Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation

An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.


Introduction
Anomalous diffusion processes in porous, fractal, biological etc. media are described by differential equations containing fractional derivatives. Depending on the nature of the process, the model may involve either fractional time or fractional space derivatives or both ones [6,19,31].
In many practical situations the properties of the medium or sources are unknown and they have to be reconstructed solving inverse problems [3,11,24,32]. Then some additional information for the solution of the differential equation is needed to recover the unknowns. If space-dependent quantities are to be determined, such additional conditions may involve instant (e.g. final) measurements or integrated measurements over time.
In case space and time variables are separable, the solution of the time-fractional diffusion equation can be expressed by a formula that is deduced by means of the Fourier expansion with respect to the space variables and integration using Mittag-Leffler functions with respect to the time variable. This formula enables to prove uniqueness of reconstruction of space-dependent factors of source terms from final overdetermination [4,13,27]. However, this method fails even in the linear case when the variables are not separable.
In the present paper we prove the uniqueness for an inverse problem to determine a space-dependent factor in the time-fractional diffusion equation in a more general case when the equation may be semilinear. Additional condition is given in a form of an integral with Borel measure over the time that includes as a particular case the final overdetermination. Such an inverse problem has possible applications in modelling of fractional reaction-diffusion processes [5,12,23], more precisely in reconstruction of certain parameters of inhomogeneous media.
Our results are global in time, but contain certain cone-type restrictions that may depend on a time interval. We will adjust a method that was applied to inverse problems for usual parabolic equations [8,9] and generalized to an integrodifferential case [10] and parabolic semilinear case [1]. The method is based on positivity principles of solutions of differential equations that follow from extremum principles. Extremum principles for time-fractional diffusion equations are proved in the linear case in [2,16,22] and in a nonlinear divergence-type case in [29]. But these results are not directly applicable in our case, because of the lack of the semilinear term. Therefore, we prove an independent positivity principle in our paper.
To the authors' opinion, such a positivity principle has a scientific value independently of the inverse problem, too. States of several reaction-diffusion models are positive functions, e.g. probability densities [12].
The stability of the solution of the inverse problem will not be studied in this paper. In the linear case the stability follows from the uniqueness by means of the Fredholm alternative [24]. The solution continuously depends on certain derivatives of the data, hence the problem is moderately ill-posed.
The plan for the paper is as follows. In the first section we formulate the direct and inverse problems. Second section contains auxiliary results about a linear direct problem that are necessary for the analysis of the inverse problem. Third section is devoted to the positivity principle. In the fourth section we prove the main uniqueness theorem in case of the general additional condition involving the Borel measure. The next section contains a particular uniqueness result in a case of the Lebesgue measure with a weight. In the last section we discuss some crucial assumptions of the uniqueness theorems.

Problem formulation
Let Ω ⊂ R n be a bounded open domain with a sufficiently smooth boundary ∂Ω. We consider the semilinear fractional diffusion equation x) + f (u(t, x), t, x) , t ∈ (0, T ), x ∈ Ω , (1.1) with the initial and boundary conditions with some x-dependent vector function ω(x) = (ω 1 (x), . . . , ω n (x)) such that ω(x) · ν(x) > 0 where ν is the outer normal to ∂Ω. Here and in the sequel ∇ denotes the gradient operator with respect to the space variables. Moreover, is the Riemann-Liouville fractional derivative of order β ∈ (0, 1) and the operator A has the form where the principal part is uniformly elliptic, i.e. n i,j=1 a ij (x)ξ i ξ j ≥ c|ξ| 2 ∀ξ ∈ R n , x ∈ Ω for some c > 0.
Note that in case of sufficiently smooth u the term in the left-hand side of (1.1) is actually the Caputo derivative of u, i.e.
We will consider the case when the nonlinearity function f has the following form: where a and b are given but the factor z is unknown. Let us formulate an inverse problem.
We note that a special case of µ is the Dirac measure concentrated at the final moment t = T . Then the condition (1.6) reads u(T, x) = d(x), x ∈ Ω.

Preliminaries
In this section we formulate and prove some auxiliary results. In addition to D β t , we introduce the operator of fractional integration of order γ > 0: In the sequel we consider the fractional differentation and integration in Bessel potential and Hölder spaces. We consider the abstract Bessel potential spaces H β p ([0, T ]; X) for Banach spaces X of the class HT that is defined in the following manner ( [28], p. 18, [21], p. 216): 1 ) and X is of the class HT . Let us formulate four lemmas that directly follow from known results in the literature.
Assertions of the lemma follow from Thms 14, 19 and the first part of Thm. 20 of [7] if we continue w(t) by zero for t < 0. Although [7] considers the case X = R, all arguments included in proofs of the mentioned theorems automatically hold in case of arbitrary Banach space X, too.
Proof. The existence and uniqueness assertions of Lemma 3 are a particular case of a more general maximal regularity result proved in Thm 4.3.1 of [28]. The proof of the cited theorem is based on approximation of the problem by a sequence of localized problems with constant coefficients. For the localized problems, existence and uniqueness theorem for an abstract parabolic evolutionary integral equation containing unbounded operator is applied. The study of the such an abstract equation is based on the construction of a solution in the form of a variation of parameters formula that contains a convolution of a resolvent (operator) of the equation with the given right-hand side. The desired assertions follow from the properties of the resolvent. A formula of the resolvent is constructed and its properties established by means of the Laplace transform. The analysis of the mentioned abstract parabolic equation is contained also in [30].
To prove the additional assertion, let us consider the problem Due to the proven properties of u and the additional as- Lemma 3 implies that the problem (2.15), (2.16) has a unique solution u 1 ∈ U p and u 1 (0, ·) = 0. The next aim is to show that u t = u 1 . This will complete the proof.

Positivity principle
In this section we prove a positivity principle for the solution of the equation (1.1) in a bit more general form. Namely, we consider the equation and the kernel k has the following properties: This generalization doesn't make the proofs more complicated. Moreover, some parts of proofs even require such a more general treatment.
Finally, we assume u 0 ≥ 0 and g ≥ 0. Then the following assertions are valid: where Recall that the cases I and II were defined in (1.4).
Before proving Theorem 2, we state and prove a lemma.
Lemma 5 (a minimum principle). Let the assumptions of Theorem 2 be satisfied, except for the conditions (3.3) and u 0 , g ≥ 0. Moreover, let instead of (3.5) the following stronger condition Proof. Suppose that the assertion of the lemma doesn't hold. Then Since x = x 1 is the stationary minimum point of u(t 1 , x) over Ω and the principal part of A is elliptic, it holds n i,j=1 [20]). Thus, Further, in view of (3.8) and u(t 1 , Observing that u(τ, x 1 ) ≥ u(t 1 , x 1 ) holds for 0 ≤ τ ≤ t 1 and taking the relation (3.2) into account we estimate the term between the brackets {} in (3.11): Thus, from (3.11) due to (3.7), (3.2) and the inequalities t The inequalities (3.12), (3.9) and (3.10) show that the left-hand side of the equation (3.1) is negative but the right-hand side is nonnegative at t = t 1 , x = x 1 . This is a contradiction. Therefore, the assertion of the lemma is valid.
Proof of Theorem 2. Firstly, let us prove (i) in case (3.5) is replaced by the stronger condition (3.8). Let again (t 1 , x 1 ) be the minimum point of u over [0, T ] × Ω. Suppose that (i) doesn't hold. Then u(t 1 , x 1 ) is negative. In case I the point (t 1 , x 1 ) is contained in the subset (0, T ] × Ω. This implies that ∇u(t 1 , x 1 ) = 0 and in view of Lemma 5 we reach the contradiction. In case II the minimum point (t 1 , x 1 ) is contained in (0, T ] × Ω. If x 1 ∈ Ω, we reach the contradiction in a similar manner. Thus, it remains to consider the case II with x 1 ∈ ∂Ω. In the minimum point at the boundary the inequality ω(x 1 ) · ∇ u(t 1 , x 1 ) ≤ 0 holds. This together with the assumption ω · ∇ u| x∈∂Ω = g ≥ 0 implies ω(x 1 ) · ∇ u(t 1 , x 1 ) = 0. Moreover, τ · ∇ u(t 1 , x 1 ) = 0, where τ is any tangential direction on ∂Ω at x 1 , because x = x 1 is the minimum point of the x-dependent function u(t 1 , x) over the set ∂Ω.
Putting these relations together we see that ∇u(t 1 , x 1 ) = 0. Again, we reach the contradiction with Lemma 5. The assertion (i) is proved in case (3.8).
Secondly, we prove (i) in the general case (3.5). Note that due to the continuity of f , the relation (3.5) can immediately be extended to f (w, t, |f (w, t, x)| and q > 0 is an arbitrary number. Let us define the new nonlinearity function: According to the definition of f , we can rewrite the equation for u (3.1) in the following form: Defineũ(t, x) = e −σt u(t, x), where σ > 0 is a number that we will specify later. Thenũ solves the following equation: , x), t, x), x ∈ Ω, t ∈ (0, T ), (3.15) and satisfies the initial and boundary conditions wherẽ We are going to show that the assumptions of the theorem are satisfied for the problem (3.15), (3.16). The smoothness conditionsk ∈ L 1 (0, T ) ∩ C(0, T ], Finally, we prove (ii). Suppose that this assertion does not hold. Then, due to the continuity of u, for some (t 0 , x 0 ) ∈ (0, T ] × Ω N , such that u(t 0 , x 0 ) = 0, it holds We have In view of (3.7) and the relation u(t 0 , x 0 ) = 0 it holds lim ǫ→0+ 1 ǫ ǫ 0 k(τ )u(t 0 − τ, x 0 )dτ = 0. Therefore, from (3.18) and (3.19) we get On the other hand, (t 0 , x 0 ) is a stationary local minimum point of u(t 0 , x), i.e. it holds ∇u(t 0 , x 0 ) = 0. This follows from the assumption u(t 0 , x 0 ) = 0 and the inequality u(t 0 , x) ≥ 0 that holds in the neighborhood of x 0 as well as from the con- Consequently, the left-hand side of (3.1) is negative but the right-hand side is nonnegative at t = t 0 , x = x 0 . This is a contradiction. The assertion (ii) is valid. Theorem is completely proved.

Uniqueness results in the general case
In this section we state and prove a uniqueness theorem for IP that is the most general in the sense of the Borel measure µ. But firstly we provide a technical lemma. and then f (u, ·, ·) ∈ C α ([0, T ]; C γ (Ω)) and where Cm ,m is a constant depending onm andm. Proof.
Further, due to the range of p and an embedding theorem it holds U p ⊂ Cᾱ([0, T ]; L p (Ω)).
We see that the assumptions of Theorem 1 are satisfied for the problems (4.26), (4.27). This implies that (4.26), (4.27) Taking into account the smoothness properties of q 0 and u ± , the relation u ± (0, ·) = 0 and performing some elementary computations we reach the following expression: By the assumptions (4.7), (4.8) and the formulas (4.14), (4.19) we see that the function q 0 is nondecreasing in t. This with the proven inequalities u ± ≥ 0 implies that D β t (q 0 u ± ) − q 0 D β t u ± ≥ 0. Moreover, by (4.13) and z ± ≥ 0 it hold (D β t − Θ)a(u 1 (t, x), t, x)z ± ≥ 0. Consequently, we have ϕ ± ≥ 0. Applying Theorem 2 with Remark 1 to the equation (4.26), we obtain the inequality v ± ≥ 0 and the relation if v s (t 0 , x 0 ) with some s ∈ {+; −} equals zero in some point (4.28) Next we are going to establish relations between v ± and u ± . To this end, we deduce and analyze problems for the functions Q ± = u ± − J β t v ± + Θu ± . Adding (4.17) multiplied by Θ to (4.26), taking the operator J β t and subtracting from (4.17) after some transformations we arrive at the following equation for Q ± : (4.29) with the initial and boundary conditions where Using (4.20) and the relation with some constantĈ 1 . Estimating the solution of (4.29), (4.30) by means of the technique that was used in derivation of the estimate (2.19) we obtain max 0≤τ ≤t Q ± (τ, ·) Lp(Ω) ≤Ĉ 2 ζ ± Lp((0,t);Lp(Ω)) for any t ∈ (0, T ) with some constant for any t ∈ (0, T ). Applying Gronwall's theorem we reach the equality max 0≤τ ≤t Q ± (τ, ·) Lp(Ω) = 0 for any t ∈ (0, T ). Therefore, Q = u ± −J β t v ± +Θu ± = 0. This yields the following relations between v ± and u ± : Let us return to the equation (4.17). We subtract the term Θu ± from both sides and integrate. Taking into account the right relation in (4.32) we obtain for any x ∈ Ω. By continuity, this relation can be extended to Ω. Due to the relations u 2 − u 1 = u + − u − and the equality in Ω. By continuity, (4.34) holds in x ∈ Ω. Define Without restriction of generality we may assume that x * ∈ Ω N . Indeed, if x * ∈ Ω \ Ω N = ∂Ω in case I, the vanishing boundary condition u ± | ∂Ω = 0 implies T 0 u ± (t, x * )dµ = 0 and since u ± ≥ 0 from (4.35) we get T 0 u ± (t, x)dµ ≡ 0, which means that we can redefine x * as an arbitrary point in Ω.
From the proved theorem we infer the following uniqueness result for a linear inverse problem.

Uniqueness in a particular case
In case the measure µ has a special form, the uniqueness can be proved under lower regularity assumptions and the cone condition (4.7) and the restriction (4.8) dropped. Let us consider the following particular case: dµ = κ(t)dt, where dt is the Lebesgue measure and κ ≥ 0, κ = 0. (5.1) Theorem 4. Let (5.1) hold, where κ ∈ W 1 s (0, T ) with some s > 1 1−β . Suppose that IP has two solutions (z j , u j ), j = 1, 2, such that with some α ∈ (0, 1). Assume that a ij , a j , ω satisfy the conditions of Theorem 1 and a, a w , b w ∈F α+β ∩ F α,α .

Additional remarks
The aim of this section is to interpret the conditions (4.8), (4.10), (4.11) in a suitable way and estimate the quantity Θ occurring in the conditions (4.12), (4.39) from below.