Stability of Conductivities in an Inverse Problem in the Reaction-diffusion System in Electrocardiology

In this paper, we study the stability result for the conductivities diffusion coefficients to a strongly reaction-diffusion system modeling electrical activity in the heart. To study the problem, we establish a Carleman estimate for our system. The proof is based on the combination of a Carleman estimate and certain weight energy estimates for parabolic systems.


Introduction.
Let Ω ⊂ R N (N ≥ 1) be a bounded connected open set whose boundary ∂Ω is regular enough. Let T > 0 and ω be a small nonempty subset of Ω. We will denote (0, T ) × Ω by Q T and (0, T ) × ∂Ω by Σ T .
To state the model of the cardiac electric activity in Ω (Ω ⊂ R 3 being the natural domain of the heart), we set u i = u i (t, x) and u e = u e (t, x) to represent the spacial cellular and location x ∈ Ω of the intracellular and extracellular electric potentials respectively. Their difference v = u i − u e is the transmembrane potential. The anisotropic properties of the two media are modeled by intracellular and extracellular conductivity tensors M i (x) and M e (x). The surface capacitance of the membrane is represented by the constant c m > 0. The transmembrane ionic current is represented by a nonlinear function h(v).
The equations governing the cardiac electric activity are given by the coupled reaction-diffusion system: where f and g are stimulation currents applied to Ω. We complete this model with Dirichlet boundary conditions for the intra-and extracellular electric potentials and with initial data for the transmembrane potential It is important to point out that realistic models describing electrical activities include a system of ODEs for computing the ionic current as a function of the transmembrane potential and a series of additional "gating variables", which aim to model the ionic transfer across the cell membrane. Assume that the intra and extracellular stimulations are equal: f χ ω = gχ ω . If M i = µM e for some constant µ ∈ R, then by multiplying the second equation in (1) by µ and adding it to the first equation in (1) one gets the first equation in the following parabolic-elliptic system: The second equation is obtained by computing the difference of the two equation in (1). Here M = M i + M e . System (4) is known as the monodomain model.
We approximate the above model (4) by the following family of parabolic equations ε is a fixed small constant. Since v = u i − u e in the bidomain model, it is natural decompose the initial condition v 0 as v 0 = u i,0 − u e,0 . Note that when ε → 0 in (5), we obtain the classical monodomain model. In this work, we study the stability result for the conductivities diffusion coefficients to the following linearized system of (5) with semi-initial conditions where a(t, x) and its derivative with respect to t exists and are bounded in Q T . For some θ ∈ (0, T ), the semi-initial conditions v θ (x), u e,θ (x) are sufficiently regular. The unknown conductivity tensors M and M e are assumed to be sufficiently smooth and shall be kept independent of time t. The existence of weak solutions of (1) is proved in [10] by the theory of evolution variational inequalities in Hilbert space. Then Bendahmane and Karlsen [2] proved the existence and uniqueness for a nonlinear version of the bidomain equations (1) by a uniformly parabolic regularization of the system and the Faedo-Galerkin method. Moreover, Bendahmane and Chaves-Silva [1] studied exact null controllability to (1) for each ε > 0 by establishing estimates for its dual system. To learn more about the cardiac problems, one can refer to the work of Bendahmane et al. [3,4]. However, it is noted that there is no stability results for the inverse bidomain model.
The paper by Cristofol et al. [11] obtains the stability results for reaction-diffusion system of two equations with constant coefficients using a Carleman estimate. Then Sakthivel et al. [21] established the stability results for Lotka-Volterra competitiondiffusion system of three equations with variable diffusion coefficients. Our inverse stability results are new because system (6) contains a strong coupling term. The technics we shall discuss are similar to the framework using Carleman estimates for inverse problems but the obtained estimates differs from those of [24], [21] because of the strongly coupled terms.
Let (ṽ ε ,ũ ε e ) be a solution of system (6) with conductivity tensors (M e ,M ) and semi-initial data (ṽ ε θ ,ũ ε e,θ ). Then setting and G = div(g 2 (x)∇ũ ε e ). Throughout the paper, we make the following assumptions: Assumption 1.1. The conductivity tensors M e (x), M i (x) and M (x) are C ∞ , bounded, symmetric, semi-definite, and elliptic matrixes (there exists β > 0 such that Σ 3 i,j M i,j ξ i ξ j ≥ β|ξ| 2 for all ξ ∈ R 3 ). All their derivatives up to the third order are respectively bounded by the positive constants γ 1 , γ 2 , γ 3 . Assumption 1.2. Assume the bounded measurements ∂ t A 1 and ∂ t A 2 in (0, T ) × ω are given. Also A i (θ, x), ∇A i (θ, x), ∆A i (θ, x) and ∇(∆A i (θ, x)) for some fixed θ ∈ (0, T ), where i = 1, 2 in Ω are given. Now the question of interest is whether we can determine the conductivity tensors M e and M by the two measurements.

2.
A Carleman type estimate. In this section, we prove the Carleman estimate based on the standard technique for general parabolic equations. In order to frame a Carleman type estimate, we shall first introduce a particular type of weight functions.
2.1. Weight functions. First, we introduce weight functions for the parabolic equations given in [12]. Letω ⊂⊂ ω be a nonempty bounded set of Ω, and ψ ∈ C 2 (Ω) such that Then we introduce another two weight functions: where λ > 1, t ∈ (0, T ) and β(t) = t(T − t). Note that the weight function α is positive, and blows up to ∞ as t = 0 or t = T . As a result, e −2sα and φe −2sα are smooth. Even they vanish when t = 0 or t = T . It can be seen that Before proving the main estimate, we give the following estimates for the two weight functions α and φ. Note that throughout the paper we will denote C as a generic positive constant. After some computations, we can obtain the following estimates: Furthermore, we also have Refer to [12] for the details.

2.2.
Main proof of a Carleman type estimate. Let us set Q ω = (0, T )×ω. For each positive integer m, we denote the Sobolev space of functions in L p (Ω) whose weak derivatives of order less than or equal to m are also in L p (Ω) with the norm denoted · L p (Ω) , by W m,p (Ω) with p > 1 or p = ∞. When p = 2, we denote W m,p by H m (Ω). Moreover, let L 2 (0, T ; H 1 (Ω)) be the space of all equivalent classes of square integrable functions from (0, T ) to H 1 (Ω). For the space L 2 (0, T ; L ∞ (Ω)), we define it in the same way. Let A 1 be the solution of the first equation of (7) with help of using Assumption 1.1. We apply the Carleman estimate (see Theorem 6.1 in [1].) derived for the parabolic equations to the first equation in (7). For λ > λ 0 ≥ 1, s ≤ s 0 (T +T 2 +T 4 ), there exists a constant C depending on Ω, ω, ψ and β so that whereω ⊂⊂ ω 1 ⊂⊂ ω, and Similarly, for λ > λ 0 ≥ 1, s ≥ s 0 (T + T 2 + T 4 ), there exists a constant C depending on Ω, ω, ψ and β satisfying with Now coupling the above inequalities (13) and (15), we have for sufficiently large s ≥ s 0 (T + T 2 + T 4 ) and λ ≥ λ 0 . From the definition of I 1 , also M i and ∇M i being bounded, we obtain Then it can be summarized as our desired Carleman estimate as follows.
Theorem 2.1. Let ψ(x), φ(t, x) and α(t, x) be defined as in the above subsection, a(t, x) is a bounded function. Moreover, Assumption 1.1 holds. Then there exist λ 0 and s 0 such that for all λ > λ 0 ≥ 1 and sufficiently large enough s > s 0 , the following inequality is true.
3. Stability of the conductivities. In this section, we study the stability of the conductivity tensors M e and M . Then an inequality is established which estimates g 1 , g 2 , ∇g 1 , ∇g 2 with an upper bound given by some Sobolev norms of the derivative of A 1 and A 2 over Q ω , certain spatial derivative of A j (θ, ·), j = 1, 2, where θ ∈ (0, T ) makes 1 β(t) attain its minimum value and the Sobolev norm of g 1 , g 2 , ∇g 1 , ∇g 2 in a small spaceω.
First, we let B 1 = ∂ t A 1 , B 2 = ∂ t A 2 . Using this and (7), we get the following system: where and Indeed, to prove the main result here we need to impose some regularity properties as follows.
Assumption 3.1. Suppose v ε θ and u ε e,θ are C 3 real valued functions. Then all their derivatives up to order three are bounded and satisfy |∇ψ · ∇v ε θ | ≥ δ > 0, |∇ψ · ∇u ε e,θ | ≥ δ > 0, on Ω \ω, whereω ⊂⊂ ω ⊂⊂ Ω. Before start proving our main conclusion, we need to give the following Lemma 3.3, which will be useful in the following part. We define the following operators P 0 and Q 0 and the initial conditions on α and φ at t = θ: Lemma 3.3. Consider the first order partial differential operator P 0 h = ∇U θ · ∇h, where U θ satisfies Assumption 3.1. Then there exists a constant C > 0, such that for sufficiently large enough λ and s, the following result holds: with θ ∈ (0, T ) and h ∈ H 1 0 (Ω).
Proof. Let B 1 = e −sα θ h, we have h ∈ H 1 0 (Ω). Then we take the square of both sides in (20), multiply 1 φ θ and integrate by parts with respect to space variable for both sides of (20) as follows: where we used Assumption 3.1 in the last step. Thus we obtain (12), we have
With the help of the Lemma 3.3, we are proving the following proposition. Proposition 1. Let (A 1 , A 2 ) be the solution of (7), and (B 1 , B 2 ) be the solution of (18). Suppose all the conditions of Theorem 2.1 and Assumption 3.1 hold. Then there exists a constant C = C(γ 1 , γ 2 , δ) > 0 such that for sufficiently large enough s and λ the following estimate is true.
for any g 1 , g 2 ∈ H 2 0 (Ω), where the functions E j , are given as follows: Proof. Due to the value of the solutions satisfying the first equation in (18) at t = θ, and F = div(g 1 (x)∇ṽ ε ), from (19) we obtain Note that we replace h by g 1 when choosing U θ asṽ ε θ . Therefore, inspired by Lemma 3.3, we get Similarly, from the value of the solutions satisfying the second equation in (18) at t = θ, and G = div(g 2 (x)∇ũ ε e ), we obtain

It leads to
On the other hand, from the expression of P 0 g 1 , we can see that, Similarly, we also have Using the same method to preceding estimates and Lemma 3.3, it follows that Combing the above three estimates (23), (24) and (25), the proof is complete.