Parameter identification and control in heat transfer processes

We study the problem of recovering a lower order coefficient depending on time together with a solution in heat transfer mathematical models. This control parameter allows to ensure a given temperature distribution at a given point of spatial domain. Let G be a bounded domain in Rn with benmdary Г arid Q = (0, T) x G. The mathematical model can be written as ut - Lоu + p(t)u = f (x,t), (t,x) G Q,

nn

L о u = A о u + B о u,A о u = ^ d x i ( a ij ( x ) U x j ) , -B о u = ^^ i ( x ) U xi + a o( x ) u. i,j =1                                  i =1

Equation (1) is furnished with the initial and boundary conditions u|t=о = uо, Bu\s = g(t, x), S = (0, T) x Г, where                       n

Bu = a(^2 aij (x)niuxj + b(x)u) + (1 — a)u, i,j=1

ni is the i-tli coordinate of the unit outward normal to Г. a(x) G C(Г) is a. continuous function taking two values 0, 1. Thus, on different connectedness components the boundary condition can be of different type (Dirichlet, Neumann, or Robin boundary condition). Let Гi = {x G Г :  a(x) = i} (i = 0, 1), Si = (0,T) x Гi. The unknowns in (1), (2) are the solution u and the function p(t). The overdetermination conditions are written as u (x о ,t )= 0 (t).                                      (3)

where - 0 ( t ) is some function specified below.

Determination of a single unknown time-dependent property such as the capacity, conductivity or diffusivity from additional local or non-local measurements of the main dependent variable at the boundary or inside the space domain represents a classical example of a coefficient identification problem (see, for instance, [1,2]). Problem (1) - (3) is classical and was studied by many authors. Numerical methods of solving the problem are developed in [3-5]. For local (in time) solvability results, see, for instance, [6,7]. The book [8, Ch. 6] contains some abstract theory of such problems and its applications. In particular, the conditions for a local (in time) solvability of (1) - (3) are presented in Corollary 9.4.2 of [8]. Moreover, similar result is also exposed in [9]. Some results on close inverse problems can be found in [10-13]. Inverse problems with integral overdetermination conditions are studied in [14-18]. The structure of the paper is as follows. In Section 1 we formulate our results. The main result is Theorem 4 which is a global (in time) existence and uniqueness theorem for solutions to the problem (1) - (3). Some stability estimates are given in Theorem 3. The solvability conditions are stated in terms of some inequalities and the proof relies on the maximum principle. Section 2 is devoted to the proofs of the main results.

• 1.    Preliminaries

Given Banach spaces X, Y, the symbol L ( X, Y ) stands for the space of linear continuous operators defined on X with values in Y. Let F be a Banach space. By L p ( G ; E ) ( G is a do main in R ⁿ ) we denote the space of strongly measurable functions defined on G with values in E endowed with the norm |||| u ( x )|| E I L p ( _G ) [19]. We employ also the spaces C k ( G ) comprising functions continuous in G with all their derivatives up to order k admitting continuous extensions on the closure G. The Sobolev space notations are conventional, i.e.. W p ( G ; E ). W S ( Q ; E ). etc. (see the definitions in [19.20]). If E = C (E = R) c>r E = C n (E = R n) then the latter sqrace is denoted by W s ( Q ). Similarly, we use the notations W p ( G ) оr C ^k ( G ) rather than W s ( G ; E ) оr C ^k ( G ; E ). Thus, the membership u G W p( G ) (c>r u 6 C ^k ( G )) for a given vector-fuiiction u = ( u 1 ,u2,...,u_k ) means that every of its component u i belongs to W s ( G ) (or C ^k ( G )). The norm of the vector is just the sum of the norms of the coordinates. Given an interval J = (0 ,T ), put W s ^,r ( Q ) = W s ( J ; L p ( G )) П L p ( J ; W_p ^r ( G ). Respectively. W s,r ( S ) = W s ( J ; L p (Г)) П L p ( J ; W_p ^r (Г)).

Consider the direct problem (1), (2). In what follows we assume that G is a bounded domain in R n with boundary Г G C ² (see the definition in [21, p. 17]). Expose the conditions on the data of the problem. All spaces below and the coefficients of equation (1) are assumed to be real. Fix p > n + 2 (this condition simplifies the arguments and it cari tie weakened). Let q = p/ ( p — 1). Deiiote B_s ( x0) = {y G R ⁿ : |y — x01 < 1}.

The conditions on the coefficients of the operators L 0, B are as follows:

aij G C 1(G), ai,aо G Lp(G), b G C 1(Г).(4)

The matrix {a ij } is symmetric and the ellipticity condition n

351 > 0 : |2 aijUj > 61К|2 7^ G Rn, x G G,(5)

holds. The conditions on the data are of the form

• uo(x) G W2-1 /p(G), g(t,x) G Wp1 -1 /2p,2-1 /p(Sо) П W^/2-1 /2p,1 -1 /p(S 1),(6)

f G Lp (Q), B (0 ,x) u01 г = g (0 ,x) Vx G Г,(7)

u0(x) > 0 (x G G), g(t,x) > 0, f > 0 ((t,x) G Q).(8)

We also use some additional conditions

Д G W^(0,T), 362 > 0 : 1Д(t)| > 62 Vt G [0,T], Д(0) = uo(xo),(9)

36o > 0 : B^OM G G, aij G W^(Bs0(xo)), ai,ao G Wp(Bs0(xo)),(10)

Vuo(x) G Wp2-1 /p(Bsо(xo)).(11)

Assume that Q_s = (0 , T ) x B_s ( x _o), Q ^Y = (0 , y ) x G . Present some auxiliary statements. Replace the equation (1) with the equation

Lu = ut — Lou = f (x,t), (t,x) G Q.(12)

Theorem 1. Assume that conditions (4) - (7) hold. Then there exists a unique solution to (2), (12) such that u G W^, ²( Q ). Under the additional conditions (10), (11), a solution u possesses the property Vu G W^, ²( Q_s ) for all 6 < 6 _o. If condition (8) is valid then the function u is nonnegative in Q.

Proof. If S _o = S or S 1 = S then we can refer to the standard theorems on solvability (see, for instance, [21, Theorem 9.1, Ch. 4] in the case of the Dirichlet conditions or [21, Theorem 10.4, Ch. 8] in the case of more general boundary conditions). Examine our case. First we take homogeneous initial and boundary conditions in (2). Let D ( A ) = {u G Wp ’ 2( Q ) ^: Bul S = 0 }■ The claim in the case of L _o = A _o results from Theorem 12.2 in [22]. In the general case the result is a consequence of Theorems 12.2 and 3.2 in [22]. To refer to Theorem 3.2 we need an additional estimate for the norm || B _o l L _p (o , _{T ; L} ( _{Eg X} , _{E 0})) for some 0 G (0 , 1) and p >  max(1 / (1 — в ) ,p ), where E _o = L p ( G ) and E^ = ( E _o ,D ( A )) 3 ^ is the space obtained by the real interpolation method (see the definitions [19]). Since the operator B _o is independent of t, it suffices to establish the estimate

|Bou^Lp(G) < c^u^Ws(G), s < 2, and use the embedding Ee,^ G Ws(G) foг в > s/2 (which follows from the embedding (1.1) in [22] and Theorem 5.2 in [23]). This estimate results from the embedding theorems and conditions (4). We infer

||B o u L ( G ) < c ll u ll w T ( G ) < ^c 1 HuH w p ( G ) , ^{s G (1}+ ^n/p, ²⁾ .

The claim of the theorem in the case of nonhomogeneous initial and boundary conditions follows from the conventional theorems on extension of the boundary conditions inside the domain (see, for instance, Theorem 7.3 in [24]).

The proof of the fact that a solution u possesses the property Vu G W^ ’ ²( Q_s ) for all 6 < 6 _o under conditions (10), (11) is realized with the finite difference method with the use of Lemma 4.6 of Ch. 2 in [25]; it is similar to that in the proof of Theorem 1.1 in [9] or in the proof of Theorem 3.1 in [26].

Different maximum principles for parabolic equations can be found in [27]. Unfortunately, they are not applicable in our case. Under stronger conditions on coefficients we can refer to the generalized maximum principle in [22, Theorem 17.1]. We use conventional arguments those involved in the proof of the maximum principle for generalized solutions. Let u =/ u(,xX) ’ ^u (^ ’x)   0’ Note that u

G W_P4Q ) (see [21.

0 ,     if u ( t,x ) >  0 .

Sect. 4 of Ch. 2]). Moreover, we have that u I S 0 =0 and u I t =₀ = 0. Multiply (11) by u

and integrate the result over G. Integrating by parts and using the boundary conditions we obtain that

2 dt У( u- ) 2 dx + $ ¹ У I^u-1 ² dx + jb ( u- ) 2 d Г - У д ( t’x ) u-d Г —

• < fu-d dx + | / ^^a i u x iu- + a ₀ |u-| ² dx|.

G              G^G i = ¹

Since the data are of constant sign, we derive that

• 2 dt У^ u- )² dx + $ ₁ У I^u-1 ² dx У ^ aiu_xi u- + a₀|u-|²dx| + | jb (u-)²d Г |. (13)

All summands on the right-hand side of (13) are estimated similarly. We use the conditions on the data, the Holder inequality, embedding theorems, and interpolation inequalities. Estimate the summands on the right-hand side under the integral sign. We have f aiuXiu dx < Ji(t) ||Vu ||L2(G) ||u ||L2p/(p-2)(G) <

G _         _                                                 (14)

< C 1 J i ⁽ t ) || V u ||_L 2 ( G ) ^|u ||ws ( G ) ’ J i ⁽ t ) = || ^ai||Lp ( G ) ’ s = ^n/ p.

Next, the inequalities (see [19])

r           r ^′

Hulks(g) — C2Wu\Wi(G)|u|L-G)’ IabI — e+        (e> 0), - + - = 1, imply that the right-hand side of (14) is estimated as eIWulL 2 ( G) + c (e) J. (t )2p/  n + 2). All summands on the right-hand side of (13) except for the last of them are estimated similarly. The estimate for the last summand is simpler. We have

I^ b ⁽ u^{- )2} d ^r | <  ^c ii ^u||L 2(_Г) <  ^c i|i ^{u - |}| ² _w s i ( g ) < e ii vu- ii L 2( g ) + ^{c ( e} ) |i ^{u - |}|L 2( g ) ⁽ s i E ⁽¹ /p’ 1)) .

In this case inequality (13) can be rewritten as

2 d Jg^ u ⁾² dx ⁺ $ 1 Jg ^{IVu 1 2} dx ~ ^C 2 ^{eHVU H} L 2 ( G )

+ C ( e ) J o ( t ) || u ^-|| L 2 ( G ) ’

where c2’C(e) are some posith'e constants and J0 G L1 (0’T). Clioosing c2e = $ 1. we arrive at the inequality y'(t) < cJo(t)У(t)’ У(0) = Hu-1|L2(g)|t=0 = 0.

Hence, we can conclude that u^- = 0 almost everywhere in Q, i.e., u >  0 a.e. in Q.      □

Corollary 1. There exists X о >  0 such that —L о + X for all X > X₀ is an isomorphism of the space {u E W^( G ) : Bui г = 0 } orito L_p ( G ) (see [22. Remark 3.1 (b)]).

In view of Corollary 1 we assume below that the problem L ₀ u = f E L_p ( G ), Bui г = 0 has a unique solution u E W_p2(G ), otherwise we make the change of variables u = ve^t in (1) and reduce the arguments to this case.

Theorem 2. Let conditions (4) - (7) hold. Then, for y >  0, fl solution u E W[^{, 2}( Q ) to (2), (12) with homogeneous initial and boundary conditions (i.e., u ₀ = 0 , g ( t,x ) = (П satisfies the estimate

\\^ue-Yt\\w[, • ²( Q ) + |Y\^^ue-Yt^L_P ( Q ) < cWfe-^Lp ( Q ) •

Fix an arbitrary 8 ₃ < 6 ₀. If conditions (10), (11) are fulfilled then the following estimate holds:

\^x^ue Y \ w 1 , 2( Q_s 3) + |^Y\\Vx^ue Y \Lp ( q , 3) <

c ( 1Р./е- *и( q , 0) + \fe^-l‘\Lp ( Q )) , y >  0 •

Tire constant c in this estimates is independent of the parameter y >  0.

Proof. Let u = veYt. Equation (12) is transformed to the equation vt — L0v + yv = e-Ytf (x, t) with v|t=0 = 0, Bv\S = 0•

Next, we refer to the estimate in Theorem 3.1 of [26] and make the inverse change of variables.

□ Remark 1. Generally speaking our reference to Theorem 3.1 in [26] is not exact, since the case of different boundary conditions on different connectedness components of the boundary is not treated there. However the proof of Theorem 3.1 remains valid in this case as well, since it is based on a partition of unity and local considerations.

Lemma 1. Let u ( t ) E W[(0,T ) a nd u (0) = 0. Then there exists a constant c >  0 independent of Y >  0 such that

\\^{e Yt}u\\_Lp (0 ,t ) <  y \\^{e Y}tut\\L_P (0 ,t ) •

The proof is elementary and we omit it.

Denote by Фа solution to (12), (2) assuming that conditions (4) - (7) are fulfilled. We impose the following additional constraints on the data:

fit <  Ф t ( t,x ₀) a.e. on (0 ,T ) , L ₀Ф <  0 a.e. on Q, 36 ₄ >  0 : fi ( t ) > 6 ₄ Vt E (0 ,T ) .  (15)

Assume that ^{R (ф)} = \ |^ф i^L^ ( q ) + \V ^ф i \ l _x ( q , ₀)■ ^{e (} t ) = ^fi’ ⁽ t ) ^{/fi (} t )■ ^fi’ ^{( T} ) = ^{fi ( T} ) ^{— ф(} т,х 0). r ⁽ t ) = ^{—e (} t ) e^-&^{e ( T} ) ^dT. ^R 0^(ф) = || r ⁽ t ) \Lp (0 ,t ).

Now we can state our main results. The former half of Theorem 3 below (the existence theorem) is known (see, for instance, [9]). However, we present here this formulation for completeness of the exposition.

Theorem 3. Let conditions (4) - (7) and (9) - (П) hold. Then there exists a constant Y0 < T such that on the segment [0, y0] there exists a 'unique, solution (u,p) to (1) -(3) with the property u E W^,2(QY0). Vs E W^,2((0,y0) x Bs(x0))) for all 5 < 50. and p(t) E Lp(0,yo) Assunle that ui (x) ,gi (t,x), fi (t,x) ,fii (t) (i = 1, 2) are two different collections of the data satisfying the conditions of the theorem and the functions Фi (i = 1, 2) are solutions to (1). (2). where, p (t) = 0. De note ri (t) = —ei (t) e- fiei(T) dT (ei (t) = (^i(t)-фt)( t’x0)))■ fix a, 'number R > 0. and. ass-ume that R(Фi) + R0(Фi) < R (i = 1, 2). Then there, exist numbers y0 n/id c0(R) > 0 such that there exist 'unique, solutions (ui,pi) (i = 1, 2) to (1) - (3) on the time segment [0,y0] satisfying the inequalities lip 1 - p 2\\lp (0 ,Yo) < c 0 (R)( 11в 1 - в 2\\lp (0 ,Yо) + Wfi1 - fi 2 ||wp1(0 ,yo) +

+ \ ^ф1 - ^ф2 \ l „ ( Q^Y 0) + \V ^(ф1 - ^ф2) \ l „ ((0 ,Y o) -B. 0 ( x o)) ■

Theorem 4. Let conditions (4) - (11), (15) hold. Then there exists a unique solution to (1) - (3) such that u E W ’ ²( Q ). p ( t ) E Lp (0 ,T ) an id Vu E W1 ’ ²( Qs ) for all 5 < 5 0. The function p ( t ) is nonnegative and

P ⁽ t ) >

c 0(Ф t ( t,x 0) - fit ( t )) fi ⁽ t )

^t E [0 ,T ] ,

where c ₀ is a positive constant depending on the norms of the data.

Remark 2. As it is easily seen, the statement of Theorem 4 remains valid if we change all signs in inequalities (8), (15), i.e., the functions —u ₀ , —g ( t,x ) , —f ( t,x ) , — Ф( t,x ) , —fi meet conditions (8), (15).

2. Proofs of the Main Results

Proof of Theorem 3. Let Ф be a solution to (2), (12). Make the change of variables u = v + Ф. Me obtain that

Lv + p ( t )( v + Ф) = 0 , v\_t =0 = 0 , vis = 0 , v ( t,x 0) = fi ( t ) — Ф( t,x 0) = fi ( t ) ■

Next, we make the following change of variables: v = ue -^p ^{( T} ) ^dT. Me infer

Lu + p(t)efitp(T)dTФ = 0, u\t=0 = 0, u\s = 0, ш(t,x0)= fi(t)efip(T)dT.(16)

Put x = x0 в (16). We arrive at the equation fi fi ft p (T)dT + fi (t) p (t) e fitp (T)dT = L 0 ш (t,x 0).

Denote a(t) = e^0p(T) dT. The equations can be rewritten as fi ‘a + fi (t) a' = L 0 ш (t,x 0) ■(17)

Expressing the function a, we arrive at the equality

a(t) = e-1te

J 0     ^{fi ( T} )

Thus, we have a‘lt\ = _Qlt\e- ft e(T) dT   L 0 w(t,xo)         f Lo^rx)   Jte(5) d,

a (t)      в (t) e            +    Д (t)       e (t) Jo     Д (T)    e              , where the function w is a solution to the problem

Lw = —a'(t)Ф, w\_t=₀ = 0, Bw S = 0.                       (20)

Hence, we infer w = L 1(—a' Ф). De note a 0( t) = a' (t). In view of (19), (18), we derive that

a o( t) = r (t) + S (a o( t)),

α

(t) =    ^t

ao (т) dr + 1,

S (a o)= ^{L 0}U⁽‘^,xo) — в (t) f LBrx- e-^ft^{в ((})d⁽dT, w = L^- 1(—a оФ) •     (->->)

^(t)             0

Demonstrate that equation (21) is uniquely solvable in the class a₀(t) G Lp(0,T). Estimate the norm of S. The embedding theorems (see [19]) ensure that

\Lоw(t,xo)\ < c\\Lоw^ws(Bg3(xo)), s G (n/p, 1), 83 < 60•(23)

Next, we consider the summands in the main part of the operator L0. Since a function of the class Cp(G) is a pointwise multiplier in WS(G) with s < p (see, for instance, the item 3.3.2 in [28]), we have a' wxixj\Wp(Bg 3(xo)) < c\\w\\w2+s ( Bg 3 (x 0)) < c 1 Ilwllw3( Bg 3 (x 0))\w\lp (Bg3 (x 0)) •(22)

Lower order summands are estimated similarly. Note that the embedding theorems ensure that ai,a₀G C^{1 -n/p}(B_s0(x₀)). The definition of the norm in Ws yields

WaiUxi\\ws(Bg3(x0)) < c 1 laillws(Bg3(x0))\^wxi\c(Bg-(x0J) + +^{c 1}laillc (Bg 3 (x 0))\^wxi IIW⁽Bg 3⁽x 0)) < ^c 2 ll^wllw1+^s (Bg₃(x 0)) <

< ^c3IMI Wp(B3(x0)) ^|w|L-B)g/3(x0))•

Estimates (23) - (25) imply that there exist constants в G (0, 1) and c₄> 0 such that \^{L 0 w(}t,^xq)\ < ^{c 4}IIMlWpT Bg 3 (x 0))ll^w|l Lp (Bs 3 (x 0)) •                           (25)

Now we estimate the norm \e^-YtS(a₀) \_Lp(₀,_T)• The above definition of the operator S yields

\^e-YtS^(a0)\lp(0,T) < c5\^e-^Yt^L0^w(t,x0)\lp(0,T) + ^c6 II^е\Lp(0,T) Jq ^e-^YT\^L0^w(T) \ d^T< < c7\e^-YtLоw(t,x0)\Lp(0,t)•

Inequalities (26), (27) and Lemma 1 imply the estimate lie YtS(a0)\Lp(0,T) < c8 lie Ytw\Lp(0,T;W3(Bg3(x0)) lie YtW\Lp(Qg3) < < Y- He ^“Wlp (0 ,T; Wp3( Bg 3 (x 0)) He YtWtI L- (Qis 3).

Next, applying Theorem 2 and the definition of ш, we obtain the estimate lle^'S(a0)IL.(от) < Y-(\e-YaоФIL.(q) + ||е-"аоVФ||L.(q,o,) <

< Y->\e^-,‘aоIL.(от)R(Ф),                             <2»)

where the constant с_1о is independent of y and the norms of the data, and it depends on the norms of the coefficients of the equation, the constants in embedding theorems, interpolation inequalities and T. Choose a constant yо such that

4^ R (Ф) = 1 / 2.                            (29)

Yо

In this case we have the estimate

II^{е Yt}S⁽aq)\l_p(от) < ^ ||e ^Y'^aо\l.(от)

for all Y > Y0' Thus, the operator S is contractive in some equivalent norm of the space L_p(0,T) and. thereby. (21) is solvable with respect to the function a₀ = a'(t). Obviously, this solution satisfies the estimate

II^{е Y}'^a'\Lp (0 T) < ²\r⁽t ) 6 ^Y'\lp (q т), ^Y> ^Yq .

In particular, we infer lla'IIL.(о,т) < 26'0т\r(t)lLp(от) = сi(R(Ф),R(Ф)).

We can restore the function a = 1 + ft a₀(т) dT. Given a function a, find a solution ш to the problem (20). Fix 1₀< T and estimate

∥

о

^a'^(T ) d^T\c ([0,10 ]) < t 0/q ||^a'||Lp (Q, to) < t 0^{/q с} 1^(R,R 0 ).

Choose 10 so that tQ/q с i (R, R 0) = 3 / 4. In th!s case a > 1 — 3 / 4=1 / 4 > 0 on [0 ,t 0] and we can construct the function f^p(т) dT = ln(1 + a(t)). Respectively, p(t) = a'(t)/(1 + a(t)). Obviously, p(t) G Lp(0,tQ). Verify that the functions p(t), ш(t) are a solution to the inverse problem (16). Integrating (19), we obtain (18) whose transformation validates equality (17) and. hence, 0'a + -0(t)a' = Lош(t,xQ) — a'Ф(t,xQ). On the other hand, taking x = xQ in (20). we have ш' (t,x Q) = L Q ш (t,x Q) — a' Ф( t,x Q). He псе. ш' (t,x Q) = (0a)' and thereby ш(t,xQ) = a0 = etp(T)dT0, i.e., equalities (16) hold. Proceed with stability estimates. Assume that we have two collections of the data ui (x) ,gi (t,x) fi (t,x) ,0i (t) (i = 1, 2) and functions Фi (i = 1, 2) are solutions to (1). (2) with these data, and p(t) = 0. The respective functions ai meet the equations ai(t) = ri(t) + Si(ai(t)), i = 1,2, where the operators Si are defined as in (22), but instead of ш we take solutions to the problem

^Lшi = ^—ai⁽t^)Фi, i = ¹,², ^шi^|t=0 = ⁰, ^Bшi\s = ⁰.                  00

Thus, ui = L¹ (-aiФi) and

_ ^L о ^ui(^t,x 0)

^{S ( a} ) = ---T7T\ ^i⁽t )

/ t L о и^(T,x о)

- ^в(t’ /о            e

/t ^ei^(€) d^ dT.

Choose the parameter yо ^as in (29), where we take the quantity R from the conditions of the theorem rather than R(Ф). Next we choose t_о as before inserting the quantity R rather than R_о(Ф) and R(Ф). In this case a solution (u_i,pi) to our problem satisfying the above initial and boundary data exists on the segment [0, yо] (yо = t_О). The corresponding functions a 1 , a₂ satisfy the estimate

ai(t) > 1 /4,  ||a||_1Ч(оT) < c(R)•

We have

‘             f       L 0^{( 2}1 -2 2⁾⁽tx 0)                 1         1 A

S^1(a 1) - S^1(a2) =     ^(t)      + L^оU² ^(t) - ^t) -

д (t) [t ^{L0(21 -22})^(Tx⁰⁾ e^-1 О        ^ i( т)

Tt в 1( €) d^ d_T -

— f t 1         V \e ^в 1⁽t)^{e Jte 1( €})

О ^Lоu2^(T,^x^ 1(T)

в₂(t) e^-sT^e2⁽j) * ^ 2^(T )

dτ.

Subtracting equalities (31) for i = 1,2, we infer

Lu 1 — Lu₂ = L (u 1 — u₂) = — (aKt) — L₂( t ))Ф1 + a ‘ (Ф1 — Ф₂).

Repeating the arguments those in the proof of (28), we obtain the inequality lie Yt L О( u 1 — u 2)(t,x О) IL (О ,yo) < Y— ( ||e Yt (a1 — a 2 ) IL (О ,y0)c(R) +

+^c 1^(R)( II ^Ф1 ^{— ф}2 ||l«> (Q^Y o) + ||V^(ф1 ^{— ф}2 ) ||l^ ((О ,yo) xB_s0 (xo)))) •

The claim of the theorem follows from (32) - (34), the equality a 1 — a'₂ = r 1 — r₂ + S 1 (a() — S 1 (a'₂), and some simplest estimates.

□

Proof of Theorem 4- Let the conditions of the theorem hold. As in Theorem 3 we reduce the problem to the study of equation (21) and justify its solvability. Since the norm of S is less than 1 and the operator is linear, a solution to (21) can be found using the method of successive approximations beginning with ао = 0. Successive approximations are written as a (t ) = r (t) + s (a- 1( t)).                               (35)

In view of (15), a¹ = r(t) > 0. Assume that the function ai^-1 is nonnegative. Demonstrate that ai is nonnegative too. The corresponding function ui is a solution to (20), i.e., we have

Lui = —ai^- 1Ф, ui|t=о = 0, Buils = 0.

Since condition (8) is fulfilled, by Theorem 1 Ф > 0 a.e. in Q. By condition L_ОФ < 0 a.e. and ai^-1> 0 a.e. Hence. L_о(—ai^- 1Ф) > 0 a.e. Consider the problem

Lu* = —a¹L_ОФ, u*|_t=_о = 0, Bu*|_s = ai¹BФ|_s = ai¹g(t,x) > 0.

By Theorem 1 a solution ш* is nonnegative in Q. Define the function Ш = L₀¹ш*. We have Ш It₌₀ = 0. ВШ IS = 0 (by construction), Ш G Wp,²( Q). aiid L₀ш G Wp^,2( Q). Moreover, we can conclude that

ш* - Lo(ш* - ai^- 1Ф) = 0.

Applying L₀¹ to this equality, уve can claim that шt — (L₀ш — ai- 1Ф) = 0. Hence. Шt — L₀ш = —ai- 1Ф. In inew of the uniqueness theorem, we derive that ш^г = ш and thus L₀ci = ш* > 0

a.e. Consider equality (35) and recall the definition of S (see (22)):

a i-1. ₌^L0^ш(t,x0) _ „, . Г‘ L0pTX«l  _Rв(_£)d_(d

S⁽°   )       V (t)       ^{в (}t) I     V (т)    e

Since в(t) A 0. every summand here is iioinnegative and in view of (35). ai > 0 a.e. Since the limit a₀ is a strong limit of the sequence ai in the space Lp(0,T). we conclude that a₀> 0 a.e. In this case the function ftp(т) dr = ln(J₀t a₀(т) dr + 1) is defined on the whole segment [0,T], respectively the function p(t) = ₁₊^t) is a solution to our problem. Establish the desired estimate. We have (see (30)) that

['p(T) dT = [' f^(T) dr < T^{1 /q}Wa'^(0,T) A c 1(R, R0)T^{1 /q}, q = p/(p - 1).

J 0              0 ¹ + ^{a (T})

From (21). it follows that a'(t) = p(t)e^o^p(T)^dT> r(t) > 0. Thus.

p(t) > r(t)/T^{1 /q}C 1(R,R0) = r(t)C0, t G [0,T].

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The authors were supported by the Russian Foundation for Basic Research and the Government of KhMAO-Yugra (Grant No. 15-41-00063, r_ural_a).