CC BY
33
MSC 35K70
DOI: 10.14529/mmp 160207
INVERSE PROBLEMS FOR SOME SOBOLEV-TYPE MATHEMATICAL MODELS
S. G. Pyatkov, Ugra State University, Khanty-Mansyisk, Russian Federation, S_pyatkov@ugrasu.ru,
S.N. Shergin, Ugra State University, Khanty-Mansyisk, Russian Federation, Ssn@ugrasu.ru
The present article is devoted to the study of mathematical models based the Sobolev-type equations and systems arising in dynamics of a stratified fluid, elasticity theory, hydrodynamics, electrodynamics, etc. Along with a solution we determine an unknown right-hand side and coefficients in a Sobolev-type equations of the forth order. The overdetermination conditions are the values of a solution in a collection of points of a spatial domain. The problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. The existence and uniqueness theorems of solutions for the linear and nonlinear cases are proven. In the linear case the result is global in time and it is local in the nonlinear case. The main spaces in question are the Sobolev spaces.
Keywords: the Sobolev-type model; Sobolev equation; existence and uniqueness theorems; inverse problem; boundary value problem; plasma waves; rotating fluid; the Boussinesq - Love model.
Introduction
We consider the problem on determining a solution U, the right-hand side, and coefficients in the equation
LoUtt + L\Ut + L2U = f, (x,t) £ Q = G x (0,T), (1)
where Li (i = 0,1, 2) are second order operators in the variables x, G is a bounded domain in Rn(n > 1) with boundary r £ C2. The equation is supplemented with the following initial an boundary conditions:
U\s = S = r x (0,T), (2)
U\t=o = Uo(x), Ut\t=o = Ui (x). (3)
We assume that the right-hand side in (1) is of the form
m
f = £ Ci(t)fi(x,t) + fo(x,t), (4)
i=1
where the functions fi (i = 0,1,... ,m) are given. The problem is to find the right-hand side f (i.e. the functions ci(t) j) and coefficients of (1) dependent on time with the use
U
Thus, the overdetermination conditions are as follows:
U(xi,t) = ^i(t), (i =1, 2, ..,m), (5)
with xi arbitrary points of G. Mathematical models relying on the Sobolev-type equations of the form (3) arise in many fields of mathematical physics, in particular, in elasticity theory, hydrodynamics, and electrodynamics. The class of equations (1) contains the classical Sobolev equation describing small oscillations of a rotating fluid [1], the Boussinesq - Love model describing the shallow-water waves [2] and longitudinal oscillations of an elastic rod [3] (see also the monograph [4]), and the models describing linear waves in a nonmagnetized plasma (see [5]). The inverse problems for the Sobolev-type equations are not studied well. The main attention in the previous articles was paid to the equations of the third order, i.e. the pseudoparabolic equations. Describe some results. The inverse problem of recovering a leading coefficient in a pseudoparabolic equation with the use of an integral overdetermination condition on the boundary of the domain is studied in several articles by A.Sh. Lyubanova and E.A. Tani [6, 7]. They establish local existence and uniqueness theorems and derive some properties of solutions to inverse problems of this type. Close results but with different overdetermination conditions are exposed in [8-10, 11]. Some inverse problems for pseudoparabolic equations are presented in the monograph [12]. The problem of recovering a kernel of an integral operator occurring in Sobolev-type equations when some functional of a solution is examined in [13]. Close results are presented in [14]. In the case of forth order models (1), only some particular results are known. Optimal control problems for the Boussinesq - Love model are examined in [15-17]. Some inverse problems, one of which is included in our class of inverse problems (1) - (5), for one-dimensional Boussinesq - Love model is treated in [18,19], where the study of existence of a solution is reduced to a system of integral equations. Similar existence theorems in the one-dimensional case are exposed in [20]. Here several inverse problems are stated. They differ in the overdetermination conditions; an integral condition, the Cauchy data, the final overdetermination condition, and the condition of the form (5) with m = 1.
In contrast to the previous results, we consider a more general class of multidimensional models with an arbitrary number of conditions (5). In this case the right-hand side can be treated as an approximation of an arbitrary function f (t,x) by a finite segment of a series. In the article we consider the linear case, i.e. the problem on determination of the right-hand side and nonlinear, i.e. the coefficient problem. The main results are Theorems 3 and 4 stating that in both cases the problem (1) - (5) is well-posed. In the linear case the result is global in time and respectively local in the nonlinear case.
1. Preliminaries
We employ the Sobolev spaces Ws(G) and the Holder spaces Ca (G) (see the definitions in [21]). The symbol Lp(0,T; H) (H is a Banach space) designates the space of strongly measurable functions defined on [0,T] with values in H. Below we assume that r G C2 (see the definition, for example, in Sect. 1, Ch. 1 of [21]), the operators Lk (k = 0,1, 2) are of the form
m m
LkU = E ag(x,t)UXiX. + E ak(x,t)Ux. + ag(x,t)U,
i,j=l i=l
with real coefficients, and L0 is an elliptic operator, i.e., there exists a constant 50 > 0 such that
m
5>0j££ > ¿o|£I2 V £ G Rn, V (x,t) G Q.
ij
i,j=l
(6)
Fix a parameter p > n and assume that
a0 e C(Q), 00,00 e C([0,T]; Lp(G)), a00(x,t) < 0 a.e. in Q aj e Lp(0, T; L^(G)), ak ,ak0 e LP(Q) (ij = 1, 2,..,n, k = 1, 2).
L
Theorem 1. The Dirichlet problem
Lou = f, u\r = 0, (7)
for every f e Lq(0,T; Lp(G)) (p > n, q e [1, o]) has a unique solution u e Lq(0,T; Wpj(G)) satisfying the estimate
\\u\\hq (0 ,T;W2(G)) < c\\f \\Lq (0,T ;Lp(G)) i
where c is independent of f.
t
uniqueness of solutions (see the maximum principle in Ch. 9 of [22]) and the Fredholm property of these problems. Estimates for solutions can be obtained with the use of the
t
of every point t0 e [0, T], we can prove an estimate of the form \\u\\W2(G) < c\\L0u\\Lp(G)
ct claim of the theorem.
□
Lemma 1. If b e Lq(G) with q > n for p < n and q > p for p > n then there exists a c > 0
\\bVU\\lp(G) < c\\U\\W2G) yu e W2(G). (8)
Ifb e Lq(G) with q > n/2 for p < n/2 and q > p for p > n/2 then there exists a constant
c > 0
\\bU\\Lp(g) < c\\U\\W2(g) yU e Wp(G). (9)
Proof. The proof coincides with that of Lemma 1 in [23].
Theorem 2. Assume that y,yt,Vtt e Lp(0,T; W^-l/p(G)) (p e (1, oo)), f e Lp(Q), Ui(x) e Wp(G) (i = 0,1), conditions (6) for the coefficients and the agreement conditions
p(x, 0)\r = Uo(x)\r, &(x, 0)\r = U1(x)\r (10)
hold. Then there exists a unique solution to (1) - (3) such that Utt e Lp(0,T; W^(G)), U, Ut e C([0, T]; W£(G)). If V = 0, Ui(x) = 0 (i = 0,1) then there exists a constant c> 0 independent of y e (0,T] and f such that a solution to the problem (1) - (3) satisfies the estimate
\\u\\lU0,t,W2(G)) + \\Ut\\LU0,r,W2(G)) + \\Utt\\Lp(0,r,W2 (G)) < c\\f \\lp(Qy ) [Q1 = G x (0,y))-
Proof. Consider the segment [0,T], Find a function $ such that
$t, $tt G Lp(0,T; W2(G)): $|s = p, $|i=c = Uo(x), $t|t=o = Ul(x), which can be constructed as follows:
t a
$ = & + Uo + tU1(x), & = J j Mx,T)drd£, A&o = 0, ^o|r = Ptt
oo
(the existence of a function follows from the well-known results, see, for instance, [22]). The function V = U — $ is a solution to the problem
LoVtt + LiVt + L2V = f — Lo$tt — Li$t — L2$ = g, V |s = 0, V |t=o = 0, V|t=o = 0.
Theorem 1 yields
t a t a
V + J j L-1(LiVt(T,x) + L2V(r,x))drd£ = j J L-1g(r,x)drd£.
o o o o
Condition (6) and Lemma 1 validate the estimate (r < T)
\\L-lLiV\\hq (O,T ;W2 (G)) < c\\V\\Lq (O,T ;W2(G)), q G [1, , (H)
which for q = to along with the fixed point theorem allows us to justify the existence theorem. The estimate from the claim is proven by analogy with that in Theorem 2 in [23].
2. Basic Results
2.1. The Linear Case
Write out the conditions on the data of the problem. In addition to the agreement conditions (10) we require that
&i(0) = Uo(xl), ^it(0) = Ui(xi), i = 1, 2,...,m. (12)
The well-posedness condition. Denote by B the matrix with the entries bij = L-1fi(xj, t) and assume that there exists 8o > 0 such that
| det B| > So yt G [0,T], (13)
where L-1 fi is a solution Ui to the problem LoUi = fi, Uj\s = 0.
Theorem 3. Assume that the coefficients of the operators Li (i = 0,1, 2) satisfy conditions (6), the agreement conditions (10), (12) and the well-posedness condition (13) are fulfilled, and
fo G Lp(Q), fi G L^(0,T; LP(G)), Uj G Wjj(G) (j = 0,1), ,ptt G Lp(0,T; W^~1/p(G)),
& G W2(0,T), i = 1, 2,... ,m, p>n. Then there exists a unique solution to (1) - (5) such that
U,Ut G C ([0,T]; W2(G)), Utt G Lp (0,T; W^(G)), a G Lp (0,T), (i = 1, 2,..,m).
7g Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2016, vol. 9, no. 2, pp. 75-89
Proof. As in the proof of Theorem 2, we construct a function $(x,t) such that $, $t, $tt e Lp(0,T; W£(G)), $\s = v, $\t=o = Uo(x), $t\t=o = Ui(x). The function V = U — $ is a solution to the problem
m
LoVtt + LiVt + L2V = fo - Lo$tt - Li§t - Ь2Ф+ f = go + f
o Vtt + LiVt + L2 V = Jo — Lo$tt — Li$t — L2$ + ci
i=1
V (xi,t) = U (xi,t) — $(xi,t) = tfi — $(xi,t) = Vi(t), V\s = 0, V\t=o = 0, Vt\t=o = 0.
Lo
i=l
(14)
Vtt + L-lLiVt + L~-1L2V = L-lgo + £ cL-fi,
i=1
(15)
where the operator L0 1 takes a function g into a solution to the problem
LoV = g, V\s = 0, V\t=o = 0, Vt\t=o = 0.
Since $ e W^(G; Wp(0,T)) and thereby, (s ince p > n) changing it on a set of zero measure if necessary, we can assume that $ e C 1(G; Wp(0,T)) (see [24]). Therefore, $(x0, •) e W2(0,T), in this case v e W2(0,T). Putting x = x0, we have
фи + L0lLiVt(xi,i) + LQlL2V(xi,t) = L- lgo(xi,t) + ^(L- lfj)(xi,t)cj(t), i = 1,
, m.
j=l
This equality and condition (13) yield
C = B
l
( фш + L-lLlVt(xl,t) + L-lL2V(Xl,t) - L-lgo(xl,t) \
\ фти + L- LlVt(xm,t) + L- L2V(xm,t) - L- go(xm,t) J
c =
f Cl\
cm
The latter equality is an operator equation for the vector-function c = (c1 ,c2,.., cm), where V = V(c) is an operator taking cinto a solution to the problem
aov = lovtt + llvt + l2v = go + E f v\s = 0, v|t=o = 0, vt\t=o = 0. (16)
i=l
The function V is representable as V = A° 1go + A° ^^m=1 ci.fi) = Vo + A° ^^m=1 ci.fi)-
c
Co = B
l
C = Co + R( C),
( фш + L-l(LlVot(xl,t) + L2Vo(xl,t)) - L-lgo(xl,t) \
фтИ + L- (LlVot(xm,t) + L2Vo(xm,t)) - L- go(xm,t)
(17)
/ m \
/ L-\Lxdt + L^A-1^ Cifi(x1,t) \
i=1
R( c) = B-1
m
11
L-1(L1&t + L2)A-1Y^ Cifi(xm, t) \ i=1 )
Note that the functions L-1 fi(xj,t) belong to L^(0,T). Indeed, fi G L^(0,T; Lp(G)) and L-1fi G L^(0,T; Wp2(G)). Therefore, for almost all t L-1f G Wp2(G). The embedding theorems (see, for instance, [21]) and the condition p > n imply that the function L-1 fi(x, t) is continuous in x after a possible modification on a set of zero measure and we have
\\L-lfi(xj ,t)\L^(o,T) < C\\L-1 fi(x,t)\L^(o,T ;W2(G)) < C1\fi\L^(o,T;Lp (G)).
As easily seen, the estimate \\B-1 g(t)\\Lp(o,Y) < C\\g(t)\\Lp(o>Y), is valid for every g G Lp(0,7) and y G [0, T] with a constant C independent of 7. Thus, we have
mm
\\R( c)\\lp (o,Y) < C2J2 \\L-\L1dt + L2)A-1^ Cifi)(xj ,t)\\Lp (o,Y). j=1 i=1
In view of embedding theorems and Theorem 2 the right-hand side is estimated by the quantity
m m m
C^ \\L-1(L1dt+L2)A-1 (£ fL (o,rw2 (G)) < Qi\\(L1dt+L2)A-1(Y; C fi)hp (o,TLP (g)) < j=1 i=1 i=1
< C371/p\(L1dt + L2)A-1(YJ Cifi)\\ L^(o,T;Lp (G)) < C471/p\\^2 Cifi \\ LP (QY).
i=1
The right-hand side here is estimated by C571/pY^m=1 \Cifi\Lp(qy) and
Y
Pi
\\Cifi\\Lp(QY) = (J Cm J ^(x^dxdt)1^ < WfihuoTLp(G))(J Cmdt)1/p.
o G o
Taking the above inequalities into account, we arrive at the estimate
\\R(C)\\ Lp (o,Y) < C771/p\\c! Lp (o,T), (18)
where C7 is independent of 7 Thereby, if C7yl/p = qo < 1 then equation (17) is uniquely solvable. Fix a constant 7 satisfying this condition and examine the equation
c — R( c) = Co. (19)
We can assume without loss of generality that all constants used in the proof of (18) are independent of 7. Let c be a solution to (17) defined on [0,7]. Construct the vector ^ f c t < 7 ^^^^
c1 =|0' t> 7 c^anSe k1 = c — C1 (k1 = (k1, k2,..., km)) in (19); if c is a
gQ Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2016, vol. 9, no. 2, pp. 75-89
Y
solution to (19) on [0, 27} then k1 = 0 for t < ^d k1 — R(k1) = c0 — c1 + R(c\). The right-hand side here vanishes on [0,7} and, respectively, R(k\) = 0 for t < 7. We can solve the system
ki — R( ki) = c0 — ci + R( ¿J (20)
on [7, 27}. Indeed, if kl E Lp(0, 27) then repeating the above argument we establish the estimate
m m
||R( ki)\\bp(Y,2Y) < С £ \\L~o\Lidt + L2 kifi)(Xj ,t)\\Lp{l,2l) <
j=l i=l mm
< C2Y, \\L-l(Lidt + L2)A-1^2 kifi)\Lp (y,2r,W?(G)) < j=l i=l
m
< C4(Lidt + L2)A-1(Y] kifi)\Lp(y,2j;Lp(G)) <
i=l
mm
< C371/p\\(Lidt + L2)A-1^2 kifi)\ < C771/PY. \ki\
i=1 i=1
The final estimate is of the form
\\R(kl)\\Lp(Y,2Y) < C771/P\\kl\\Lp(Y,2Y),
C7
in (18). Thus, equation (20) has a unique solution k1 E Lp(0, 27) such that k1 = 0 for t E (0,7). Hence, the function c = c1 + k1 is a solution to (17) from Lp(0, 27). Repeating the arguments, we can construct a solution c to equation (17) от the whole segment [0, T}.
Construct a function V as a solution to (16). Demonstrate that V satisfies (14). Inverting L0 we obtain that V satisfies (15). Taking x = xi in (15), we infer
m
Vtt(Xi,t) + L-\LA + L2)V(Xi,t) = L^1g0(Xi,t)^Y, L-1 j(t)fj(xi,t). (21)
j=1
c
m
Pitt(t) + L-\LA + L2)V (Xi,t) = L-1 g0(xi,t) + £ L-1 j (t)fj (xi,t). (22)
j=1
Subtracting these equalities we obtain Vtt(xi,t) — pitt(t) = 0 or d2(V(xi}t) — фi(t)) = 0.
t
V (xi} t) — <pi(t) = 0. Thus, V (xi} t) = <pi(t) Vi = 1, 2,...,m.
□
2.2. Nonlinear Case
Let the right-hand side in (1) and the operators Ls (s = 1, 2) be of the form
ro
f = Y1 ci(t)fi(x,t) + fo(x,t), (23)
i=1
LsU = LosU + £ ck(t)LkU, LkU = £ aj{x,t)UXiX. + £ ak(x,t)UXi + ak0(x,t)U.
k=rs-1+l i,j=1 i=1
The overdetermination conditions are as follows:
Ulx=Xi = ®i(t), ai(t) E Wp2(0,T), Xi E G, (i =1, 2,..,T2). (24)
The agreement conditions are written as
ai(0) = Uo(xi,t), ait(0) = Ui(xl,t) (i = 1, 2,..,t2). (25)
Let a function $ E C 1([0,T]; W2(G)) (p > n) is such that $tt E Lp(0,T; W2(G)), $|t=o = Uo(x), $tIt =o = Ui(x), $|t=o = Uo(x), $|s = V- Construct a matrix B with rows
L-fi (xj, t), L-1f2(xj ,t),..., L-1 fr0 (xj ,t), -L-1Lr°+1$t(xj ,t),..., —L-1^1 $t(x3 ,t), —L-1Lri+1$(xj ,t),..., —L-1Lr2 $(xj ,t), j = 1, 2,..,T2, and assume that there exists a constant 6o > 0 such that
| det BI > So yt E [0,T], (26)
where L-1 fi is a solution Ui to the problem L0Ui = fi, Uj\s = 0. It is easy to justify that locally in time condition (26) is independent of the choice of $ under the conditions of the theorem below.
Lo
coefficients satisfy the first condition in (6), the operators L01 ,L02 are of the same form as L1 , L2
The coefficients of the operators Lk (k = t0 + 1,..., t2) satisfy the relations
aj E L^(Q), (i,j = 1, 2,..,n), ak,ak E L^(0,T; LP(G)), k = To + 1,...,r2. (27)
Theorem 4. Assume that condition (A), the agreement conditions (25) and (10), and the well-posedness conditions (26) hold and
fo E Lp(Q), fi E L^(0,T; Lp(G)), U3(x) E W^(G) (j = 0,1),
V, Vt, Vtt E Lp (0, T; W2-1/p (G)), ai E W^(0,T ),i = 1, 2,...,T2, p > n.
Then there exists a constant yo > 0 such that there exists a unique solution (U,c1,... ,cr2) to (1) - (3), (24) for t E [0, Yo] such that
U,Ut E C ([0, Yo]; W2(G)), Utt E Lp (0,Yo; W?(G)), c() E Lp (0,Yo) (i = 1, 2,...,T2). Proof. Let $ be a solution to the problem (see Theorem 2)
Lo$tt = fo, $|t=o = Uo, $t|t=o = U1, $|s = V.
We have that $, $t E C([0, T]; Wp2(G)), $tt E Lp(0, T; W£(G)). Without loss of generality we can assume that (26) is fulfilled with this function $ Then the function V = U — $ is a solution to the problem
ro ri r2
LoVtt + LV + L2V = £ ci(t)fi(x,t) — £ ci(t)Li$t — £ ci(t)Li$, (28)
i=1 i=ro+1 i=ri+1
V|t=o = 0, VIs = 0, V(Xj,t) = a(t) - Ф(х3,t) = à3(t).
(29)
We have that $ G W2(G; W2(0,T)) and thus $ G C1 (G; W2(0,T)) after possible modification on a set of zero measure (see, for instance, (5.4) in [25]). In particular, $(xi, •) G Wj2(0,T ) and there by àj G Wj2(0,T ) (j = 1, 2r2). Inverti ng Lo in (28), we arrive at the equality
ro
ri
Г2
Vtt+L-l(Lldt+L2)V = Y, cl(t)L^1fl(x,t)- Ф)Ц1Пф- £ ф)Ь-1ПФ. (30)
i=1
i=ro+1
i=ri+1
Note that the traces L0 Li$t(xj ,t) (i = ro + 1,..., r^, L0 Li$(xj ,t) (i = T1 + 1,... ,r2) are defined for almost al t and every of these functions belongs to L^(0,T). Indeed, we have L-1Li$t E Wj2(G; Lp(0,T)) and thus there exists the trace [25, (5.4)], L-1Li$t(xj, •) E Lp(0,T). For almost all t, we have
L^L^t (Xj ,t)I < c\\L^t(x,t)\\bp (G) < ci№ (x,t)\\w2(G).
Therefore,
\\L-1L^t(Xj ,tWb^(0,T) < cW^tWb^(0,T ;W2(G)), i = Го + l,...,ri. 1
(31)
Similarly we can prove that L0 Li$(xj, •) E L^(0, T) (i = r1 + 1,..., r2) and L0 fi(xj, •) E L^(0,T). Take x = xj in (30). In view of (24), we obtain that
ri
r2
àjtt + L-\Lo1dt + Lo2)V (Xj ,t)+ E Ci(t)L-LiVt(xj ,t) + E cl(t)L-xLiV (xj ,t)
i=ro+1
i=ri + 1
ro ri r2
E Ci(t)L--1 fi(xj,t) - E Ci(t)L-1 L^t(Xj,t) - E cl(t)^-1Li^(Xj,t),
i=1 i=ro+1 i=ri+1
where j = 1, 2,.., r2. Equalities (32) can be written in the form
( /3i + Ri (c) \ ( $i \
(32)
Be
в
V Pr2 + Rr2 (c), j
Pi = àitt,
V Pr2
where Rk ( c)
ri
L-1(Lo1dt + Lo2)V(xk,t) + E ci(t)L-1LiVt(xk,t) +
i=ro+1
r2
E ci(t)L0 LiV(xk,t). Thus, we arrive at the operator equation
i=ri+1
i в1 + R1( c) \
c = c = B
1
R( c).
(33)
\ Pr2 + Rr2 (c),
The right-hand side is an operator R(c) taking the vector c(t) into the right-hand side of (33), with V(x, t) a solution to (28). Study the properties of this operator. Fix Ro = 2\\B~l[\\Lp(o,T). By construction, aitt E Lp(0,T). Since the entries of the matrix B belong
to L^(0,T), we have B 1[ G Lp(0,T). Theorem 2 implies that, for every vector-function c(t) G Br0 = { c G Lp(0, T) : ||o\\lp(o,t) < Ro}, the problem
ro ri r2
LoVtt + (Ld + L2)V =£ Ci(t)fi(x,t) - E Ci(t)L^t - E ct(t)L^,
i=1 i=ro+1 i=ri+1 (34)
V |t=o = 0, Vt\t=o = 0, V\s = 0, has a unique solution. For X> 0, we have the estimates
1 ( fT \1/p
sup e-xt\V(t,x)lW2{G) I \Vt\pW2{G)e-Xtpd^ , 1/p +1/q = 1. (35)
In view of (35) and (34), we infer
\\Vttlw2(G)e-xt\\Lp{o,T) < c(\l\l(Lidt + L2)Vhp{G)e-xthp{o,T) +
ro ri r2
+ lll\E Ci(t)fi(x,t) - E Ci(t)Li^t - E Cirn^hp{G)e-xthp{o,T)) < (36)
i=1 i=ro+1 i=ri+1 ^ '
< (lC^\Lp(0 ,T ) + 1)llVtte-Xt\\Lp (0, T ;Wp2(G)) + c1\\c\Lp(0,T),
where the constant c0 is independent of A. Choosing sufficiently large A > 0 and estimating the quantity \\c\\Lp(0,T) by R0, we obtain that
\\ Vtt \\ Lp(0 ,T ;Wp2(G)) + |Vt\Lp(0 ,T;W2(G)) + \\V \Lp(o,T;W2(G)) < c2 \\ c\\Lp(0, T ) < c(R0), (37)
where c(R0) is independent of c G BRo. Proceed with further estimates. Let c1,c2 G BRo and let V1, V2 be respective solutions to (34). Thus, we infer
LoVtt + (Lo1dt + Lo2)Vi + Er=ro+1 j (t)Lj Vti + E^i+1 j (t)Lj Vi =
= j cij(t)fi(x, t) -Ej=r0+1 cij(t)Lj*t -Y,?=ri+1 cj(t)Ljm
Note that the functions Vi satisfy (37). Subtracting equalities (38) for i = 1, 2 and denoting u = V1 - V2, we derive that
ri r2
Loutt + (Lo1dt + Lo2)u + E c1(t)Lj ut + E c1(t)Lj u =
j=ro+1 j=ri+1
ro ri
= E(j - c2)(t)fi(x,t) - E (c1 - cj)(t)LV2- (39)
j=1 j=ro+1
r2 ri r2
- E (c1 - cj)(t)LV2 - E (c1 - cj)(t)L*t - E (c1 - cj)(t)L
j=ri + 1 j=ro+1 j=ri+1
Inequality (37) and the previous arguments validate the estimate
\utt\Lp(0,T;W2(G)) + \u\Lp(0,T;W2(G)) + \ut\Lp(0,T;W2(G)) < c2||c1 - c2\Lp(0,T)- (40)
The arguments of Theorem 2 imply the estimate
\utt\Lp(0,r,W-2(G)) + \ut\Lp(0,r,W22(G)) + \u\Lp(0,T,W-2(G)) < c3\cl - c2\Lp(0,1) (41)
valid for all 7 G [0,Т]. The corresponding estimate of the form (37) is also true with a constant c3 independent of 7. The estimate is valid for all c(t) G BRo,Y = {c(t) : IIc\\Lp(0,Y)<Ro}■ Proceed with the estimates for the operator R. The estimate (41) yields
Г2
\\R(cl) - R(c2)\\lp(o,y) < cE \\L- (Loidt + Lo2)(V1 - V2)(Xj,t) +
j=i
ri ri Г2
+ E ciL^iLiVt1(xJ,t) - E c2L-lL%2(Xj,t) + E clL-lUV l(Xj,t)-
i=ro+l i=ro+l i=ri+i
Г2 r2 ri
- E c2L-iLiV2(x3,t)IlLpio,Y) <E(II E c - c2)(t)L-1 LiVi(Xj,t) + (42)
i=ri+1 j=i i=ro+i
r2 ri
+ E c - c2)(t)L-iLiV l(Xj ,t)IlLp(0,Y) + IE c2 (L-lLi(Vi - V2)(Xj ,t)) +
i=ri+l i=ro+l
r2
+ E (c2L^lLi(V1 - V2)(Xj ,t))\Lp(0Y))+ c4\\cl - c2 || Lp(0,Y) ■
i=ri +l
Estimate the summands on the right-hand side as follows:
\l(cl - c2)L-lLiVt\Xj,t)\\Lp(or() < ic - c2llLp(0r()I\L-lLiVtl(XjMl^),
||(cl - c2)L-lLiVl(Xj,t)I\Lp(0,Y) < ic - c2lLp(0,Y)\L^lLiVl(Xj,t)\\LU0,Y)■ The embedding theorems and Theorem 1 imply that
\L^lLiVt1(Xj ,t)\L^(0,Y) < c\Vt1(X,t)\L^(0Yr,W2(O)), < cryl/q\\Vtl\\lp{0,y;W2p (G)) (l + p = 1).
\L-lLiV l(Xj ,t)\L^(0,Y) < c\\V l(X,t)I\LU0,Y;Wt2(G)), < cryl/q\Vt1\Lp(0,Y;W2 (G)) (± + p = 1). Hence, we have the estimates
lic - c2)L-lUVt\Xj,t)I\Lp(0,Y) < clYl/ql\cl - ¿\\Lp{o,Yb (43)
lic - c2)L-lLiVl(Xj,t)\\Lp(0,y) < clYl/ql\cl - ¿\\Lp{o,Yb (44)
where the constant cl depends on R0 and is independent of 7. Similarly, employing (41) we obtain that
\\c2(L-lLi(Vt1(Xj,t) - V2(Xj,t)))\\Lp(o,y) < Roc\\Vtl(x,t) - V2(xMl^^g)) <
< RocYl/q V - v2t \l Lp(0,Y;Wp2(G)) < clYl/q l\cl - C2\Lp(0,Y).
(45)
I\c2(L-1 Li(Vl(Xj,t) - V2(xj,t)))\\Lp(0,Y) < Roc\\Vl(x,t) - V2(xMl^^g)) <
< RocYl/q V - V2\\lp(0 ,Y;Wp2(G)) < clYl/q l\cl - ¿\\lp(0 y).
(46)
In accordance with (42) - (46), we conclude that
IlRc) - R(c2)\\Lp(0,Y) < cyl/ql\cl - C2\ lLp(0,Y). (47)
Choose y0 : cY^/q = 1/2. In this case for cl,c2 G BRo,Y with y < Y0 we have inequality (47). Moreover, R(c) = R(c) - R(0) + R(0) and
I\R(c)hq(0,y) < I|R(0)||L,(0 , y) + 2IIciL(0,y) < Y + Y = Ro.
Thus, R takes the ball BR0r(0 into itself and contractive. The fixed point theorem ensures the solvability of (34). The facts that V is a solution to (28) and satisfies conditions (29) can be proven as in the linear case.
□
The authors were supported by RFBR (Grant no. 15-01-06582). References
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Received March 3, 2016
УДК 517.95 DOI: 10.14529/mmpl60207
ОБРАТНЫЕ ЗАДАЧИ ДЛЯ МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ СОБОЛЕВСКОГО ТИПА
С. Г. Пятков, С.Н. Шергин
Работа посвящена рассмотрению обратных задач для ряда математических моделей, основанных на уравнениях и системах Соболевского типа, возникающих в динамике стратифицированной жидкости, теории упругости, гидродинамике, электродинамике и других областях. Вместе с решением определяются неизвестная правой части и коэффициенты в уравнении типа Соболева четвертого порядка. В качестве условия переопределения берутся значения решения в отдельных точках пространственной области. Задача сводится к некоторому операторному уравнению, разрешимость которого устанавливается при помощи априорных оценок и теоремы о неподвижной точке.
Доказываются теоремы о существовании и единственности решения поставленной задачи для линейного и нелинейного случая. В линейном случае результат является глобальным по времени, а в нелинейном локальным по времени. В качестве основных пространств рассматриваются пространства С.Л. Соболева.
Ключевые слова: модели соболевского типа; уравнение Соболева; математическая модель; теорема существования и единственности решения; обратная задача; краевая задача; волны в плазме; вращающаяся жидкость; модель Буссинеска - Лява.
Литература
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10. Кожанов, А.И. О разрешимости коэффициентных обратных задач для некоторых уравнений соболевского типа / А.И. Кожанов // Научные ведомости Белгородского государственного университета. Математика. Физика. - 2010. - Т. 18, № 5. - С. 88-98.
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Сергей Григорьевич Пятков, доктор физико-математических наук, профессор, кафедра «Высшая математика:», Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), S_pyatkov@ugrasu.ru.
Сергей Николаевич Шергин, аспирант, кафедра «Высшая математика», Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), ssn@ugrasu.ru.
Поступила в редакцию 3 марта 2016 г.
