Determination of time dependent factors of coefficients in fractional diffusion equations

We consider fractional diffusion equations and study the stability of the inverse problem of determining the time-dependent parameter in a source term or a coefficient of zero-th order term from observations of the solution at one point in a bounded domain.


Introduction
Let Ω be a bounded domain of R d , d = 1, 2, 3, with C 2 boundary ∂Ω. We set Σ = ∂Ω×(0, T ) and Q = Ω × (0, T ). We consider the following two initial-boundary value problem (IBVP in short) for the fractional diffusion equation with 0 < α < 1. Here we denote by ∂ α t the Caputo fractional derivative with respect to t: The differentiual operator A is defined by where the coefficients satisfy a ij = a ji , 1 ≤ i, j ≤ d, and a ij (x)ξ i ξ j ≥ µ|ξ| 2 , x ∈ Ω, ξ ∈ R d for some µ > 0. Moreover B σ is defined as and ν = (ν 1 , . . . , ν d ) is the outward normal unit vector to ∂Ω. Here σ is a C 2 function on ∂Ω satisfying For the regularity of a ij , we assume Note that the regularity for a ij depends on whether σ ≡ 0 or not, which is due to condition (2.3) in the next section.
In the present paper, we consider the inverse problem which consists of determining the function {f (t)} t∈(0,T ) in (1.1) and (1.2) from the observation of the solution at a point x 0 ∈ Ω for all t ∈ (0, T ).
The partial differential equations with time fractional derivatives such as (1.1) and (1.2) are proposed as new models describing the anomalous diffusion phenomena. Adams and Gelhar [1] pointed out that the field data in a highly heterogeneous aquifer cannot be described well by the classical advection diffusion equation. Hatano and Hatano [11] applied the continuous-time random walk (CTRW) as a microscopic model of the diffusion of ions in heterogeneous media. From the CTRW model, one can derive a fractional diffusion equation as a macroscopic model (see e.g., Metzler and Klafter [15] and Roman and Alemany [18]). In particular, the fractional diffusion equation can be used as a model for the diffusion of contaminants in a soil. Therefore the inverse problem considered in this paper means the determination of the time evolution of pollution source in (1.1) and reaction rate of pollutants in (1.2) respectively. In this paper, we consider such problems assuming the boundedness of the time-dependent parameter {f (t)} t∈(0,T ) (see (2.1)).
The remainder of this paper is composed of four sections. In Section 2, we state our main results. In Section 3, we study the forward problem for the IBVPs (1.1) and (1.2) and prove the unique existence and regularity of the solutions. In Sections 4 and 5, we complete the proof of our main results-Theorems 2.1 and 2.2 respectively.
Then there exists a constant C > 0 depending on M, T , Ω, δ and q L ∞ (0,T ;H 2 (Ω)) such that In Theorem 4.4 of Sakamoto and Yamamoto [21], a similar problem to Theorem 2.1 is considered, but our result is more applicable in the point of view that the factor R(x, t) is also allowed to depend on t. Moreover, we may assume less regularity for R in Theorem 2.1. The arguments of Theorem 2.2 can also be applied to parabolic equations in order to consider the result of Theorem 1.1 in [7] with observations of the solution at a point x 0 ∈ Ω when Ω ⊂ R d , d = 1, 2, 3.
For such inverse problems with α = 1, we can also refer to Section 1.5 of Prilepko, Orlovsky and Vasin [17], Cannon and Esteva [6] and Saitoh, Tuan and Yamamoto [19,20], for example. In our main results, we assume conditions (2.4) and (2.7), which means that the observation point cannot be far from the source. On the other hand, in [6] and [19,20], the determination of time dependent factor in the source term is considered without assuming such conditions and the logarithmic type and Hölder type estimates are proved respectively. However, the results for fractional diffusion equations without these conditions have not been obtained yet. Here we restrict ourselves to the case with assumptions (2.4) and (2.7), and show the Lipschitz type stability.
Let us remark that the results of this paper can be extended to the case d ≥ 4. For this purpose additional conditions such as more regularity for a ij and ∂Ω are required. In order to avoid technical difficulties, we only treat the case d ≤ 3.

Forward problem
This section is devoted to the proof of unique existence and regularity of the solution of the IBVPs (1.1) and (1.2).
with C > 0 depending on Ω, T and γ satisfying If all coefficients are independent of time variable t, then we can apply eigenfunction expansion and the problems can be reduced to ordinary differential equations of fractional order (e.g. [21]). However, since we consider the determination of the time dependent factor of coefficients, we apply fixed point theorem to show the unique existence of the solutions to (1.1) and (1.2) as in Beckers and Yamamoto [4].
In order to prove these results, we consider the IBVPs with more general data in the next subsections.
From these three lemmata we deduce easily Propositions 3.1 and 3.2.

Preliminary.
We define the operator A as A + 1 in L 2 (Ω) equipped with the boundary condition B σ h = 0; Then A is a selfadjoint and strictly positive operator with an orthonormal basis of eigenfunctions (φ n ) n≥1 of finite order associated to an increasing sequence of eigenvalues (λ n ) n≥1 . We can define the fractional power A γ , γ ≥ 0, of A by (3.12) Since D(A) is continuously embedded into H 2 (Ω) with norm equivalence (see Theorem 5.4 in Chapter 2 of [13] for example), we see by interpolation that In order to prepare for the arguments used in this paper, we consider the following Cauchy problem in L 2 (Ω); We define the operator valued function the Mittag-Leffler function given by .
Let p ∈ (1, ∞] be as in (3.6). Noting that A and A −1 S ′ A (t) commute, we see that for F ∈ L p (0, T ; D(A)), By p > 1/α and (3.15), the mapping t → A −1 S ′ A (t) belongs to L q (0, T ; B(L 2 (Ω))) where q ∈ [1, ∞) satisfies 1/p + 1/q = 1. Therefore by Lemma 3.6, u belongs to C([0, T ]; D(A)) and satisfies By using these estimates, we will show the unique existence of the solution applying the fixed point theorem. where u(t) := u(·, t) and F (t) := F (·, t). Noting that F ∈ L p (0, T ; D(A)) by (3.6), we see from (3.14) that the solution u of (3.19) satisfies where the map H is defined by (3.18). Therefore we will look for a fixed point of the map Repeating the similar calculation, we get .

By induction, we have
Therefore we obtain
By the original equation ∂ α t u = −Au + F , we see that ∂ α t u belongs to L p (0, T ; H 2γ (Ω)) with the estimate; which implies (3.8). Thus we have completed the proof.
For the proof of Lemma 3.4, we prepare the following fact; Lemma 3.7. Let u, v ∈ H 2 (Ω) and d ≤ 3, then uv ∈ H 2 (Ω) with the estimate For this lemma, see Theorem 2.1 in Chapter II of Strichartz [24].

Proof of Theorem 2.1
In this section, we prove Theorem 2.1. To this end, we prepare the following lemmata with Gronwall type inequalities; Lemma 4.1. Let C, α > 0 and u, d ∈ L 1 (0, T ) be nonnegative functions satisfying For the proof, see Lemma 7.1.1 p.188 of [12].
Lemma 4.2. We take 2 ≤ p ≤ ∞ and µ > 2/p. Let f ∈ L ∞ (0, T ) and u, R ∈ L p (0, T ) be non-negative functions satisfying the integral inequality (4.1) Then we have where the constant C depends on p, µ, T and R L p (0,T ) .
Proof. We set d(t) := f p L p (0,t) . From equation (4.1), we have Now we estimate the right-hand side of the above. By the Cauchy-Schwarz inequality, Therefore if p > 2, then Lemma 3.6 yields that where r ∈ [1, ∞) satisfies 2/p + 1/r = 1. For p = 2, we also have Thus for any p ≥ 2, we have where C depends on T , p, µ and R L p (0,T ) . By (4.3) and (4.4), we have Hence by the Gronwall inequality, we have , t ∈ (0, T ) with C depending on p, µ, T and R L p (0,T ) . Thus we have proved (4.2). Now we are ready to prove Theorem 2.1.