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Inverse Problems for a Generalized Subdiffusion Equation with Final Overdetermination.

1 Introduction

Anomalous diffusion processes are described by different models [6]. Among them stands out the time (or space-time) fractional diffusion equation that is the most common way to represent a subdiffusion. For some situations such approach does not work [19]. Therefore, more general models that unify wider range of subdiffusion processes are introduced [19,25].

In this paper we use an operator that is more general than the fractional time derivative:

[D.sup.{k}.sub.t] v = d/dt k * v, (1.1)

where * denotes the time convolution, i.e. [mathematical expression not reproducible]. Taken k = [t.sup.-[beta]]/[GAMMA] (1-[beta]), (1.1) transforms into a well-known Riemann-Liouville fractional derivative. The operator corresponding to the Caputo fractional derivative denoted as [D.sup.C.sub.(k)]v was introduced by Luchko and Yamamoto in [23] and also in [15].

The toolkit for treating such a type of derivative have been developed by Pruss et al. [5]. They have created a setting to introduce the operator inverse to [D.sup.(k).sub.t] through the concept of Completely Positive kernels [5]: a kernel M [member of] [L.sub.1, loc] ([R.sub.+]) is called completely positive if there are [k.sub.0] [greater than or equal to] 0 and nonnegative and nonincreasing [k.sub.1] [member of] [L.sub.1, loc]([R.sub.+]) such that M * ([k.sub.0][delta] + [k.sub.1]) = 1 holds. The applications of this concept can be found in [1,33,34]. Another approach to this issue has been developed by Kochubei [19].

Often parameters of models are unknown. Then additional observations are performed and inverse problems solved to reconstruct unknown quantities [12,13,16,17, 20, 21]. In the present paper we consider two inverse problems (IPs) that use final observation data: IP1 is to identify a space-dependent factor f of a source term g(t,x)f (x); IP2 is to reconstruct a coefficient r(x) of a linear reaction term.

IP1 for fractional and perturbed fractional diffusion equations is studied in several papers. Theoretical and numerical results are obtained in the particular case g = g(t) [7,17,18,26] and in the case g = g(t,x) [30,32]. In latter papers the existence and uniqueness of solutions are proved for almost all scalar diffusion coefficients. IP1 for a semilinear fractional diffusion equation is considered in [15]. Uniqueness of the solution is proved.

In this paper we consider IP1 for a more general diffusion equation that includes the operator (1.1) instead of the fractional derivative. We prove the uniqueness of the solution to IP1 by applying a modified version of the positivity principle from [15]. That falls into category of maximum principle results [13,20,22]. Similar approaches to the inverse problems are well-known in the domain of parabolic equations [2,12]. Next we prove the existence and stability of the solution of IP1 by means of the Fredholm alternative. The uniqueness of solution of IP2 follows from the IP1-results. Finally, we prove local existence and stability of the solution to IP2 by means of the contraction argument.

2 Formulation of direct and inverse problems

Let us consider the generalized subdiffusion equation

[U.sub.t](t, x) = [(M * LU).sub.t](t, x) + Q(t, x), (2.1)

where U physical state, t is the time, x [member of] [R.sup.n] is a space variable, Q is a source term, the operator L = L(x) is such that

[mathematical expression not reproducible]

and I is the unity operator. The kernel M is a memory function related to a non-locality of the diffusion process.

There are two ways to derive the equation (2.1) from physical laws. One method consists in modelling continuous time random walk processes in microlevel and taking a continuous limit in a macro-level [4] and another one uses conservative laws and specific constitutive relations with memory [27].

Real world applications of the equation (2.1) include diffusion in fractal and porous media, e.g. propagation of pollution, heat flow in media with memory, dynamics of protein in cells, transport in dielectrics and semiconductors, usage of optical tweezers, Hamiltonian chaos etc. [3,4,6,27,31].

Let us assume that there is a function k such that k * M = M * k =1. Then if we apply k * to (2.1), we obtain an equation that contains the explicit differential operator L and is called the normal form of (2.1): k * Ut(t,x) = LU(t, x) + H(t, x), where H(x, t) := k*Q(t, x). The term k*[U.sub.t] can be rewritten in the form [D.sup.{k}.sub.t](U--U(0, *)) that does not contain the 1st order derivative of U. Therefore, we get the equation

[D.sup.{k}.sub.t](U--U(0, *)) = LU(t,x) + H(t, x). (2.2)

Conversely, in case of sufficiently regular U, the equation (2.1) follows from (2.2) by means of the application of the operator [partial derivative]/[partial derivative]t M *.

The equation (2.1) and its analogue (2.2) incorporate the following possibilities:

1. The kernel M(t) = [t.sup.[beta]-1/[GAMMA] ([beta]), 0 < [beta] < 1, represents a power-type memory. Then (2.1) becomes the celebrated time fractional diffusion equation [U.sub.t] = [??][D.sup.1-[beta]]LU + Q, where [D.sup.1-[beta]]v = ([t.sup.[beta]]/[GAMMA] ([beta]) * v) is the Riemann-Liouville fractional derivative of the order 1--[beta] [4,17,20,26]. For such M, it holds k = [t.sup.-[beta]]/[GAMMA](1-[beta]) and [[k * (v-v(0)].sub.t] = k * [v.sub.t] = [[partial derivative].sup.[beta]].sub.t]v is the Caputo fractional derivative.

2. The kernel M or its associate k is a linear combination of power functions [25,31]:

[mathematical expression not reproducible]

3. The kernel M has the form [mathematical expression not reproducible] where p [greater than or equal to] 0 is a nonvanishing integrable function (cf. [3,25,31]). Such a kernel stands for the distributed order fractional derivative that is used for modeling diffusion with a logarithmic growth of the mean square displacement [19].

4. Tempered fractional calculus [29], that is another way to generalize a fractional calculus, falls into the case

[mathematical expression not reproducible]

This type of kernel is used for modelling the transition from anomalous to normal diffusion.

Every presented example of M (or k) has a completely monotonic associate k (or M) that solves k * M = 1 (see Section 3).

Let [OMEGA] [member of] [R.sup.n] be an open bounded domain with the boundary [partial derivative][OMEGA]. In direct problem we have to find a function u that solves the initial-boundary value problem

[mathematical expression not reproducible], (2.3)

Here [PHI] and b are given functions and

Bv(x) = v(x) or Bv(x) = w(x) * [nabla]v(x),

with [omega] * V > 0 and v(x) denoting the outer normal of [partial derivative][OMEGA] at x [member of] [OMEGA]. An important particular case is [omega] = ([summation].sup.n.sub.j=1] [a.sub.ij][v.sub.j]|i = 1, ..., n. Then the condition [B(U--b)|.sub.(t,x)[PHI](0,T)x[partial derivative][OMEGA]] = 0 corresponds to the flux specified at [partial derivative][OMEGA].

Let us proceed to inverse problems. To this end we introduce the condition

U (T, x) = [PSI] (x), x [member of] [OMEGA], (2.4)

with a given observation function [PSI]. Firstly, we formulate of an inverse source problem. Let

H (t, x) = g(t, x)f (x) + [h.sub.0](t,x), (2.5)

where the components [g.sub.f] and [h.sub.0] may correspond to different sources or sinks. The factor f is unknown and to be reconstructed by means of the data (2.4). Since the whole function U is also unknown, the first inverse problem consists in determination a pair of functions (f, U) that satisfies (2.3), (2.4) and (2.5).

In the second inverse problem, our aim is to identify the coefficient r of the linear reaction term rU. In the mathematical formulation, the problem consists in finding a pair (r, U) that satisfies (2.3) and (2.4). We can handle the case of zero initial condition [PHI] = 0 (for details, see the end of Section 6).

Methods to be used in this paper require homogeneous boundary conditions. Therefore, we perform the change of the second unknown u = U - b in our problems. It brings along shifts of data by addends containing b.

Firstly, from (2.3) we obtain the following problem for u = U - b:

[mathematical expression not reproducible], (2.6)

where

[psi](x) = [PHI](x)--b(0, x), (2.7)

F (t, x) = H (t,x) + Lb(t, x) - [D.sup.{k}.sub.t](b-b(0, *))(t,x). (2.8)

The overdetermination condition (2.4) in terms of u has the form

u(T,x) = [psi](x), x [member of] [beta], (2.9)

where [psi](x) = [PSI](x)-b(T, x). Plugging (2.5) into (2.8) we obtain

F(t, x) = g(t,x)f (x) + h(t, x), (2.10)

where h(t,x) = [h.sub.0] (t,x) + Lb(t, x) - [D.sup.{k}.subt](b-b(0, *))(t,x).

In the reformulated first inverse problem (IP1), we seek for the pair of functions (f, u) that satisfies (2.6), (2.9) and (2.10).

Let us reformulate the second inverse problem, too. From the relations (2.3), (2.4) with [PHI] = 0 by means of the change of variable u = U--b, we obtain the following problem for the pair (r, u):

[mathematical expression not reproducible], (2.11)

where b(0, x) = 0, x [member of] [beta], the function [psi] is expressed by [psi](x) = [PSI](x)--b(T, x) and [F.sub.1](t,x) = H(t,x) + [L.sub.1]b(t,x)--[D.sup.{k}.sub.t]b(t,x).

Thus, the reformulated second inverse problem (IP2) is to find the pair of functions (r, u) that satisfies (2.11).

3 Basic assumptions

In this section we collect basic conditions on the domain, operator L and kernels k and M that will be assumed throughout the paper.

We assume that [partial derivative][OMEGA] is uniformly of the class [C.sup.2] and [omega] [member of] [([C.sup.1]([partial derivative][OMEGA])).sup.n]. Moreover, we assume that [a.sub.ij], [a.sub.j] r [member of] C([bar.[OMEGA]]) and the principal part of L is uniformly elliptic, i.e. [n.summation over (i,j=1)] [a.sub.ij](x)[[xi].sub.j] [[xi].sub.j] [greater than or equal to] c[[absolute value of [xi]].sup.2] [for all][xi] [member of] [R.sup.n], x [member of] [OMEGA] for some c > 0.

Concerning the function k, we assume that

1. k belongs to [L.sub.1,loc](0, [infinity]) and is a solution of the equation M * k =1 with a kernel M [member of] [L.sub.1,loc] (0, [infinity]) that satisfies the conditions

[mathematical expression not reproducible] (3.1)

2. k has the following properties:

[mathematical expression not reproducible], (3.2)

[exist][t.sub.k] > 0 : k(t) is strictly decreasing in (0, [t.sub.k]). (3.3)

The assumptions (3.1) ensure the existence of a sufficiently regular solution of the direct problem (see Lemma 3) and the assumptions (3.2), (3.3) are needed for the application of a positivity principle to this solution.

We mention that restricting generality a bit it is possible to reduce all conditions 1 and 2 to the single kernel M. Firstly, M [member of] [L.sub.1,loc] (0, [infinity]) and (3.1) imply the existence of a unique solution k [member of] [L.sub.1,loc] (0, [infinity]) of the equation k * M = 1 ([10], Ch. 5, Corollary 5.6). Secondly, all properties (3.2), (3.3) follow from conditions that are a bit stronger than (3.1). It is shown in the following lemma. Proof is in Appendix.

Lemma 1. Let M [member of] [L.sub.1,loc] (0, [infinity]) satisfy (3.1) and M' < 0, log M - convex, log(-M') - convex. Then the solution of M * k = 1 satisfies (3.2), (3.3).

The imposed assumptions on M and k hold for weakly singular completely monotonic kernels from

[mathematical expression not reproducible]

For M and k satisfying M * k = 1, it holds M [member of] CM if and only if k [member of] CM ([9], Theorem 3).

All examples of M and k given in Section 2 belong to CM.

4 Preliminaries

4.1 Functional spaces

Let X be a Banach space. Since k * M = 1, we have

[D.sup.{k}.sub.t](M * v) = d/dt k * M * v = d/dt 1 * v = v, [for all]v [member of] [L.sub.1]((0, T); X), (4.1)

where [L.sub.1]((0, T); X) is the space of functions u : (0, T) [right arrow] X that are integrable in the Bochner sense on (0,T). This means that the operator M * is a one-toone mapping from [L.sub.1]((0,T); X) to {M * v : v [member of] [L.sub.1]((0,T); X)} and D{k} is the inverse of [M.sub.*].

As usual, let C([0, T]; X) stand for the Banach space of functions u : [0,T] [right arrow] X that are continuous on [0,T] with the norm [mathematical expression not reproducible] and [C.sub.0]([0,T]; X) = {u [member of] C([0,T]; X) : u(0) = 0}. Based

on the relation (4.1), we introduce the functional space

[C.sup.{k}.sub.t]([0, T]; X) := M * C([0,T]; X) = {M * v : v [member of] C([0,T]; X)}.

It is a Banach space with the norm

[mathematical expression not reproducible]

Since M * [laplace(C ([0,T]; X), [C.sub.0]([0,T]; X)), it holds

[C.sup.{k}.sub.0]([0,T];X) [??] [C.sub.0]([0,T];X).

We also define the space

[mathematical expression not reproducible] (4.2)

that is a Banach space with the norm

[mathematical expression not reproducible]

Next we introduce the abstract Holder spaces with corresponding norms

[mathematical expression not reproducible]

where 0 < [alpha] < 1, and define the Banach spaces with norms

[mathematical expression not reproducible], (4.3)

Let us establish some connections between the spaces (4.2), (4.3) and the usual C, [C.sup.1]- and Holder spaces. For [C.sup.{k}]([0, T]; X) the embeddings

[C.sup.1] ([0, T]; X) [??] [C.sup.{k}]([0, T]; X) [??] C([0,T]; X) (4.4)

are valid. The right embedding follows from M * [member of] [laplace](C([0, T]; X)) (1). To prove the left embedding, we choose some u [member of] [C.sup.1]([0, T]; X). Then

[mathematical expression not reproducible]

and since k * [member of] [laplace](C([0,T]; X), [C.sub.0]([0,T]; X)), the left relation in (4.4) follows.

Analogous relations for the space [C.sup.{k},[alpha].sub.0([0,T ]; X) are

[mathematical expression not reproducible] (4.5)

where

[C.sup.1+[alpha].sub.0]([0,T]; X) = {u : u, u' [member of] [C.sup.[alpha].sub.0]([0,T]; X)}.

The right embedding in (4.5) is a consequence of the fact that M [member of] [laplace]([C.sup.[alpha].sub.0] ([0, T]; X)) (see Lemma 4.2 in [14]) and the left embedding in (4.5) can be proved similarly to the left embedding in (4.4).

Under additional assumptions on M it is possible to show that the operator M * increases the order of Holder continuity of a function. Namely, the following lemma is valid. Its proof is deferred to Appendix.

Lemma 2. If M (t) [less than or equal to] [c.sub.1][t.sup.[beta]-1], [absolute value of M'(t)] [less than or equal to] [c.sub.2][t.sup.[beta]-2], t [member of] (0, T) for some [c.sub.1], [c.sub.2] [member of] [R.sub.+], 0 < [beta] [less than or equal to] [alpha] < 1 then M * [member of] [laplace]([C.sup.[alpha]-[beta].sub.0] ([0,T]; X), [C.sup.[alpha].sub.0]([0,T]; X)).

Under conditions of Lemma 2, [C.sup.{k},[alpha]-[beta].sub.0([0,T]; X) [??] [C.sup.[alpha]sub.0]([0,T]; X). In the particular case M (t) = [t.sup.[beta]-1].sub.1]/[GAMMA]([beta]) (then M * is the fractional integral of the order [beta]), it holds the equality [C.sup.{k}, [alpha]-[beta].sub.0]([0,T];X) = [C.sup.[alpha].sub.0]([0,T];X) [15].

By exchanging M and k in above relations, we obtain definitions and embeddings of spaces that contain {M} instead of {k} in the superscript.

4.2 Abstract Cauchy problem

Let A : D(A) [greater than or equal to] X be a linear densely defined operator in a Banach space X. We say that A belongs to the class S([eta], [theta]) for n [member of] R, [theta] [member of] (0, [pi]) if

[mathematical expression not reproducible]

An operator A [member of] S ([eta], [theta]) is closed. This implies that [X.sub.A] := D(A) is a Banach space with the graph norm [mathematical expression not reproducible].

O bviously, S ([eta], [[theta].sub.1]) [subset] S ([eta], [[theta].sub.2]) for [[theta].sub.1] > [[theta].sub.2]. Operators of the class S ([eta], [theta]), [theta] [member of] ([pi]/2, [pi]), are the sectorial operators that generate analytic semigroups.

Now let us consider the Cauchy problem

[D.sup.{k}.sub.t](u - [psi])(t) = Au(t)+ F (t), t [member of] [0, T ], u(0) = [psi], (4.6)

with given F : [0, T] [greater than or equal to] X and [psi] [member of] X.

Lemma 3. Let A [member of] S ([eta], [pi]/2) for some [eta] [member of] R. Then the following statements are valid..

(i) (uniqueness) Let u [member of] [C.sup.{k}]([0,T]; X) [intersection] C([0,T]; [X.sub.A]) solve (4.6) and [psi] = 0, F = 0. Then u = 0.

(ii) Let F [member of] [C.sup.[alpha].sub.0]([0, T]; X) and [psi] = 0. Then (4.6) has a solution u in the space [C.sup.{k}, [alpha].sub.0]([0,T]; X) [intersection] [C.sup.[alpha].sub.0]([0, T]; [X.sub.A]). This solution satisfies the estimate

[mathematical expression not reproducible]. (4.7)

(iii) Let F [member of] [C.sup.[alpha]]([0,T];X) and [psi] [member of] [X.sub.A]. Then (4.6) has a solution u in the space [C.sup.{k}]([0, T]; X) [intersection] C([0, T]; [X.sub.A]). This solution satisfies the estimate

[mathematical expression not reproducible]. (4.8)

The constants [C.sub.1] and [C.sub.2] depend on M and A.

Proof. The change of variable v = [D.sup.{k}.sub.t](u - [psi]) [??] u = M * v + [psi] reduces (4.6) of the integral equation

v (t) = A(M * v)(t)+ F (t) + A[psi], t [member of] [0, T]. (4.9)

Provided F [member of] C([0,T]; X), [psi] [member of] [X.sub.A], the function u [member of] [C.sup.{k}]([0,T]; X) [intersection] C([0, T]; [X.sub.A]) solves (4.6) if and only if v [member of] V := {v [member of] C([0,T]; X) : M * v [member of] [C.sub.0]([0,T]; [X.sub.A])} solves (4.9). Similar one-to-one correspondence holds for u [member of] [C.sup.{k}.sub.0]([0, T]; X) [intersection] [C.sup.[alpha].sub.0]([0, T]; [X.sub.A]) and v [member of] [V.sup.[alpha]] := {v [member of] [C.sup.[alpha].sub.0]([0,T]; X) : M * v [member of] [C.sup.[alpha].sub.0]([0, T ]; [X.sub.A])} in the particular case F [member of] [C.sup.[alpha].sub.0]([0,T]; X), [psi] = 0.

Since M satisfies the conditions (3.1) and A [member of] S ([eta], [pi]/2), we can apply results of Ch. 3 of [28] to (4.9).

(i) Theorem 3.2 with Corollary 1.1 and Proposition 1.2 in [28] implies that there exists a family of operators S : [0, [infinity]) [greater than or equal to] [laplace](X) (called resolvent of (4.9)) so that a solution v [member of] V (if it exists) is represented by the formula v = d/dt S * F. By assumptions of (i), (4.9) has a solution v [member of] V. Since F = 0, we have v = 0. Thus, u = 0.

(ii) Theorem 3.3 (i) [28] implies that for F [member of] [C.sup.[alpha].sub.0]([0,T]; X) there exists a solution v [member of] [V.sup.[alpha]] of (4.9). This proves the existence of the solution u [member of] [C.sup.{k},[alpha].sub.0]([0,T]; X) [intersection] [C.sup.[alpha].sub.0]([0, T]; [X.sub.A]) of (4.6). The estimate (4.7) follows from the bounded inverse theorem.

(iii) It is sufficient to prove this assertion in case F(t) = [xi] [member of] X, because the problem with given pair of data (F, [psi]) can be splitted into two problems with the data (F--F(0), 0) and (F(0), [psi]), respectively. For the first problem, the assertion (ii) applies. Having proved (iii) for the second one, u is expressed as the sum of solutions of these two problems and satisfies (iii), too.

Thus, let us assume that F(t) = [xi] [member of] X. Due to Proposition 1.2 (ii) [28], (4.9) has the solution v = S([xi] + A[psi]) [member of] V. This implies the existence assertion of (iii). Due to the strong continuity of S(t) [28], [[parallel]S(t)[parallel].sub.[laplace](X)] < [C.sub.3], t [member of] [0,T], where [C.sub.3] is a constant. Thus, [[parallel]v[parallel].sub.C([0,T],S)] [less than or equal to] [C.sub.3] ([[parallel][xi][parallel].sub.X] + [[parallel]A[psi][parallel].sub.X]). Extracting the term A(M * v) from (4.9) and estimating it we obtain [mathematical expression not reproducible]. Consequently,

[mathematical expression not reproducible]

with a constant [C.sub.4]. This implies (4.8).

4.3 Statements on direct problem

In order to apply Lemma 3 to the direct problem (2.6), we must introduce appropriate Banach spaces of x-dependent functions and define realizations of the operator L in these spaces so that they belong to S ([eta], [pi]/2). Let us introduce the following spaces and operators:

1. [mathematical expression not reproducible].

2. [mathematical expression not reproducible].

Corollary 1. Operators [A.sub.p], p [member of] {0} [union] (1, [infinity]), are sectorial. Thus, Lemma 3 holds in cases X = [X.sub.p], A = [A.sub.p], p [member of] {0} [union] (1, [infinity]) and applies to problem (2.6).

Proof. It follows from Theorems 3.1.2, 3.1.3 and Corollary 3.1.24 (ii) in [24].

Lemma 4. Let [mathematical expression not reproducible], K be non increasing and [[there exists]t.sub.k] > 0 : K is strictly decreasing in (0, [t.sub.K]). Moreover, let F [member of] C([0,T] x [bar.[OMEGA]]). Assume that u solves the problem

[mathematical expression not reproducible]

and satisfies the smoothness conditions u [member of] C([0, T] x [bar.[OMEGA]]), [mathematical expression not reproducible]; [W.sup.2.sub.p]([OMEGA])) for some p > n, [L.sub.1]u [member of] C((0,T] x [bar.[OMEGA]]), [D.sup.{K}.sub.t](u - [psi]) [member of] C((0, T] x [bar.[OMEGA]]). Finally, let

[mathematical expression not reproducible]

If [psi] [greater than or equal to] 0, F [greater than or equal to] 0 and Bu|[partial derivative] [OMEGA] [greater than or equal to] 0 then the following assertions are valid.

(i) u [greater than or equal to] 0;

(ii) if u([t.sub.0], [x.sub.0]) = 0 in some point ([t.sub.0], [x.sub.0]) [member of] (0,T] x [[OMEGA].sub.N], where

[mathematical expression not reproducible]

then u(t, [x.sub.0]) = 0 for any t [member of] [0, [t.sub.0]].

This lemma is a slight modification of a positivity principle that was proved in [15] for a semilinear equation in case of a more smooth solution u [member of] C((0, T]; [C.sup.2]([bar.[omega])]) and strictly decreasing in (0, T) kernel K.

To prove Lemma 4, we need the following auxiliary result. It is proved in Appendix of the paper.

Lemma 5. Let w [member of] [W.sup.2.sub.p]([OMEGA]) for some p > n, [mathematical expression not reproducible].

In case [x.sup.*] [member of] [partial derivative][OMEGA] we also assume that ([omega] * [nabla]w)([x.sup.*]) [greater than or equal to] 0. Then [L.sub.1]w([x.sup.*]) [greater than or equal to] 0.

Proof of Lemma 4. Without a restriction of generality we assume that r [less than or equal to] 0. Otherwise it is possible to define [??] = [e.sup.-[sigma]t] u as in [15] and to consider the corresponding problem for [??]. Such a problem also satisfies the assumptions of Lemma 4 and has the coefficient [mathematical expression not reproducible] in place of r. Since [mathematical expression not reproducible], for sufficiently large [sigma], [??] [less than or equal to] 0.

Let us suppose that (i) does not hold. Then there exists ([t.sub.1], [x.sub.1]) [member of] (0, T] x [bar.[OMEGA]] such that u([t.sub.1], [x.sub.1]) < 0 and [mathematical expression not reproducible]. It was shown in [15] (formula (37)) that the assumptions [D.sup.{K}.sub.t](u - [psi]) [member of] C((0,T] x [bar.[OMEGA]]), (4.10), K > 0 and K - nonincreasing together with the relations u(t, [x.sub.1]) [greater than or equal to] u([t.sub.1], [x.sub.1]) and u([t.sub.1], [x.sub.1]) < 0 imply [D.sup.{K}.sub.t](u - [psi])([t.sub.1], [x.sub.1]) < 0. On the other hand, Lemma 5 applies to the function w = u([t.sub.1], *) at [x.sup.*] = [x.sub.1]. We obtain [L.sub.1]u([t.sub.1], [x.sub.1]) > 0. Also r([x.sub.1])u([t.sub.1], [x.sub.1]) > 0 and F > 0. Thus, the left-hand side of the equation [D.sup.{K}].sub.t](u - [psi])([t.sub.1], [x.sub.1]) = [Lu + F]([t.sub.1], [x.sub.1]) is negative, but the right-hand side is nonnegative. We have reached a contradiction. The assertion (i) is valid.

Let us prove (ii). Let u([t.sub.0], [x.sub.0]) = 0 at ([t.sub.0],[x.sub.0]) [member of] (0,T] x [[OMEGA].sub.N]. Define [[??].sub.0] = inf{t : t [less than or equal to] [t.sub.0], u([tau], [x.sub.0]) = 0 for [tau] [member of] [t, [t.sub.0]]}. If (ii) is not valid, then [[??].sub.0] > 0 and u(t, [x.sub.0]) [greater than or equal to] [delta], t [member of] ([t.sub.2], [t.sub.3]) for some [delta] > 0 and ([t.sub.2], [t.sub.3]) [subset] (0, [[??].sub.0]) such that [[??].sub.0]- [t.sub.2] < [t.sub.K]. Then, similarly to the proof in [15] p.138, from the assumptions [D.sup.{K}.sub.t](u - [psi]) [member of] C((0,T] x [bar.[OMEGA]]), (4.10), K > 0, K - nonincreasing and relations u [greater than or equal to] 0, u(t, [x.sub.0]) [greater than or equal to] [delta] > 0, t [member of] ([t.sub.2], [t.sub.3]), we derive

[mathematical expression not reproducible]. (4.11)

Since 0 < [[??].sub.3] - [t.sub.3] < [[??].sub.0] - [t.sub.2] < [t.sub.K] and K is strictly decreasing in (0, [t.sub.K]), (4.11) implies [D.sup.{K}.sub.t](u - [psi])([[??].sub.0], [x.sub.0]) < 0. On the other hand, from u([[??].sub.0], [x.sub.0]) = 0 and u(t,x) [greater than or equal to] 0, (t,x) [member of] (0,T] x [beta], we conclude that [mathematical expression not reproducible].

By Lemma 5, [L.sub.1]u([[??].sub.0], [x.sub.0]) [greater than or equal to] 0. Moreover, (ru)([[??].sub.0], [x.sub.0]) = 0 and F [greater than or equal to] 0. Lefthand side of the equation [D.sup.{K}.sub.t](u - [psi])([[??].sub.0], [x.sub.0]) = [Lu + F]([[??].sub.0], [x.sub.0]) is negative, but right-hand side is nonnegative. Again, we have reached the contradiction. Thus, (ii) holds.

At this point we present somewhat more concrete assumptions on the input data of the direct problem (2.6) that imply the assumptions of Lemma 4 and Lemma 3.

Corollary 2. Let F [greater than or equal to] 0, [psi] = 0 and one of the assumptions (a1)-(a3) hold:

(a1) F [member of] [C.sup.{M},[alpha]]([0, T]; [X.sub.0]) for some 0 < [alpha] < 1 and F(0, *) = 0;

(a2) F [member of] [C.sup.[alpha].sub.0]([0, T ]; [X.sub.0]) and M (t) [greater than or equal to] [ct.sup.[gamma]]-1], t [member of] (0,T) for some c [member of] [R.sub.+], 0 < [gamma] < [alpha] < 1;

(a3) F [member of] [C.sup.[alpha]-[beta].sub.0] ([0, T ]; [X.sub.0]) and [c.sub.1][t.sup.[gamma]-1] [less than or equal to] M (t) [less than or equal to] [c.sub.2][t.sup.[beta]-1], |M/(t)| [less than or equal to] [c.sub.3][t.sup.[beta]-2], t [member of] (0, T), for some [c.sub.1], [c.sub.2], [c.sub.3] [member of] [R.sub.+], 0 < [beta] [less than or equal to] [gamma] < [alpha] < 1.

Then assertions Lemma 4 are satisfied by solution of the problem (2.6).

Proof. Defining X = [X.sub.0], Lemma 3 with Corollary 1 implies that the solution of (2.6) exists and satisfies the smoothness conditions of Lemma 4. It remains to show that (4.10) holds.

The case (at). The relations F [member of] [C.sup.{M},[alpha]]([0, T]; [X.sub.0]), F(0, *) = 0 mean that F = k * [??], where [??] [member of] [C.sup.[alpha]]([0,T]; [X.sub.0]). Thus, it follows from Lemma 3 that the function [??] that solves (2.6) with F, [psi] replaced by [mathematical expression not reproducible] belongs to the space [C.sup.{k}]([0, T]; [X.sub.0]). Next, after convolving equation for u with k it is easy to see that u = k * u solves (2.6) with F = k * F. Therefore, u [member of] k * [C.sup.{k}]([0, T]; [X.sub.0]), that is u = k * M * v = 1 * v, v [member of] C ([0, T ]; [X.sub.0]). This allows us to conclude that u [member of] [C.sup.1] ([0, T]; [X.sub.0]). Hence,

[mathematical expression not reproducible].

The case (a2). Again, by Lemma 3 (ii), u [member of] [C.sup.{k},[alpha].sub.0]([0,T]; [X.sub.0]) and by (4.5), u [member of] [C.sup.[alpha]].sub.0]([0, T]; [X.sub.0]). The relation (4.10) follows from the estimate

[mathematical expression not reproducible]

The case (a3). According to Lemma 3 (ii), F [member of] [C.sup.[alpha]-[beta].sub.0] [beta]([0,T]; [X.sub.0]) implies that u [member of] [C.sup.{k}, [alpha]-[beta].sub.0]{k}, ([0, T]; [X.sub.0]) = M * [C.sup.[alpha]-[beta].sub.0]([0,T]; [X.sub.0]). By Lemma 2 it holds u [member of] [C.sup.[alpha].sub.0]([0,T]; [X.sub.0]). This enables us finish the proof as in case (a2).

5 Results on IP1

We will study IP1 in context of Huolder spaces with respect to t. For the sake of generality, we will assume different orders of spaces related to g and h: for g we use [[alpha].sub.1] and for h we use [[alpha].sub.2].

Theorem 1. Let one of the following assumptions be valid: (A1) [mathematical expression not reproducible]

(A2) [mathematical expression not reproducible]

(A3) [mathematical expression not reproducible]

Additionally, we assume that g [greater than or equal to] 0, [g.sub.1] := [D.sup.{k}.sub.t] g - [R.sub.g] [greater than or equal to] 0 where [mathematical expression not reproducible]

and

a.e. x [member of] [OMEGA] [[there exists]t.sub.x] [member of] (0,T]: g([t.sub.x], x) > 0. (5.1)

In case B = I we also assume that [for all]x [member of] [partial derivative][OMEGA], either g(T, x) > 0 or g(*, x) = 0.

Finally, let (f, u) [member of] C ([bar.[OMEGA]]) x [C.sup.{k}.sub.0]([0, T]; C ([bar.[OMEGA]])) [intersection] [C.sub.0]([0,T]; [W.sup.2.sub.p]([OMEGA]))) for some p > 1 solve IP1 for [psi] = 0, [psi] = 0, h = 0. Then (f, u) = (0, 0).

Proof. We start the proof by showing that in case B = I, for any x [member of] [partial derivative][OMEGA] such that g(T, x) > 0, the equality f (x) = 0 is valid. To show this, we consider the equality

[D.sup.{k}.sub.t] u(T, x) = f (x)g(T, x), x [member of] [bar.[OMEGA]],

that follows from equation (2.6) in view of [psi] = 0. If x [member of] [partial derivative][OMEGA] and B = I then the left-hand side of this equality equals zero. Thus, f(x)g(T, x) = 0 and provided g(T, x) > 0 we obtain f(x) = 0.

Let us introduce the functions [f.sup.+] = [absolute value of f] - f/2 and [f.sup.-] = [absolute value of f]+f/2. Due to the definition, [f.sup.[+ or -]] [member of] C([bar.[OMEGA]]) and [f.sup.[+ or -]] [greater than or equal to] 0. Moreover, in case B = I, for any x [member of] [partial derivative][OMEGA] such that g(T, x) > 0, it holds [f.sup.[+ or -]] (x) = 0. (5.2)

Firstly, we consider the problems

[mathematical expression not reproducible] (5.3)

By assumptions of the theorem and (5.2), g(t, *)[f.sup.[+ or -]] [member of] [X.sub.0], t [member of] [0, T]. Therefore, in cases (A1) and (A2) due to (4.5) we have [mathematical expression not reproducible] and [mathematical expression not reproducible] ([0, T]; [X.sub.0]), respectively. Similarly, in case (A3) due to (4.5) and Lemma 2 we obtain [mathematical expression not reproducible]. Moreover, g [f.sup.[+ or -]] [greater than or equal to] 0. The assumptions of Corollary 2 are satisfied for the functions F = g [f.sup.[+ or -]]. Hence, the solutions [u.sup.[+ or -]] of (5.3) satisfy the assertions of Lemma 4.

Secondly, let us consider the problems

[mathematical expression not reproducible] (5.4)

v[+ or -] (0, x) = 0, x [member of] [beta], Bv[+ or -] (t, x) = 0, x [member of] d[beta], t [member of] (0,T).

In case (A1) we have [mathematical expression not reproducible]. Thus, [mathematical expression not reproducible]. From [mathematical expression not reproducible] we immediately get [g.sub.1] (t, *) [f.sup.[+ or -]] [member of] [X.sub.0], t [member of] [0,T]. Therefore, [mathematical expression not reproducible].

Using similar reasoning, we deduce [mathematical expression not reproducible]. in cases (A2) and (A3), respectively. Moreover, [g.sub.1][f.sup.[+ or -]] [greater than or equal to] 0. Again, the assumptions of Corollary 2 are satisfied for F = [g.sub.1][f.sup.[+ or -]]. The solutions v[.sup.[+ or -]] of (5.4) satisfy the assertions of Lemma 4.

Let us point out that the problem for M * [v.sup.[+ or -]] is equivalent to the problem for [u.sup.[+ or -]] - RM * [u.sup.[+ or -]]. Thus,

[v.sup.[+ or -]] = [D.sup.{k}.sub.t][u.sup.[+ or -]] - [Ru.sup.[+ or -]]. (5.5)

Moreover, since f = [f.sup.+]-[f.sup.-], we have u = [u.sup.+]-[u.sup.-]. Thus, = u(T, *) = 0 implies that [u.sup.+](T, *) = [u.sup.-](T, *). Let us denote [mathematical expression not reproducible]. By definition, either [f.sup.+]([x.sup.*]) = 0 or [f.sup.-]([x.sup.*]) = 0. Let us assume that [f.sup.+] ([x.sup.*]) = 0 (the situation when [f.sup.-]([x.sup.*]) = 0 can be considered in a similar manner).

Let us suppose that either [x.sup.*] [member of] [OMEGA] or B = [omega] * [nabla] (the case [x.sup.*] [member of] [partial derivative][OMEGA] and B = I will be considered later separately). Then we can apply Lemma 5 to the function w = -[u.sup.+](T, *). We get [L.sub.1][u.sup.+](T,[x.sup.*]) [less than or equal to] 0. Thus, from (5.3), (5.5) and [u.sub.+] [greater than or equal to] 0, r [less than or equal to] R it follows:

[v.sup.+](T, [x.sup.*]) = [L.sub.1][u.sup.+](T, [x.sup.*]) + (r([x.sup.+]) - -R)[u.sup.+](T, [x.sup.*]) < 0. (5.6)

Due to Lemma 4 (i),

[v.sup.+](t,x) [greater than or equal to] 0, (t,x) [member of] (0,T) x [beta]. (5.7)

Hence, (5.6) and (5.7) imply [v.sup.+](T, [x.sup.*]) = 0. Thus, by Lemma 4 (ii), [v.sup.+](t,[x.sup.*]) = 0, t [member of] [0,T]. By formula (5.5) it means [D.sup.{k}.sub.t][u.sup.+](t,[x.sup.*])--[Ru.sup.+](t, [x.sup.*]) =0, t [member of] [0, T]. Applying M * to to this equality, we obtain the following homogeneous Volterra equation of the second kind:

[u.sup.+](t,[x.sup.*]) - RM * [u.sup.+](t,[x.sup.*]) = 0, t [member of] [0, T].

It has only the trivial solution [u.sup.+](t, [x.sup.*]) = 0, t [member of] [0, T]. Hence, [u.sup.+](T, [x.sup.*]) =0.

Since [x.sup.*] is a maximum point of [u.sup.+](T, x) and [u.sup.+](T, x) > 0, we also get

[u.sup.+] (T, x) = 0, x [member of] Q. (5.8)

Now we consider the case x * [member of] [partial derivative][OMEGA], B = I, too. Then by [Bu.sup.+]|[partial derivative] [OMEGA] = 0, immediately [u.sup.+](T, [x.sup.*]) = 0 and again we have (5.8).

Since u = [u.sup.+] - [u.sup.-] and [psi] = u(T, *) = 0 holds, from (5.8) we get [u.sup.[+ or -]](T, x) = 0, x [member of] [OMEGA]. Lemma 4 (ii) implies [u.sup.[+ or -]](t, x) = 0, (t, x) [member of] [0,T] x [OMEGA]. Therefore, u(t,x) = 0, (t, x) [member of] [0, T] x [OMEGA]. From the differential equation for u we obtain f (x)g(t, x) = 0, (t, x) [member of] [0, T] x [OMEGA]. Finally, (5.1) yields f = 0.

Next we provide simple sufficient conditions that imply the assumption [D.sup.{K}.sub.t] g - Rg [greater than or equal to] 0 in Theorem 1. For this we need the following lemma.

Lemma 6. Let w [member of] [C.sup.{k}]([0,T]; R) be nonnegative and nonincreasing. Then [D.sup.{k}.sub.t] w [greater than or equal to] k(T)w.

Proof. The assertion follows from the estimate

[mathematical expression not reproducible].

Due to that Lemma 6, [D.sup.{k}.sub.t] g - Rg [greater than or equal to] 0 holds provided along with other assumptions on g in Theorem 1, g is nondecreasing in t and k(T) [greater than or equal to] R in case R > 0.

Theorem 2. Let g, M satisfy the assumptions of Theorem 1 and the inequality g(T, x) > 0, x [member of] [bar.[OMEGA]], hold. If [mathematical expression not reproducible] and [mathematical expression not reproducible], where p [member of] {0} [union] (1, [infinity]), 0 < [[alpha].sub.2] < 1, then IP1 has a unique solution [mathematical expression not reproducible] and the following estimate holds:

[mathematical expression not reproducible]. (5.9)

If additionally [psi] = h(0, *) = 0, then [mathematical expression not reproducible] where [mathematical expression not reproducible] and the estimate

[mathematical expression not reproducible] (5.10)

is valid. The constants C5 and C6 depend on the parameters M, L, g,p, [[alpha].sub.2].

Proof. Firstly, we are going to replace the overdetermination condition (2.9) by a fixed-point equation with respect to f. n

Suppose that [mathematical expression not reproducible] solves IP1. Then, since (2.9) holds, the equation (2.6) at t = T with F = fg + h yields

[mathematical expression not reproducible], (5.11)

where [eta] is chosen so that 0 [member of] [rho] ([A.sub.p] - [eta]I).

Let us split u into the sum of two functions: u = [u.sub.1] + [u.sub.2], such that

[mathematical expression not reproducible]. (5.12)

In the context of IP1, [u.sub.2] is a known function. According to Lemma 3, the solution to (5.12) belongs to [u.sub.2] [member of] [C.sup.{k}]([0, T]; [X.sub.p]). Thus, [v.sub.2] := [D.sup.{k}.sub.t]([u.sub.2] - [psi]) - [eta][u.sub.2] [member of] C([0, T]; [X.sub.p]). Next we formulate the following problem:

[D.sup.{k}.sub.t][v.sub.1] = [A.sub.p][v.sub.1] + f ([D.sup.{k}.subt] g- [eta]g), [v.sub.1](0, *) = 0. (5.13)

Due to the assumptions (A1)-(A3) and (4.5), it holds [D.sup.{k}.sub.t] g [member of] [C.sup.[??].sub.0]([0, T]; C([bar.[OMEGA])) where

[mathematical expression not reproducible], (5.14)

Thus, f([D.sup.{k}.sub.t] g - [eta]g) [member of] [C.sup.[??].sub.0]([0,T]; [X.sub.p]). According to Lemma 3, (5.13) has

a solution. It is easy to check that [v.sub.1] = [D.sup.{k}.sub.t] [u.sub.1] - [eta][u.sub.1].

The notations introduced allow us to rewrite (5.11) in the form

f = Ff + G, (5.15)

where

g(x) = [v.sub.2](T,x)--([A.sub.p] - [eta]) [psi] (x) - h(T,x)/g(T, x), x [member of] [OMEGA], (5.16)

(F f)(x) = [v.sub.1][f](T,x)/g(T,x) (5.17)

and [v.sub.1][*] stands for the operator that assigns to f the solution [v.sub.1] of (5.13). Thus, (2.6), (2.9), (2.10) imply (5.15). On the other hand, taking into account all the substitutions performed, we can move back from (5.15) to (5.11). Together with (2.6) at t = T and (2.10) it implies ([A.sub.p] - [eta])u(T, x) = ([A.sub.p] - [eta])[psi](x). Since ([A.sub.p] - [eta]) is injective, it yields (2.9). Consequently, IP1 is in the space [mathematical expression not reproducible] equivalent to the problem of finding the pair of functions (f, u) that solves (2.6), (2.10), (5.15).

We point out that (5.15) is an independent equation for the first component f of the solution of IP1. Let us analyse properties of the operator F involved in this equation. By Lemma 3, [mathematical expression not reproducible]. Thus, [mathematical expression not reproducible].

Furthermore, [mathematical expression not reproducible]. In case p [member of] (1, [infinity]) it is a direct consequence of [W.sup.2.sub.p] ([OMEGA]) [??] [??] [L.sub.p]([OMEGA]). In case p = 0 it follows from the continuous embedding

of [mathematical expression not reproducible] (see Theorems 3.1.19, 3.1.22 in [24]) and [C.sup.1.sub.B]([OMEGA]) [??] [??] [X.sub.0].

Therefore, [v.sub.1][*](T, *) : [X.sub.p] [right arrow] [X.sub.p] is compact. Since 1/g(T, *) [member of] C([bar.[OMEGA]]) due to the assumptions of this theorem, F : [X.sub.p] [right arrow] [X.sub.p] is also compact.

Next, let us show that 1 [??] [sigma] (F). Firstly, let us consider the case p = 0. Suppose that 1 [member of] [sigma] (F). Then the equation f = Ff has a solution f [member of] [X.sub.0], f [not equal to] 0. This means that the problem (2.6), (2.10), (5.15) with homogeneous data [psi] = 0, [psi] = 0, h = 0 has the nontrivial solution (f, [u.sub.1]) in the space [mathematical expression not reproducible]). But due to the Theorem 1, IP1 with a homogeneous data has only the trivial solution in such a space. We came to a contradiction. Consequently, 1 [??] [sigma] (F).

Secondly, let us consider the case p [member of] (1, [infinity]). We again suppose that 1 [member of] [sigma] (F), hence the equation f = F f has a nontrivial solution f [member of] [X.sub.p]. The idea is to show that this solution actually belongs to [X.sub.0]. Then we can apply the arguments from the previous case to show that 1 [member of] [sigma](F) leads to a contradiction.

If [mathematical expression not reproducible]. Thus, f = F f = 1/g(T, x) [v.sub.1][f](T, *) [member of] [X.sub.0]. If p [less than or equal to] n/2, then according to embedding theorems, [mathematical expression not reproducible] where [p.sub.1] = np/n-2p > p. Therefore, [mathematical expression not reproducible]. After a finite number of iterations we obtain [mathematical expression not reproducible], where [p.sub.i] = np/n-2ip > n/2 (works for i > n/2p -1). Next iteration gives f [member of] [X.sub.0].

We have shown that the first case of Fredholm alternative is satisfied for the equation (5.15). Consequently, the solution to (5.15) exists and is unique for any g [member of] [X.sub.p] and [(I - F).sup.-1] [member of] [laplace]([X.sub.p]).

Since F = fg + h is Holder-continuous with values in [X.sub.p], Lemma 3 implies that the problem (2.6), (2.10) has unique solution [mathematical expression not reproducible]. This completes the proof of the existence and uniqueness assertion of the theorem.

In the rest of the proof, C stands for a generic constant depending on the parameters M, L, g, p, [[alpha].sub.2]. Let us deduce the stability estimate (5.9). We obtain

[mathematical expression not reproducible]. (5.18)

Further, we note that g [member of] [C.sup.[gamma].sub.0]([0, T]; C([bar.[OMEGA])) for any [gamma] [member of] (0,1) in case (A1) and for [gamma] = [[alpha].sub.1] in cases (A2), (A3). Using Lemma 3 we have

[mathematical expression not reproducible]

Together with the estimate of f (5.18) it implies (5.9).

In case [psi] = h(0, *) = 0, the solution of (2.6), (2.10) belongs to the space [mathematical expression not reproducible] and can be estimated as

[mathematical expression not reproducible]

This with (5.18) implies (5.10).

We point out that in case p = 0 and B = I, the assumptions of Theorem 2 allow to recover f [member of] [X.sub.0] = [C.sub.0]([bar.[OMEGA]]) only. In order to fix that in the following theorem we provide some additional conditions that are sufficient to restore f [member of] C([bar.[omega]]) in case B = I.

Theorem 3. Let g, M satisfy the assumptions of Theorem 2. If [mathematical expression not reproducible] for some [mathematical expression not reproducible] where [mathematical expression not reproducible] then IP1 has a unique solution (f, u) [member of] [mathematical expression not reproducible]. Moreoverp Lu [member of] C([0,T]; C([bar.[OMEGA]])) and the estimate

[mathematical expression not reproducible] (5.19)

holds. If additionally [mathematical expression not reproducible] and the estimate

[mathematical expression not reproducible] (5.20)

is valid, where [alpha]' = min{[??]; [[alpha].sub.2]} and [??] is given by (5.14). The constants [C.sub.7] and [C.sub.8] depend on M, L, g, p, [[alpha].sub.2].

Proof. Throughout the proof, [??] denotes a generic constant depending on M, L, g,p, [[alpha].sub.2] and RHS stands for the expression in brackets at the righthand side of (5.19). By Theorem 2, IP1 has a unique solution [mathematical expression not reproducible]. Let us consider the problem

[mathematical expression not reproducible]. (5.21)

Under the assumptions of this theorem, Lemma 3 implies that (5.21) has a unique solution [mathematical expression not reproducible]. Moreover, due to (4.7) and (4.8), [mathematical expression not reproducible] It is easy to check that [w.sub.2] = [D.sup.{k}] * ([u.sub.2] - [psi]) and [u.sub.2] = M * [w.sub.2] + [psi] where [u.sub.2] solves (5.12). Therefore, we have [mathematical expression not reproducible] and

[mathematical expression not reproducible]. (5.22)

Let us consider the function g given by (5.16). (Recall that there [v.sub.2] = [w.sub.2] - [eta][u.sub.2].) Due the proved properties of [w.sub.2] and [u.sub.2] and the assumptions of the theorem, it holds g [member of] C([bar.[OMEGA]) and [parallel]G[parallel]C([bar.[omega]] < [??]RHS.

Now, let us provide an estimate for [parallel]f[parallel]C([bar.[OMEGA]]) using the formulas (5.15) and (5.17). Since 1/g(T,) [member of] C([bar.[OMEGA]]) and [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

Since (I--F) is invertible in [X.sub.p], the estimate holds

[mathematical expression not reproducible].

Thus, we obtain

[parallel]f[parallel]C([bar.[OMEGA]]) < [??]RHS. (5.23)

Finally, let us derive an estimate for u and finish the proof of the first part of the theorem. We have u = [u.sub.1] + [u.sub.2], where [u.sub.1] = M * [w.sub.1], [w.sub.1] = [D.sup.{k}.sub.t][u.sub.1] and [w.sub.1] solves the problem

[D.sup.{k}.sub.t][w.sub.1] = [A.sub.p][w.sub.1] + f[D.sup.{k}.sub.t]g, [w.sub.1] (0, *) = 0.

Since f[D.sup.{k}.sub.t] g [member of] [C.sup.[alpha]]'.sub.0] ([0,T]; [X.sub.p]), Lemma 3 implies [mathematical expression not reproducible] and [mathematical expression not reproducible] Using here (5.23) we have

[mathematical expression not reproducible]. (5.24)

From (5.22) and (5.24) we obtain for u = [u.sub.1] + [u.sub.2] the estimate

[mathematical expression not reproducible]. (5.25)

It remains to estimate Lu in the space C([0, T]; C([bar.[OMEGA]])). Using (5.25) we deduce

[mathematical expression not reproducible].

From the expression Lu = [D.sup.{k}.sub.t](u - [psi]) - fg - h due to the proved estimates for [D.sup.{k}](u - [psi]) and f we obtain

[mathematical expression not reproducible]. (5.26)

Summing up, (5.23), (5.25) and (5.26) imply (5.19).

Now let us focus on the second part of this theorem that is concerned with the particular case [psi] = h(0, *) = [D.sup.{k}]h(0, *) = 0. Then RHS reduces to the expression in brackets at the right-hand side of (5.20). Lemma 3 implies that the function [w.sub.2] which solves (5.21) belongs the space [mathematical expression not reproducible], the function [u.sub.2] = M * [w.sub.2] belongs to [mathematical expression not reproducible] and [mathematical expression not reproducible]. This relation by u = [u.sub.1] + [u.sub.2] and the estimates (5.23), (5.24) and (5.26) implies (5.20).

Provided the assumptions of Theorem 3 hold and B = I, an explicit expression of the unknown function f at the boundary can be derived. Namely, setting t = T and x [member of] [partial derivative][OMEGA] in (2.6) and taking the relations F = fg + h and u(T, *) = [psi] into account we obtain f (x) = -1/g(T, x) [L[psi](x) + h(T, x)], x [member of] [partial derivative][OMEGA].

6 Results on IP2

In the context of IP2 let us introduce the following sets for the coefficient r:

[K.sub.R] = {r [member of] C([bar.[OMEGA]]) : r(x) [less than or equal to] R, x [member of] [bar.[OMEGA]]}, where R [member of] R.

Theorem 4. Let R be some real number and IP2 have two solutions (r, u), ([r.sub.1], [u.sub.1]), such that

[mathematical expression not reproducible]

for some p > 1 and the function U = u + b (and M) satisfy one of the following assumptions:

(A4) [mathematical expression not reproducible]

(A5) [mathematical expression not reproducible]

(A6) [mathematical expression not reproducible]

Additionally, we assume that

U [greater than or equal to] 0, [D.sup.{k}.sub.t]U - RU [greater than or equal to] 0, (6.1)

a.e. x [member of] [OMEGA], [[there existst.sub.x] [member of] (0, T] : U([t.sub.x], x) > 0.

In case B = I we also assume that [for all]x [member of] [partial derivative][OMEGA], either U (T, x) > 0 or U (*, x) = 0. Then (r1,[u.sub.1]) = (r, u).

Proof. The difference [mathematical expression not reproducible] solves the problem

[mathematical expression not reproducible], (6.2)

The inequalities (6.1) imply that [D.sup.{k}.sub.t]U - [R.sub.r],U [greater than or equal to] 0, where [mathematical expression not reproducible] R. Consequently, the assumptions of Theorem 1 are satisfied for the problem (6.2) and we obtain [mathematical expression not reproducible].

Let us formulate a problem that contains approximate data:

[mathematical expression not reproducible] (6.3)

We are going to prove an existence and approximation theorem for this problem in case its data vector [mathematical expression not reproducible] is close to the data vector D = (b, [F.sub.1], [psi]) of the exact problem IP2.

Theorem 5. Assume that R [member of] R and IP2 has a solution (r, u) [member of] [K.sub.R] x [D.sup.{k}.sub.0] ([0,T]; [L.sub.1]([OMEGA])) [intersection] [C.sub.0]([0,T]; [W.sup.2.sub.1]([OMEGA])) such that U = u + b (and M) satisfy one of the assumptions (A4)-(A6), the inequalities (6.1) and U(T, x) > 0, x [member of] [bar.[OMEGA]]. Then the following statements are valid.

(i) Let p [member of] {0}U (n/2 [infinity]), [[alpha].sub.2] [member of] (0,1). There exist constants [[delta].sub.1] > 0 and [K.sub.1] > 0 depending on M, [L.sub.1], r, U, p, [[alpha].sub.2] such that if

[mathematical expression not reproducible], then

problem (6.3) has a unique solution in the set

[mathematical expression not reproducible]

(ii) Let p [member of] (n, [[alpha].sub.2] [member of] (0, 1). There exist constants [[delta].sub.2] > 0 and [K.sub.2] > 0 depending on M, [L.sub.1], r, U, p, [[alpha].sub.2] such that if

[mathematical expression not reproducible]

and [mathematical expression not reproducible] where [mathematical expression not reproducible], then the problem (6.3) has a unique solution in the set

[mathematical expression not reproducible]

where [mathematical expression not reproducible]

We mention that in this theorem, the operator [A.sub.p] and the space [mathematical expression not reproducible] defined on the basis of L = [L.sub.1] + rI depend on the component r of the solution of the exact problem IP2.

Proof. Let us denote the difference [mathematical expression not reproducible]. Then the problem for the pair [mathematical expression not reproducible] reads

[mathematical expression not reproducible] (6.4)

This problem can be treated as IP1 with [mathematical expression not reproducible]. Therefore, applying the solution operator of IP1 A to (6.4), it is reduced to the operator equation

[mathematical expression not reproducible], (6.5)

where [mathematical expression not reproducible].

We are going to show that [F.sub.2] is a contraction in a ball [mathematical expression not reproducible] with a suitable chosen [rho] > 0. Firstly, we have to prove that this ball remains invariant with respect to the operator [F.sub.2]. Let ||(r,u)|xi < p. According to (5.10),

[mathematical expression not reproducible]

Let [c.sub.p] be an embedding constant such that [mathematical expression not reproducible]. Then [mathematical expression not reproducible].

Therefore,

[mathematical expression not reproducible]

where [mathematical expression not reproducible] in case p [member of] (n/2, [infinity]), and [R.sub.1] = [[parallel]r[parallel].sub.C]([bar.[OMEGA]]) in case p = 0. Now let us take [mathematical expression not reproducible] with a constant [K.sub.1]. Then

[mathematical expression not reproducible]

In case [mathematical expression not reproducible] we have

[mathematical expression not reproducible]

Let us define the constants as follows: [mathematical expression not reproducible] Then [mathematical expression not reproducible]. Consequently, for [mathematical expression not reproducible] we have [mathematical expression not reproducible].

Secondly, inside the set [mathematical expression not reproducible] let us consider the difference of [F.sub.2] at [mathematical expression not reproducible] and [mathematical expression not reproducible]. Assuming [mathematical expression not reproducible], we deduce the estimate

[mathematical expression not reproducible]

It shows that the operator [F.sub.2] is a contraction in the ball [mathematical expression not reproducible]. According to the Banach fixed point theorem there exists a unique solution to the equation (6.5) in that ball. This proves the assertion (i).

(ii) The proof of (ii) repeats the proof of (i) with appropriate changes of spaces and norms. For A, the estimate (5.20) is used instead of (5.10).

Remark 1. In case the data of (6.3) are close to data of a process without reaction (i.e. r = 0), Theorem 5 implies the existence of the reaction coefficient [??] in small.

Remark 2. Supposing the existence of a solution (r, u) of IP2, we ask: what are sufficient conditions on the data that guarantee the validity of inequality-type conditions (6.1) and U(T, x) > 0, x [member of] [bar.[OMEGA]] in Theorems 4, 5? To answer this question, we return to the problem (2.3) for U and set there [PHI] = H(0, *) = 0. Let us suppose that U is sufficiently smooth. Then constructing a corresponding problem for [D.sup.{k}.sub.t]U--RU and assuming [D.sup.{k}.sub.t]H--RH [greater than or equal to] 0, ([D.sup.{k}.sub.t]Bb--RBb)|[partial derivative] [OMEGA] [greater than or equal to] 0, Lemma 4 (i) implies the inequality [D.sup.{k}.sub.t]U -RU [greater than or equal to] 0. Next, we consider the conditions U [greater than or equal to] 0 and U(T, x) > 0, x [member of] [bar.[OMEGA]]. Let us assume that

[mathematical expression not reproducible],--nondeCTeasing : [mathematical expression not reproducible]

Define V = U--[delta]1 * [mu] with [delta] > 0. The function V solves the problem

[D.sup.{k}.sub.t]V = LV + [H.sub.1], V(0, *)=0, B(V--(b--[delta]1 * y))|[partial derivative] [OMEGA] = 0,

where [H.sub.1] = H + [delta](r1 * [mu]-- [D.sup.{k}.sub.t] 1 * [mu]). Since [D.sup.{k}.sub.t] 1 * [mu] = k * [mu], we get that for

sufficiently small [delta],

[mathematical expression not reproducible]

and BV|[partial derivative][OMEGA] = B(b - [delta]1 * [mu])[partial derivative] [OMEGA] [greater than or equal to] 0. Lemma 4 (i) yields V [greater than or equal to] 0. Thus, [mathematical expression not reproducible].

At the end of this section, we make some general remarks. We applied results on IP1 to analyze IP2. In a similar manner, results on IP1 can be applied to study inverse problems to determine other coefficients of L, too.

The basic set of assumptions (A1)-(A3) for g involves the restriction g(0, *) = 0. This is due to the fact that in case g(0, *) [not equal to] 0 we cannot ensure sufficient regularity of u to apply the positivity principle in the proof of Theorem 1. In IP2, the function u + b = U works as g. For that reason, we consider the case [PHI] = U(0, *) = 0 in IP2.

In the beginning of the proof of Lemma 4 we showed that the direct problem with r > 0 can be reduced to a problem with r [less than or equal to] 0 by the change of unknown [??] = [e.sup.[sigma]t]u, where [sigma] > 0. This suggests a possible exponential growth of u and a related time limitation of the linear reaction model in case r > 0. For bigger T, nonlinear reaction models are more relevant [6].

Solutions of IP1 and IP2 depend continuously on derivatives of the data of finite order. This means that these problems are moderately ill-posed. In case approximate data are given with errors, regularization procedures can be effectively applied (cf. e.g. [17] for IP1 with g = g(t)).

https://doi.org/10.3846/mma.2019.016

Acknowledgements

The research was supported by the Estonian Research Council, Grant PUT568. Authors thank the referee whose valuable comments led to the improvement of the paper.

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Appendix: Proofs of Lemmas 1, 2 and 5

Proof of Lemma 1. Theorems 3 and 4 of [9] guarantee that k is nonnegative, nonincreasing and convex. Convexity implies the continuity of k. From the equation M * k = 1 we easily deduce [mathematical expression not reproducible], because in the opposite case k is bounded from which it follows that [mathematical expression not reproducible].

Let us prove k > 0. Suppose that it is not true. Then in view of proved properties of k, [[there exists]t.sub.0] : k(t) > 0, t < [t.sub.0] and k(t) = 0, t > [t.sub.0]. For t > [t.sub.0] from M * k = 1 we get [mathematical expression not reproducible]. Therefore, [mathematical expression not reproducible].

The last equality contradicts to the assumptions k(t) > 0, t [member of] (0, [t.sub.0]) and M' < 0. Thus, k > 0.

Finally, let us prove (3.3) Let us choose some [t.sub.3] > 0. Since [mathematical expression not reproducible] there exists an interval (0,[delta]), [delta] < [t.sub.3], such that k(t) > k([t.sub.3]) for t [member of] (0,[delta]). Suppose that (3.3) is not true. Then we can find two points [t.sub.1] < [t.sub.2] in (0, 5) so that k([t.sub.1]) = k([t.sub.2]). Consequently, for [t.sub.1] < [t.sub.2] < [t.sub.3] we have k([t.sub.1]) = k([t.sub.2]) > k([t.sub.3]). Obviously, it contradicts to the convexity of k. Therefore, (3.3) is valid.?

Proof of Lemma 2 is similar to proof of Theorem 14 in [11] that is concerned with the case M (t) = [t.sup.[beta]-1.[GAMMA] ([beta]). Let z [member of] [C.sup.[alpha]-[beta].sub.0]([0,T]; X). Then [[parallel]M * z(t)[parallel].sub.X] [less than or equal to] const [t.sup.[beta]-1] * [t.sup.[alpha]-[beta]] = O(t[.sup.[alpha]]). Secondly,

(M * z)(t)-(M * z)(t-h) = [J.sub.1] + [J.sub.2] + [J.sub.3],

where

[mathematical expression not reproducible]

Immediately, [[parallel][J.sub.2][parallel].sub.x] [less than or equal to] const [mathematical expression not reproducible]. Moreover,

[mathematical expression not reproducible]

Further estimation of [J.sub.1] and [J.sub.3] can be performed exactly as in [11]. As a result, we get [[parallel][J.sub.1][parallel].sub.X], [[parallel][J.sub.3][parallel].sub.X] = O([h.sup.[alpha]]). This completes the proof. ?

Proof of Lemma 5. Firstly, we point out that the assumption w [member of] [W.sup.2.sub.p][OMEGA]), p > n implies w [member of] [C.sup.1]([bar.[OMEGA]]). We will use maximum principles for elliptic equations in Sobolev spaces to prove the lemma. Let us consider the case x * [member of] [beta]. Suppose that [L.sub.1]w([x.sup.*]) < 0. Then there exists a ball B([x.sup.*],[epsilon]) [subset] [OMEGA] and [delta] > 0 such that [L.sub.1]w(x) [less than or equal to] -[delta] < 0 for x [member of] B([x.sup.*], [epsilon]). Let us define the auxiliary function

z(x) = [alpha][[absolute value of x--[x.sup.*]].sup.2] with [alpha] > 0 (7.1)

such that [L.sub.1](w + z) [less than or equal to] 0 in B([x.sup.*], [epsilon]). Since w([x.sup.*]) [less than or equal to] w(x) and z([x.sup.*]) < z(x) for x [member of] [partial derivative]B([x.sup.*], [epsilon]), we get

(w + z)([x.sup.*]) < (w + z)(x), x [member of] [partial derivative]B ([x.sup.*], [epsilon]). (7.2)

On the other hand, due to [L.sub.1](w+z) < 0 it follows from the Theorem 9.1 [8] that min [mathematical expression not reproducible], that contradicts (7.2). Therefore, the supposition [L.sub.1]w([x.sup.*]) < 0 was wrong.

Next let us consider the case [x.sup.*] [member of] [partial derivative][OMEGA]. Again, suppose [L.sub.1] w([x.sup.*]) < 0. Then there exists B([x.sup.*], [epsilon]) and [delta] > 0 such that [L.sub.1] w(x) [less than or equal to] -[delta] < 0 for x [member of] B([x.sup.*], [epsilon]) [intersection] [OMEGA]. Similarly to the previous case we define z by (7.1) so that [L.sub.1](w + z) [less than or equal to] 0 in B([x.sup.*], [epsilon]) [intersection] [OMEGA]. Then (w + z)([x.sup.*]) < (w + z)(x) for x [member of] B([x.sup.*], [epsilon]) [intersection] [OMEGA]. Hence, Lemma 3.4 [8] is applicable and yields [partial derivative]w/[partial derivative]v ([x.sup.*]) = [partial derivative](w+z)/ [partial derivative]v ([x.sup.*]) < 0. That contradicts to [partial derivative]/ [partial derivative]v w([x.sup.*]) [greater than or equal to] 0 following from the assumption [partial derivative]/ [partial derivative]w w([x.sup.*]) [greater than or equal to] 0. Therefore, [L.sub.1]w([x.sup.*]) [greater than or equal to] 0 holds.

Nataliia Kinash and Jaan Janno

Department of Cybernetics, Tallinn University of Technology

Ehitajate tee 5,19086 Tallinn, Estonia

E-mail: nataliia.kinash@taltech.ee

E-mail(corresp.): jaan.janno@taltech.ee

Received November 22, 2019; revised February 4, 2019; accepted February 6, 2019

(1) The symbol L stands for the space of linear and bounded operators.
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