porous medium, the elasticity, etc.). In particular, the equation (0.1) considered in [9,11,12] describes the filtration of a liquid in a fissured medium [1]. The coefficient k(t) is in inverse proportion to the total effect of compressibility of the liquid and the fissured medium. Since the natural stratum is involved, the parameters in (0.1) should be determined on the basis of the investigation of its behaviour under the natural non-steady-state conditions. This leads to the interest in studying the inverse problems for (0.1) and its analogue.
The study of inverse problems for pseudoparabolic equations (0.1) goes back to 1980s. The first result [16] refers to the inverse problems of determining a source function f in (0.1) with Li = L2. Most of the results on inverse problems are concerned with the identification of an unknown source f and coefficient in the lowest order term u as in [4-6,15]. The work [8] is devoted to the inverse problem on reconstruction of the kernels in the integral term of the integro-differential operator L2. We should mention also the results [5,14,15] concerning with coefficient inverse problems for (0.1). In [14], the uniqueness theorem is obtained and an algorithm of determining a constant a in the second order term is constructed. In [5], the solvability is established for two inverse problems of recovering the unknown coefficients in terms u (the lowest term of L2u) and ut of (0.1). In [15], an inverse problem of recovering time-depending right-hand side and coefficients of (0.1) is considered. The values of the solution at separate points are employed as overdetermination conditions. The existence and uniqueness theorems are proven for this problem.
The paper is organized as follows. Section 1 presents the formulation of the inverse problem and certain preliminary results concerning the direct initial boundary value problem for (0.1). In Section 2, the existence and uniqueness of the strong solution of the inverse problem are proved.
1. The statement of the problem and preliminaries
Let dQ € C2, Q be the closure of Q. QT = Q x (0,T) is a cylinder with the lateral surface ST = (0, T) x dQ, QT is the closure of QT and the pair (t, x) is a point of QT.
Throughout this paper we use the following notation: || • and (■, -)R are the norm and the inner product of R"; || ■ || and (■, ■) are the norm and the inner product of L2(Q), respectively; || ■ ||j and (■, ■) are the norm of W2j(Q) and the duality relation between Wj(Q) and W-j (Q), respectively (j = 1, 2).
We introduce the operator M by the following rule: for every u € W^Q) the element Mu
gives the functional
J (v) = (Mu,v)m + / a(x)uvdx
JdQ
defined for all v G W2(Q>) where
(Mu,v)m = / {(M(x)Wu, wv)r + m(x)uv}dx.
JQ
Here M(x) = ((mij(x))) is a matrix of functions mij(x), i,j = 1, 2,... ,n, m(x) and a(x) are scalar functions. We assume that the following conditions are fulfilled.
I. mij (x), dmij/dxi, i,j,l = 1, 2,...,n, and m(x) are bounded in Q. M is an elliptic operator, that is, there exist positive constants m0 and mi such that for any v g W2(Q) and almost all x G Q
m0||v||2 < (Mv,v)m +/ a(x)uvdx < mi ||v||2. (1.1)
JdQ
II. mij (x) = mji(x), i,j = 1, 2,...,n for x G Q. We are studying the following inverse problem.
Problem 1. For a given constant n and functions f (t,x), g(t,x), u0(x), pi(t,x), p2(t,x), a(x), w(t,x), yi(t), y2(t) find the pair of functions (u(t,x), k(t)) satisfying the equation
ut + nMut + k(t) Mu = f (t,x), (t,x) G QT, (1.2)
and the conditions
u|t_o = uo(x), x G Q, (1.3)
{n^^t + k(t) ^ + °(x)(rwt + k(t)u)} + k(t)pi(t,x)= H2(t,x), (1.4)
^ dN dN > st
f (nut + k(t)u)w(t,x) dS + yi(t)k(t) = y2(t), t G (0,T). (1.5)
JdQ
d
Here = (n, A4(x)V) and n is the unit outward normal to dQ.
dN v y ' '
The conditions (1.4) and (1.5) may seem peculiar. However, such formulation of this conditions are rather natural for pseudoparabolic equations. For a deeper discussion of the conditions (1.4), (1.5) we refer the reader to [10].
We introduce functions b(t, x) and hn (t, x) as the solutions of the boundary value problems
Mb = 0 in Q, i^L + a(x)b\ = u(t,x), ' IdN J dQ
{8hv } hn + nMhn = 0 in n, |-= + a(x)hnj ^ = u(t,x).
dhn dN
The existence and uniqueness results for Problem 1 rely upon two propositions for the direct problem (1.2)-(1.4) with the known function k(t).
The first proposition concerns with the existence and uniqueness of the solution of the direct problem (1.2)-(1.4), which follows from the the results of [19] in the case of the constant coefficient k(t) = k.
Lemma 1.1. 1) Let the assumptions I-II be fulfilled, n> 0, k G C([0,T]), f G C([0,?1; L2(Q)), u0 G W2 (Q) and G C 1([0,T]; W12(dQ)). Then there exists a unique solution u(t,x) of
problem (1.2)-(1.4) such that u G C 1([0,Tj; W2(Q)).
Proof. Let us consider the problem
{ vt + G(v) = f, (t,x) € Qt,
\ v\t=0 = (I + nM)u0, x € ft, (L6)
where v = (I + nM)u and the operator G acting from C([0, T]; L2(ft)) into itself is defined as
G = k(t)M(I + nM)-1 = ^ (I — (I + nM)-1) .
n
The function v is the solution of the problem (1.6) if and only if the function u = (I + nM)-1v is a solution of problem (1.2)-(1.4). Therefore Theorem 1.1 will be proved once we prove the existence and uniqueness of the solution of the problem (1.6). From the hypotheses of the lemma it follows that G is a Lipschitz-continuous linear operator. Then, by Theorem 1.2 of [3, Chapter V], the problem (1.6) has a unique solution v € C 1([0,T]; L2(ft)). The lemma is proved. □ The next lemma is a maximum principle (a comparison theorem) for the direct problem (1.2)-(1.4). In general, the maximum principle does not hold for the pseudoparabolic equation [17]. However it has been possible to prove such assertions under certain additional assumptions on the input data of the initial boundary value problems. In [17, 18,20], comparison theorems were proved for the first initial boundary value problem for (0.1) with the general linear elliptic operators L1 = L2 = L of the second order and k = 1. In [2], the comparison theorem is proved for (0.1) with the constant coefficient k, L1 = L2 = —A in the case of the mixed boundary conditions.
Lemma 1.2. Under the assumptions of Lemma 1.1, let u(t,x) be the solution of the problem (1.2)-(14) in C1([0,T]; W22(ft)). In addition, let f (t,x) > 0 for almost all (t,x) € QT, u0(x) > 0 almost everywhere in ft, k(t) ^ 0 for t € [0, T], ¡i1 ^ 0 and n2 ^ 0 for almost all (t,x) € ST. Then
1 't
i(t,x) ^ uo(x) exp (--[ k(0)d0]
v n J 0 1
n J 0
for almost all (t,x) € QT.
Proof. It is sufficient to prove the assertion of Lemma 1.2 for the smooth solution of problem (1.2)-(1.4) since the following arguments can be justified by using the method of difference quotients or mollifiers.
The function v = u — u0 exp (--f k(Q)d,o\ is the solution of the equation
n0
v + nMv = T [f + v] exp ( — T de)dr + uo exp ( — f ^ do), (1.7)
Jo 1 n J v Jr n J v Jo n J
and obeys the boundary condition
d= + a(x)v =— i (u2 — k(r)^1)exp(--[ kdO^idr. (1.8)
L dN J St n J 0 v n J T '
Let us define the functions v1 = mm(ytxX)eQT{v, 0} and v2 = m&x(ttxX)eQT{v, 0}. We multiply (1.7) by v1 in terms of the inner product of L2(ft) and integrate by parts in the second summand of the left side of the resulting equation. In view of (1.8) this gives
+ n{Mv1,v1)M + n i w2ds — i i (f +—v2)exp( — i —d^\drv1dx— Jsq JnJo n v Jt n j
f f 2 — k^i)exp( — i —d^drvids — exp ( — i —d9^ i uovidx
Jan Jo v Jt n J v Jo n J JQ
k it1 k \ —vi exp ( — —d9)drvi dx. n V it n J
t k ( t1 k -vi exp — -In Jo n v Jt n
In the hypotheses of the lemma this relation implies the inequality
r t
■2,
Ml2 < C ||vi||2dr.
o
with the constant C > 0 depending on T, n, maxtE[0,T] k(t), whence by Gronwall's lemma it follows that vl = 0, that is, v > 0 almost everywhere in QT. The lemma is proved. □
2. The existence and uniqueness
In this section we discuss the sufficient conditions for the solvability and the uniqueness of the solution of Problem 1. By a solution {u, k} of Problem 1 we mean that
1) u G C1 ([0,T]; W2(i)), k(t) G C([0,T]);
2) the pair {u, k} obeys the equation (1.2) almost everywhere in QT and the conditions (1.3) for almost all x G l, (1.4) almost everywhere in St and (1.5) for all t G [0,T].
The main result of this article is established by our next theorem.
Theorem 2.1. Let the assumptions I—II be fulfilled, dll G C2 and n be a positive constant. Assume that
(i) f G C([0,T]; L2(i)), 1^1,^2 G C ([0,T]; Wl/2(di)), uo G W$(Q), a G C(di), u G C 1([0,T]; Wl/2(di)), G C([0,T]);
(ii) the assumptions of Lemma 1.2 are fulfilled and there exist constants a0 > 0 and $0 > 0 such that
yi — iiuds ^ ao, (2.1)
Jen
$(t) = ^(t) — (f, hn) — i i2hnds > (2.2)
Jen
Then Problem 1 has a solution {u, k} and this solution is unique. Moreover, u ^ 0 almost everywhere in QT and the estimates
0 <k0 < k(t) < $a-1, (2.3)
||u||2 + \\uth < Ci (2.4)
hold with positive constants k0 and C1. Here $ = maxt£[0]T]
Proof. Following the idea in [11,13], we reduce Problem 1 to an equivalent inverse problem with a nonlinear operator equation for k(t). To do this, we multiply (1.2) by hn in terms of the inner product of L2(Q) and integrate by parts twice. This yeilds
(ut, hn) + (nMut + kMu, hn) = (ut, hn) — f (ndu= + k^u) hnds+
JanK dN dNJ
+ (nMut + kMu,hn)M = (ut,hn)+ k/ ¡j1hnds - ¡j2hnds+
Jan Jan
r r dhv
+ a(x)(nut + ku)hn ds + (nut + ku)^= ds + (nut + ku,Mhn ) = Jan Jan dN
- yi +/ Mihnds --(u,hn)! -f ds + y2 = (f,hn),
Ian V
whence we obtain that
k
-i ¡j1hv ds + -(u,hn) = <£(t) Jan V
We define the operator A which maps every element y € C+([0,T]) = {y | y e C([0,T]),y > 0} into the element Ay e C([0,T]) by the rule
r f 1 ] -1
Ay = ®(t) - ds + -(uy,hn) (2.5)
1 Jan V -1
where uy is the solution of the problem (1.2)-(1.5) with k(t) = y. The element Ay is meaningful for every y e C+([0,T]). Indeed, the direct problem (1.2)-(1.4) with k = y has a unique solution uy e C 1([0, T]); W2(Q)) for every y € C+ ([0,T]) by Lemma 1.1. Moreover, by Lemma 1.2 and the maximum principle for elliptic equation (uy, hn) ^ 0, which implies in view of (2.1) that
-i nihPds +-(uy,hn) > ao. (2.6) Jan V
It can be shown that Problem 1 is solvable if and only if the operator equation
y = Ay (2.7)
has a solution in C+([0, T]). Really, the deduction of the equation (2.7) shows that if {uy, y} is a solution of Problem 1, then y is a fixed point of the operator A by (2.5). On the other hand, let y* is a solution of equation (2.7) and u* is a solution of (1.2)-(1.4) with k(t) = y*(t). Multiplying (1.2) by hn in terms of the inner product of L2(Q) and integration by parts twice in the second and third summands implies in view of (1.4), (2.5), (2.7) that the pair {u*(t,x),y*(t)} obeys the condition of overdetermination (1.5).
The relations (2.2), (2.5) and (2.6) imply the estimate
0 < Ay < $a-1.
Let us prove that there exists such yo > 0 that the operator A maps the set
Y = { y I y e C ([0,T ]),yo < y < Za-1}
into itself. We multiply (1.2) with k = y € Y by the solution uy of the problem (1.2)-(1.4) in terms of the inner product of L2(Q) and integrate by parts in the second and third summands. It gives
A \\uy||2 + n(Muy, uy)m + n au2yds\ + y(Muy,uv)m + y au2yds - Jan } Jan
= (f,uy) - (y^i - H2)uyds.
Jan
We next integrate the last relation with respect to t from 0 to t, 0 <t < T, and rewrite as
\\uy||2 + n (Muy,uy)m + vu2yds
+ n
On
(muo,uo)m + auOds + 2 \(f,uy) — (yii — l2)uy dsldt. (2.8)
Jan J Jo ^ Jon J
By (1.1), the left side of this equality is estimated from below as
\\uy I + nmo\\uy\\1 + / au2 ds.
Jen
One can estimate the right term of (2.8) with the help of the embedding theorem and the Cauchy inequality. This gives
+ 2 y Jo
(Muy ,Uy )m + / &u2y ds
Ida,
dt
\\uo\\2 + n
(muo,uo)m + I auyds
/ auOfds + 2 \(f,uy ) - (ymi - H2)uy dsldt <
Jda J Jo 1 Jda J
<
nm0
1\\^i\\l2(st ) + \\^2\\l2(st )
+
\L2(Qt )
+ \\uo\\2+
+n
(Mu0,u0)M + o\\uo\\2L2(dQ,) + (\\uy\\2 + nmo\\uy\\2) dt
where a = \a\C(n), c > 0 is the constant of the embedding W2(l) ^ L2(dl). Thus, we obtain from (2.8) that
\\uy\\2 + nmo\\uy\\1 < — Uaol\^1\L2(ST ) + \\V2\\L2(ST )) + \\f WliQ
+
+ N^0N2 + n (Muo,uq)m + v\\uo\\2L2{dn) + / (IKII2 + Vmo\\uy\\1)dt,
L J Jo
whence by Gronwall's lemma we have
„2 , N 2
\\uy\\2 + nmo\\uy\\21 < [-m ($ao1\\Mi\\L2(ST ) + \\V2\\l2(st )) + \\f \\h(Q i nmo v /
)+
+ \ \ u0 \2 + n( (Mu0,u0)M + v\\uo\\2L2(ddn)
Coming back to (2.5) we can determine y0. By (2.2), (2.9),
Ay > $o pi + max {\\mi\\L2(da)\\u\\L2(da) + n^CW\\ L te[o,T]
eT = C2. (2.9)
= yo,
(2.10)
where tp1 = \\^>i\\c([0,T])- Thus, the operator A maps the set Y with y0 defined by (2.10) into itself.
Let y1,y2 G Y and uyi ,uy2 be the solutions of the problem (1.2)-(1.4) with k = y1 and k = y2, respectively. In view of (2.5)
\Ayi - Ay2\ = n 1$(t)|(uy2 - uy! ,hn )|{ pi + / mW ds + n 1(uy1,
1 Jda
1
+ nihv ds + n 1(uy2,
^ Jda
<
$maxi£[o,T] \\hri\\
\\uyi - uy2\\. (2.11)
T
2
2
)
T
1
x
i
X
On the other hand, the difference wy = uyi — uy2 satisfies the equation
wyt + nMwyt + y1Mwy = (y2 — y1)Muy2 (2.12)
and the conditions
wy (0, x) = 0,
V duy
= (y2 — y1) '
St
We multiply (2.12) by wy in terms of the inner product of L2 (Q) and integrate by parts in the second and third summands and the right side. In view of (2.13) this gives
dwyt dwy ] r (ouy2 \ "1
n^W + yi^W + a(x)(nWyt + yiwy) ^ =(y2 - yi) —=f + auy2j ^ + Mi . (2.13) dN dN -St A dN / ST
1 d I ,, ||2 . w"
2 dt 1 11 y
1 \\wy\\2 + n (Mwy,wy)m + / vw2yds |+ yA (Mwy,wy)m + / aw2ds\ = L L Jan J J L Jan J
= (y2 — y1){ (Mu2y,wy)M + J (vu2y + m)wyd^. (2.14) By (1.1), (2.9) and the embedding theorem,
(y2 - yi )j (Mu2y,wy)m + (au2y + Mi)wyds I L Jan J
<
< c\y2 - yi\{(mi + C(j)\\uy2\\i + \\Mi\\L2(an)} \\wy\\i <
<
\y2 - mn+ßC + M "L2(dn^2 + mK\\2. (2.15)
2nm0 \ (nm0)1/y ( j 2
Integrating (2.14) with respect to t from 0 to t, 0 < t < T, and estimating the left side of the result with the use of (1.1) we obtain in view of (2.) that
2 y c2 fT y ( (m1 + ca)Cy 1
\\wy\\ + nmo\\wy\\1 < —]0 1 y2 — y1 1 \ {nm0)1/2 + W^Wl2(anH dt+
+ ! (WwyW2 + nmoWwyWl) dt.
o
Accordig to Gronwall's lemma the last relation implies the estimate
2 c2eT ((m1 + ca)C2 \2 fT 2
K \ < n^Kim)^ + Wc(-[o,T];L2(an))j ^ y — yA dt. (2.16)
Combining (2.11) and (2.16) we are led to the inequality
/ r \ 1/2
IAy1 — Ay2l < J Iy1 — y2l2dtj
where _
c$maxte[o,T] \\hn\\ f (m1 + ca)C2 ..........N t/2
K = ^m^a2 I (nmo)1/2 + ^^«W««»; e
Let us introduce an equivalent norm in C([0,T]) as
I • I v = max {e-vt\ • \} - 668 -
C2
with a positive constant v to be determined later. Then
| Ayi — ÄV2\v <
Choosing v = 2K 2, we obtain the inequality
K
(2v )i/2
\yi — y2 \ v .
\Ayi — Äy2 \ v < 2 \yi — y2 1
(2.17)
This means that the operator A : Y ^ Y is a contraction. Thus, in accordance with the principle of contracting mappings the operator A has a unique fixed point k* G Y. The pair {u*, k*} gives the solution of the inverse problem (1.2)-(1.5) where u* satisfies (1.2)-(1.4) with k = k* and u* G C 1([0, T]; W22(i)) by Lemma 1.1. Moreover, the estimates (2.3) and (2.9) are valid for k* and u* .
We are coming now to the estimates for u and ut in W2?(Q). Multiplying (1.2) by ut in terms of the inner product of L2(Q) and integrating by parts in the second and third terms of the left side yields
+ n{Mut,ut)M + n a(x)u2ds = —k(Mu,ut)M — kl a(x)uutds + futds + (f,ut).
Jen Jen Jen
By (1.1), (2.3), (2.9) and the embedding theorem, the last equality implies the relation
+ nmo ||ut|2 + n a(x)utds << 'en L
2 a ^ { ®('mi + c2^)
ao
l|u||i + 4ß ||L2(Sn)}
L +
ut || < 1 { °2®(mi + c2a) + c ^ 2nm0\ a0(nm0)i/2
21
+ 2 |
2 +
1
+- (|ut|2 + nmoHutHl)
2
which implies the estimate
2 1 { C2$(mi + c2a)
|U||2 + nmo|M2 < — { C2 r \ Cina) + 4ßh2m)}
nm0 I a0(nm0)l/2 >
+ Uf ||2 = C2.
(2.18)
We are now in a position to get the estimates for u in W2(i). To do this we rewrite the boundary condition (1.4) in the form
" du
+ au
dN
ß
(2.19)
where
ß=
'duo
+ auo
-dN
St
exp
(--I k dQ^ +— I (n2 — kni)exp(--[ kdQ^dr.
V n J 0 J n J 0 V n Jt J
Multiplying (1.2) by Mu in terms of the scalar product of L2(Q) gives
2 dt1
UMuf + k(t)UMuU2 = (f — ut, Mu).
By (2.3) and (2.18),
2 dt UMuf + yoUMuf < 2y~o (Uf || + o3)2 + y2 UMuf
v
2
2
s
t
whence
1 ft
\\Mu\\2 < \\Muo\\2e-nt + - (\\f\\ + C3)
yo Jo
2 (t-t)
2 e n K 1
dr.
The inequality [7, Chapter 2]
IM|2 < KMMvW + \\p\\wi2/2{dn) + llvHi)
(2.20)
(2.21)
is valid for all v e W22(Q) satisfying the boundary condition (2.19) where the constant K1 > 0 depends on m0, m1 and mesQ. In view of (2.9), (2.18)-(2.21) we have
\\u\\2 < K [\\Muo\\ + (
yo o
\1/2
+ Cs)2dr) + \\ß\\wi/2{aQ) + C2(nmo)-1/2 = C4. (2.22)
The estimates (2.18) and (2.20) enable one to conclude from (1.2) that
¥
nWMutW < WfII + C3 + — C5.
ao
(2.23)
The positive constant C5 depends on C2, C3, yo, T, n, Wf Wl2(qt). By (1.4), (2.18) and (2.23), Wnut + k(t)uW2 < K^WM(nut + k(t)u)W + WP — a(nut + k(t)u)WWi/2 (8Q) +
+ \\vut + fc(i)u\\i) < Ki
- ( $
- ut\\ + \\ß\\w1i/2{an) + (cov+l)[v\\ut\\i + —\\u\\i
<
^ Ki
+ C3 +
i1/2(dn) +
whence
(coä+1)f„ $
(nmo)1/2
$ ,, ,, „ $
w2 m) 1 (nmo)i/2
Lc3 + — C2 \ ao
= Ca
n\\ut\\2 < Ce + — \\u\\2 < Ce + — CA.
ao ao
The last inequality and (2.22) implies the estimate (2.4).
The uniqueness of the solution {u,k} follows from (2.17). Really, let {u1,k1} and {u2,k2} be two solutions of Problem 1. Hence, k1 and k2 satisfies the operator equation (2.7) and the inequality (2.17) is valid. Then
| ki - k2 | v = I Aki - Ak2\v < 2 I ki - k2 | v ,
which proves that k1 = k2. From this in turn follows by (2.16) that u1 = u2. □
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-02-2023-936).
References
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Обратная задача для псевдопараболического уравнения со смешанным граничным условием
Анна Ш. Любанова
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. В данной статье исследуется обратная задача идентификации старшего коэффициента в псевдопараболическом уравнении со смешанным граничным условием. Неизвестный коэффициент восстанавливается по дополнительным интегральным граничным данным. Доказано существование и единственность сильного обобщенного решения. Результат связан с идентификацией гидравлических свойств трещиноватой среды.
Ключевые слова: фильтрация, обратная задача, псевдопараболическое уравнение, существование, единственность.
