On some mathematical models of filtration theory

Together with a solution U, we determine an unknown right-hand side and coefficients of the equation

LUt + MU = f, (x,t) G Q = G x (0,T),(1)

where L,M are the second-order differential operators in the variables x and G is a bounded domain in R ⁿ ( n >  1) with boundary Г G C 2. The equation is complemented with the initial and boundary conditions

U\s = Ф, S = Г x (0,T),(2)

U\t=o = Uo (x).(3)

We employ the values of a solution U at separate points as overdetermination conditions. So, our overdetermination conditions are as follows:

U(xi,t) = ai(t),     (i = 1, 2,..,r),(1)

with xi being arbitnary points in G.

Mathematical models based on the pseudoparabolic equations arise when describing heat and mass transfer, filtration, wave processes, and many other processes [1, 2]. Many articles are devoted to the study of solvability of boundary value problems for pseudoparabolic equations (see, for instance, [3, 4]). In particular, initial, initial-boundary, and periodic problems, the questions of global (in time) solvability and blow-up of solutions are studied. In the case of global solvability some investigations are devoted to the questions of asymptotic behavior of solutions to the direct problems, scattering theory, and stability of soliton-type solutions to both

S.G. Pyatkov, S.N. Shergin one-dimensional and multidimensional equations, in particular, to the Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations [5, 6]. The semigroup approach to singular Sobolev-type equations was developed by Sviridyuk and Fedorov [4]. Some results devoted to the theory of pseudoparabolic equations with an indefinite or noninvertible operator at the higher-order derivative with respect to time can be found in [7]. The questions of local solvability for nonlinear pseudoparabolic equations are examined in [8].

The pseudoparabolic equations with a monotone nonlinearity are studied by R.E. Showalter in [9], where the classical monotonicity method in an expanded form was applied to various classes of equations of mathematical physics, in particular, to nonlinear Sobolev-type equations with monotone nonlinearities. The existence of global (in time) solutions for the Boussinesq equation with a source and finite-time blow-up solutions were studied by Kozhanov in [10]. In his articles the blow-up of solutions to the first boundary value problem is proven with the use of the comparison principle for these equations. In particular, the blow-up of a positive solution is proven and some existence and nonexistence theorems are exhibited.

The question of uniqueness of solutions to the Cauchy problem for the quasilinear pseudoparabolic equation ut = cAut + ф (u)

is studied in [11] in a class of growing functions ф ( u ), with u from some well-posedness class. The maximum principle for pseudoparabolic equations is also presented in [12]. The method of the proof of nonexistence of solutions to some boundary value problems relying on the maximum principle is developed in the articles by Yu.V. Egorov and V.A. Kondratev. The methods of complex analysis are employed in the study of pseudoparabolic equations in [13]. A principally new approach called the method of test functions was proposed in [14, 15]. The existence and nonexistence questions for different mathematical models on the base of Sobolev-type equations are presented in the well-known monograph [2], where the necessary bibliography can be found.

The inverse problems for Sobolev-type equations are not studied well. In [1, 16] the authors consider a model that discribes filtration of a fluid in a fractured media. They state some inverse problems, including a problem close to that of out article, and establish local existence and uniqueness theorems under an integral overdetermination condition on the boundary. Moreover, some properties of solutions to inverse problems of this type are derived there. The problem of recovering of a kernel of an integral operator occurring in Sobolev-type equations when some functional of a solution is given is examined in [17]. The coefficient inverse problems are considered in [18]. The uniqueness theorem is proven and an algorithm for solving an inverse problem for the equation ut - A ut = a Au + b (y) uy + c (y) + 6 (t, x, y), (x, y) G R2, t > 0, is specified. The unknowns are u(t,x), b(y), c(y), and a constant a and the functions u (t,x, 0) ,uy (t,x, 0), a nd u (0 ,x,y) are given. Here 6 (t,x,y) is the Dirac delta function.

One more problem is considered in [19], where the value of a functional of a solution allows us to recover a scalar function depending on t which is a factor before an element of a given Banach space on the right-hand side of an abstract Sobolev-type equation. A similar problem is treated in [20], where an element of a Banach space is recovered under the integral overdetermination condition. We can refer also to [21], where some inverse problems for composite type equations are considered.

Point out a series of monographs and articles [22-26], where essential advancements in the theory of inverse problems for parabolic equations and systems are made. In particular, the inverse problems in the same statement but for parabolic equations and systems are treated in [27-29].

The main result of the article is Theorem 3, where existence and uniqueness of a solution to the problem (1) - (4) are proven and a stability estimate is obtained. The problem of determination

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ a right-hand side (a source function) is linear and the result in this case is global in time (see [30]).

1.    Preliminaries

In the article we employ the Sobolev spaces WpS ( G ) and the Ho Ider spaces C^a ( G ). The space of strongly measurable functions on [0 , T ] with values in a Banach space H is denoted by L_p (0 ,T ; H ). The condition Г G C ² m means that, for every x о G Г, there exists a neighborhood U (the coordinate neighborhood) and a coordinate system y (local coordinate system), obtained after a rotation and translation of the initial system in which

U П G = {y G R n : y‘ G Br,ш ( y‘ ) < yn < ш ( y‘ ) + 6},

U П (R n \ G ) = {y G R n : ш ( y‘ ) - 6 < y_n< ш ( y‘ ) },

Г n U = {y G Rn : y‘ G Br., yn = ш(y‘)}, where y‘ = (y 1 ,y 2 ,...,yn-1). Br = {y' : \y'\ < r}. 6 > 0 is a cons taut, and ш G C2 m (Br). Without loss of generality, we can assume that the axis yn of the local coordinate system is directed along the normal to Г at xо.

Let L and M be second-order operators of the form:

LU = £ aij(x,t)UxiXj + £ ai(x,t)Uxi + ao(x,t)U, i,j=1

MU = £ bij ( x,t ) Ux i X j + £1 bi ( x,t ) U + b 0( x,t ) U.

Here L is assumed to be elliptic. Tims, there exists a constant 6 >  0 such that

m

E aij'' > 6о^\2 V e G Rn, V (x, t) G Q.(5)

Write out the conditions on the coefficients of L, M. Fix a parameter p > n and assume that aij G C (Q), ai,a о G C ([0 ,T ]; Lp (G)) (i,j = 1, 2 ,..,n), ao(x, t) < 0 a.e. in Q, bij G Lp(0,T;L.(G)),bi,bо G Lp(Q) (i,j = 1,2,..,n).(7)

Under these conditions on the coefficients of L, the following theorem is valid.

Theorem 1. For every f G L_p ( Q ), the Dirichlet problem

Lu = f,   u\г = 0,(S)

has a unique solution u G Lp(0,T; Wp2(G)) satisfying the estimate llullw2(G) < cllfIIlp(G) almost everywhere, on (0. T).

where c is a constant independent of f and t.

Proof. The solvability of problems (8) (depending on a parameter t ) results from the uniqueness of solutions (see the maximum principle in [31, Chap. 9]) and the Fredholm property of these

S.G. Pyatkov, S.N. Shergin problems. An estimate for solutions can be justified with the use of the continuity of coefficients in t; indeed, in some neighborhood of every point 10 G [0,T] there is an estimate of the form ||u||w2(G) < c^Lu^Lp(G) with a constant c independent of t. These estimate ensure the presence of a global estimate which is claimed.

□ Lemma 1. If b G Lq ( G ) with q > n for p < n and q > p for p > n then there exists a constant c >  0 such that for all functions u G Wp2(G ) the following estimate is valid:

Il bVU I Lp ( G ) < C^b^Lq ( G )|| U^IW2(G ) .                                  (9)

If b G Lq ( G ) with q > n/ 2 firr p < n/ 2 a:id q > p for p > n/ 2 then there extsts a constant c >  0 such that, for all functions u G Wp2(G ), we have

^IIbUHLp ( G ) < c^Hb^HL_q ( G ) ^HU I W A G ) .                                (ДО)

Proof. Prove the former statement, the latter is established by analogy. Fix i = 1 , 2,... ,n. The Holder inequality yields

IbUx i IL ( G ) = ( J |b|pUx i Ipdx )1 ^/P<  ( J bqdx ) q ( J lUx i I - dx ) q - ■

G

G

G

By the embedding theorems (see [34]), we have the estimate

^IIUX i II l qp ( G ) < ^cIU'W(G ) , q>ⁿ, p < n. q - p

Thus

^IbUX i ^HL p ( G ) < cWU^HW2(G ) , ^c = Il b l L q ( G ) .

The estimate (10) is obtained similarly.

□

Let Q^Y = G x (0 ,y ).

Theorem 2. (solvability of the direct problem). Assume that f G L_p ( Q ), U o( x ) G Wp2 ( G ) and ф_t G L_p (0 ,T ; Wp ^{1 /p}(G )) , where p G (n, ж )₇ and conditions (6), (7) for the coefficients hold together with the consistency conditions

Ф ( x, 0) | г = U o( x ) | г .

(И)

Then there exists a unique solution to problem (1) - (3) such that

U,Ut G L_p (0 ,T ; W_p ²( G )) , U ( t ) G C ([0 ,T ]; W _p²( G )) .

If ф = 0 , Uo ( x ) = 0 , then there exists a constant c >  0 independent of g G [0 ,T ] such that a solution to problem (1) - (3) meets the estimate

^IIU H l^ (0 ,Y ; W2( G )) + ll ^Ut^HL p (0 ,Y ; Wf( G )) < cf Il L p ( Q Y ) .

Proof. Consider the segment [0 ,T ]. Find a function Ф such that

Ф t G Lp (0 ,T ; Wp ²( G )): Ф |_s = Ф, Ф It =o = U o( x ) .

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

It can be constructed as follows:

t

Ф = -0 + U 0 , 0 = У - 0 o( x, T ) dT, △ 0 0 = 0 ,   0 0 1 г = фt

(the existence of a function 0 o follows from the known results (see, for instance, [31]). Make the change V = U — Ф. Function V satisfies the conditions

LVt + MV = f — L Ф t — M Ф = g, V I s = 0 ,   V It =o =0 .

Using Theorem 1 we can obtain that

t

V + У L- ¹ MV ( T,x ) dT = L- ¹ g,

Conditions (6) and Lemma 1 justify the estimate ( t < T )

t

I I У ^{L 1 MV dT} ^C ([0 ,T ]; W2( G )) < ^{CT 1} /q ^^V li e ([0 ,t ]; W2( G )) ,                     (12)

where 1 /p + 1 /q = 1, which together with the fixed point theorem allows us to prove the claim on solvability on some time interval [0 ,t 0] (ct J ^/q<  1). The global theorem results by repetition of the arguments at the segment [ t 0 , 2 t 0], [2 t 0 , 3 t 0], etc. It is easy to see that a solution meets the estimate from the theorem.

Consider the interval (0,7) (y < T) and prove the estimate of the theorem. Consider problem (1) - (3). (4). where U0 = 0, ф = 0. and f0={f;   t< Y     G Lp(Q’■

There exists a unique solution U to problem (1) - (3) satisfying the estimate:

II U II l^ (0 ,T ; W2( G )) + ll ^Ut ^HL p (0 ,T ; W'²( G )) < cf 0 1 1 L p ( Q ) = cf 1 1 L p ( Q^Y ) ■              (13)

By the uniqueness theorem, a solution U agrees with a solution U to the problem (1) - (3), with U0 = 0 and ф = 0 on [0 ,Y ]• Thus, we can rewrite (13) as follows

II U H l^ (0 ,Y ; W^(G )) + hUt^L p (0 ,Y ; W^(G )) < c^Hf b p ( Q y ) ■

□

Remark. We can easily formulate an analogue of this theorem for any p G (1 , от ). But to obtain the main results we need a condition p > n for p. Hence, we state the above theorems in this case.

2.    The Main Results

We consider the inverse problem of determining the unknown functions occurring in the righthand side of the equation, and in the operator itself. In this case the problem is nonlinear. We assume that the right-hand side of (1) is represented as f = ji^ Ci (t) fi (x,t) + f0(x,t), fi G L. (0 ,T; Lp (G))     (i = 1, 2 ,..,r 0),         (14)

i =1

S.G. Pyatkov, S.N. Shergin where the functions fi are given. We suppose also that some coefficients of the operator M depending on t are unknown arid operator M has the form

MU = Mr 0 U + £ Ck ( t ) Mk U, k = r о + 1

MkU = E bj ( x,t ) Ux i X j + E bk ( x,t ) Ux i + b^kk ( x,t ) U. i,j =1                    i =1

Consistency conditions:

ai (0) = U o ( xi,t ) , ( i = 1 , 2 ,..,r ) .                                  (15)

Construct the function Ф G C ([0 ,T ]; W _p( G )) ( p > n ) such that Ф _t G L_p (0 ,T ; W _p( G )). Ф |_t =o = U o( x )■ Ф | s = ф Gee the proof of theorem 2). Construct also the matrix B with the rows

L-1 f1( xj ,t) ,L-1 f2( xj ,t), ...,L-1 fr о (xj ,t), -L-1 Mr о+1Ф( Xj ,t), -, —L—1 Mr Ф( Xj ,t), where j = 1, 2, ...,r. arid assume that there exists So > 0 such that

| det B| > So a.e. on [0,T].(1G)

Here L-1 fi is a solution Ui to the jproblem LUi = fi. Uilt=o = 0. Uils = 0. We assume that the coefficients of the operator nn

LU = ^ aij(x, t)UxiXj + ^ ai(x, t)Uxi + ao(x, t)U, i,j=1

meet conditions (6) and the coefficients of Mi the conditions bj G L. (Q), bj ,bk G L. (0 ,T; Lp (G)) (i,j = 1, 2 ,..,n,k = ro ,...,r).(17)

In this case, locally in time, conditions (16) does not depend on the choice of function Ф. Indeed, let Ф1 and Ф2 be such functions. Consider lL-1 MiФ1(xj,t) - L-1 MiФ2(xj,t)| < c\\MiФ1(x,t) - MiФ2(x,t)^Lp(g) <

< ^c\\ ^Ф1^{( x,} t ) ^{- Ф}2⁽ x, t ) || w_p ²( G ) < ^{ct 1} /q | | ^Ф1 1 ^{- Ф}2 t\L p (o ,T ; W²( G )) •

This inequality shows that if B 1 and B ₂ are matrices constructed using Ф1 and Ф₂, respectively, arid | det B 1 1 > S o >  0. then we can find y o >  0 such tliat. for t G [0 , y o]- | det B ₂ 1 > S o / 2 >  0 and thus on the interval [0 , у o] | det B_i| > S o / 2 >  0 fc>r i = 1 , 2.

Theorem 3. Let conditions (5), (6), (15) - (17) be fulfilled. Then there exists a constant у o >  0 such that on the interval [0 ,y o] problem (1) - (4) has a unique solution ( U,c 1 ,...,c_r ) such that

U G C ([0 ,y o]; Wp 2( G )) ,  Ut G Lp ([0 ,y o]; W_p ²( G )) ,  ci ( t ) G L_p (0 ,y o) ( i = 1 , 2 ,...,r ) ,

Proof. Let

Ф G C ([0 ,T ]; W _p²( G )) : Ф _t G Lp ([0 ,T ]; W _p²( G ))

be a solution to the problem (see Theorem 2)

L Ф t + Mr о Ф = f o ,  Ф |t =o = U o ,  Ф | s = ф.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

In this case the function V = U — Ф is a solution to the problem

LV + MV = £ a(t)fi(x,t) — £ ci(t)MiФ, VIt=0 = 0, VIs = 0.(18)

i =1                  i = r o+1

There exists y 0 >  0 such that condition (1G). with S о / 2 rather■ than S o- dole Is on [0 ,y о ]■ Note that U is solution to problem (1) - (4). In this case

• V(Xj, t) = aj(t) — Ф(Xj,t) = aj(t).(19 )

The function Ф G Wp2(G; Wp(0, T)) after a possible modification on a set of zero measure possesses the property Ф G Ca(G; Wp(0,T)). a < 2—n (see (5.4) in [35]). In particular. Ф(xi,t) G Wp1 (0,T), and thus a j (t) G W(0,T) (j = 1, 2 ,..,r).(20)

Inverting L in (18). we have

Vt + L-1 MV = ]Tai(t)L-1 fi(x,t) — £ ci(t)L-1 MiФ.(21)

i =1                       i = r o+1

Note that L- ¹ Mi Ф( Xj ,t ) ar id L- ¹ Mi Ф( Xj ,t ) G L^ (0 ,T ) for almlost all t. Actually. L- ¹ Mi Ф( x,t ) G Wp( G ; L_p (0 ,T )) and tl iereby L- ¹ Mi Ф( Xj ,t ) G L_p (0 ,T ) (see [35. (5.4)]). In this case we have

I L^{- 1} Mi ^{Ф( X}j,t ) \\ l _x (0 ,T ) < \\^Mi ^{Ф( X}j,t ) ^L_M (0 ,T ; L p ( G )) <

c] ^{Ф( X},t ) \ l^ (0 ,T ; W p 2(G )) < ^c\ ^Ф \ l^ (0 ,T ; W_p ²( G )) ■

Thus,

I L ¹ Mi ^{Ф( X}j ,t ) W l^ (0 ,T ) < ^c\\ ^Ф He ([0 ,T ]; W2( G )) ■                          (22)

Similarly, we can prove that L- ¹ fi ( Xj,t ) G L^ (0 ,T ) and this trace is defined. Let x = Xj in (21). We have

Vt ( Xj ,t )+ L- ¹ MV ( Xj ,t ) = ^£ a ( t ) L- ¹ fi ( Xj ,t ) — ^£ a ( t ) L- ¹ Mi Ф( Xj ,t ) . i =1                       i = r o+1

The overdetermination condition yields ajt + L-1 Mr 0 V (Xj ,t)+ £ Ci (t) L-1 MiV (Xj ,t ) = £ Ci (t) L-1 fi (Xj ,t) — i=r o+1                           i=1

r                                                                         (23)

• — £ Ci ( t ) L ¹ Mi Ф( Xj ,t )       ( j = 1 , 2 ,..,r ) .

i = r o+1

We can rewrite equation (23) in the form a11 + L 1 Mr0V(x 1 ,t)+ £ ci(t)L 1 MiV(x 1 ,t) i=r o+1

Be =

art + L- ¹ Mr o V ( Xr ,t )+ £ Ci ( t ) L- ¹ Mi V ( Xr ,t ) i = r o+1

S.G. Pyatkov, S.N. Shergin

As a result, we infer

« 1 1 + L ¹ Mr о V ( x i ,t ) + £ c ( t ) L ¹ MiV ( x i ,t ) i = r о+1

c = B

= R (c).

к

art + L- ¹ Mr о V (Xr ,t )+ E Ci (t ) L- ¹ Mi V (Xr ,t ) i = r о+1

/

The right-hand side can be viewed as an operator R(c) taking the vector c ( t ) in a solution V ( x,t ) to problem (18) and then taking V ( x,t ) into the right-hand side of (25). Investigate properties of this operator. Fix

( « 1 1 \

I •

«rt /

By construction, a it G Lp (0 ,T ). Due to the fact that tl ie entries of the matrix B belong to L^ (0 ,T ), we have that B- ¹ a G Lp (0 ,T ). By Theorem 2, for every vector-function

C(t) G BRо,Y = {c G Lp(0, Y) : IlcL(0,Y) < R0}, Y < T, the problem r0                        r

LVt + MV = £c. ( t ) fi ( x,t ) -  E   ci ( t ) L^{- 1} Mi Ф ,  VIt =o = 0 ,  V I s = 0 ,       (2G)

i =1                  i = r о+1

is uniquely solvable. For X > 0, we have that sup e-X\\V (t.x) |\lVp2( G) < XL (Y I\Vi\\WYG) e-^pdt) ‘ 'p .              ria

From (26) it follows that ll\Vt\wp2(G)e ^Wbp(0,y) < c(\\MV\wp2(G)e ^Wbp(0,y) + l\ll E ci(t)fi(x.t) - E c.(t)L-1 MiФUl,(G)e-X‘«Lr(0,Y)) < i=1                i=r о+1

< X/(\cllLp(0,Y) + !)\Vte-Xt\Lp(0,Y;wphg))) + c 1II cL(0,y), where constant c0 is independent of X > 0, y < T- Choosing a sufficiently large X > 0 and estimating || c\lp (0 ,Y) thro ugh R 0 we obtain

\^Vt\L p (0 ,y ; W_p ²( G )) + \^V\L p (0 ,y ; W_p ²( G )) < ^c 2 II cL (0 ,y ) < ^{c ( R} 0) ,                  (29)

where c ( R 0) is independent of c G B r 0,_Y. Obtain the remaining estimates. Let c ¹ , c ² be a vectorfunction from Lp (0 , y ) and V ¹ , V ² be corresponding solutions to (26). Thus, we infer

	r                        r 0                          r LVti + Mr 0 Vi + £ cj ( t ) MjVi = £ cj ( t ) fj ( x,t ) - £ cj ( t ) Mj Ф .        (30) j = r 0+1                j =1                 j = r 0+1
П2	Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 2, pp. 105-116

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Note that the functions Vi satisfy estimate (29). Subtracting equations (30) for i = 1 , 2 and denoting ш = V ¹ — V ², we arrive at the relation

L* + Mr о ш + 52 jt )( Mj ш )= £ ( cj — c ²)( t ) fi ( x,t ) — j = r о + 1                 j =1

— ^£ ( cj — c2)(t)MjV² - £ ( c 1 - j)(t)Mj Ф .

j = r о + 1                        j = r о + 1

Inequality (29) and the previous arguments yield

⃗⃗

IHl L p (0 ,Y ; W p2( G )) + I M I l _p (0 ,Y ; W_p ²( G )) < ^c 2ll ^{c 1 — c 2} II l _p (0 ,y ) •                    (°^

Also we have estimate (29), where the constant c 3 is independent of 7 . The estimate holds for all c ( t ) G B r ₀ ,7. Proceed with the estimate for the operator R. In view of (32) we have that

I I R ( c 1) — R ( c ²) lL p (0 ,Y ) <

• < c ^£ WL- ¹ Mr о ( V ¹ — V ²)( Xj ,t ) + ^£ ci L^{- 1} MiV 1( Xj ,t ) — c2 L^{- 1} MiV ²( Xj ,t ) H l _p (0 ₎₇ ) < j = 1                                    i = r о + 1

• < ^£ ( W ^£ ( c .¹ — c 2)( t ) L^{- 1} MiV ¹( Xj,t)WL. (0 ,Y ) +

j = 1 i = r о + 1

+ ^Wc ²⁽ L^{- 1} MiV ^{1 (} xj ,t ) ^— L^{- 1} Mi V ²⁽ xj ,t )) ^WL p (0 ,Y )) + ^c 4 ^Wc 1 ^— c ^{2 W}L p (0 ,Y ) ■

We estimate the first summands on the right-hand side as follows:

^{W ( c}i ^— ci) L^{- 1} MiV ¹⁽ xj ,t ) ^WL p (0 ,Y ) < ^Wci ^{1 — c}i ^WL p (0 ,Y ) ^WL^{- 1} MiV ¹⁽ xj ,t ) I L~ (0 ,y ) ■

By the embedding theorems and Theorem 1, we derive that

WL-1 MiV 1(Xj,t)Hl^(0,y) < WMiV1 Hlx(0,y;Lp(G)) < cWV1(X,t) IIl^(0,Y;Wp2(G)) < cY 1 /q IIVt1 WLp(0,Y;Wp2(G)) ( 1 + p = 1) ■

Next, we have the estimate

W(c1 — c2)L-1 MiV 1(Xj,t)IL(0,Y) < c 1Y1 /qWc1 — c2IL(0,Y),(34)

where constant c 1 depen ds on R 0, but independent of 7 . Similarly, we justify the inequality

Il c K ^{L 1} Mi ⁽ V ^{1( X}j,t ) ^— V ^{2( X}j,tL ^WL q (0 ,Y ) < ^R 0 ^WV ^{1( X,t} ) ^— V ^{2( X,t} ) II l^ (0 ,Y ; W ²( G )) <

_(35)

< ^R 0 ^c7 ^{1 /q} II Vt^{1 —} Vt^{2 W}L p (0 ,Y ; W2( G )) < ^c 1 7 ^{1 /q W}c ^{1 — c 2 W}L p (0 ,y ) ■

From (34), (35), and (33) it follows that

WR(c 1) — R(c2)WLp(0,Y) < c71 /qllc1 — c2Wl,(0,Y).061

Choose 70 so that c7 01 /q = 1 / 2.

c , c ∈ BR _{0 ,γ}

In this case, for all

пз

S.G. Pyatkov, S.N. Shergin with y < Y0; we have inequality (36). Moreover,

R(?) = R(?) - R (0) + R (0)

«R(C)\L (0 ,Y ) < «R (0) »Lq (0 ,Y ) + 2 M kq (o ,y ) < R ⁰ + R = R o -

Thus, R takes the ball B r 0,_Y0 into itself and is contractive. By the fixed-point theorem, equation (26) is solvable. By construction, V isa solution to (18). The fact that V meets (19) is proven by analogy with arguments those of in the linear case (see [30]).

□

Conclusion. Thus, we proved that the inverse problems under consideration are well-posed at least locally in time. The results obtained allow to construct new numerical algorithms for solving problems of form (1) - (4).

The authors were supported by the Russian Foundation for Basic Research (Grant 15-0106582).