# Mathematical analysis of a 1d-3d coupled problem in cardiac electrophysiology

Khouloud Kordoghli, Saloua Mani Aouadi, Nejib Zemzemi

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Khouloud Kordoghli, Saloua Mani Aouadi, Nejib Zemzemi. Mathematical analysis of a 1d-3d coupled problem in cardiac electrophysiology. Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, 2024, 76 (1), pp.23. 10.1007/s00033-024-02371-z. hal-04878589

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Distributed under a Creative Commons Attribution 4.0 International License
MATHEMATICAL ANALYSIS OF A 1D-3D COUPLED PROBLEM IN CARDIAC
ELECTROPHYSIOLOGY

KHOULOUD KORDOGHLI¹, SALOUA MANI AOUADI¹, & NEJIB ZEMZEMI²

ABSTRACT. The objective of this paper is to analyze a coupled problem that describes the propagation of the electric wave in the heart. The problem comprises coupled partial differential equations posed on a three-dimensional domain representing the heart and on a one-dimensional tree representing the Purkinje network. Each system of PDEs is itself coupled to ordinary differential equations that describe the electrical activity at the cellular level. We establish the existence of a unique solution, utilizing a fixed-point approach with a judicious and non-conventional choice of functional spaces and contraction.

# 1. INTRODUCTION

The electrical behaviour of the heart has been extensively studied in various mathematical works [1, 5, 11, 12, 17, 18]. The Purkinje system, initially identified by physiologist J. Purkinje (1787-1869), plays a significant role in cardiac activity and is represented as a one-dimensional tree-like structure [1, 5]. In a normal heart rhythm, the electrical current originates from the sinus node, travels through the two atria, inducing their contractions, and then converges at the atrioventricular node. Subsequently, it travels along the His bundle and the Purkinje network to reach the ventricular walls, leading to their contraction. The interaction between the myocardium and the Purkinje system has been explored in various ways [1, 5, 15]. In [19], the authors introduced a model of the junction between the Purkinje network and the myocardium, presenting a mathematical framework for continuous-level coupling conditions. The propagation of the electric wave is described by a non-linear reaction-diffusion equation coupled to an ordinary differential equation. The impact of the Purkinje system on the myocardium is represented through a source term, while the counter-effect is captured by a Robin-like boundary condition applied to the terminals of the Purkinje network (see Figure 1). Numerical calculation was developed with this model and numerical stability was proved under various ionic current dynamics in [15].

Our contribution here is to establish a rigorous existence and uniqueness result for this model. The difficulties arise from the support for 3D/1D couplings of PDEs, for PDE/ODE coupling, and the non-regularity of the one-dimensional domain.

In the paper by Bordas et al. [18], a model for cardiac activity is analyzed, and an existence result is proven using the fixed-point approach from [2]. This approach was also adapted for the myocardium-Purkinje system in [16]. The authors utilized cross-sectional averaging of Purkinje fibers to formulate a one-dimensional model of the coupling conditions and prove the existence of a solution for the time semi-discretized problem. Within the framework of reaction-diffusion equations, Veneroni et al.'s papers [13, 14] studied the existence and uniqueness for parabolic equations coupled to ODEs, representing a general formalism of the electrical activity of the cell membrane. In [3], the authors proved the existence of a maximal solution for 1D blood flow in a network of vessels. Moreover, the wave equation on 1D networks has been examined in previous works, specifically in [9]. Despite the abundance of literature, and to the best of our knowledge, no existence result has been proven for the aforementioned model.

The paper is outlined as follows. In Section 2, we describe our coupled problem. In Section 4, we prove the

<footnote>Date: November 4, 2024.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. Primary 35Q92, Secondary: 35R30.</footnote>

<footnote>Key words and phrases. 1D/3D coupling, monodomain model, cardiac electrophysiology, fixed point method, mathematical analysis.</footnote>
existence and uniqueness of a solution of the coupled problem. For that, we employ the strategy introduced in [13] with fundamental differences in functional spaces and coupling conditions.

## 2. PRESENTATION OF THE MATHEMATICAL MODEL

We denote $\Omega \subset \mathbb{R}^3$ as the myocardium domain, and $\Lambda$ as the Purkinje network, composed of the Purkinje fibers, the His bundle, and side bundles. We assume that $\Lambda$ consists of $N_{bran} \in \mathbb{N}$ regular curves: $\Lambda = \cup_{n=1}^{N_{bran}} \{\Lambda_n\}$, and it has $N_{ter}$ terminal points $x_1, \dots, x_{N_{ter}}$. Each terminal is connected to the myocardium in a small subdomain denoted $\Omega_i \subset \Omega$, referred to as the Purkinje Muscle Junction (PMJ) (see figure 1). The boundary of each branch is either a terminal point or a branching node. Let's consider $B_{br} = \{b_1, b_2, \dots, b_{p_{bran}}\}$ as the set of Purkinje branching nodes. Each node $b_j$ serves as a boundary for a set of branches, and we denote $I_j$ as the set of indices for these branches.

For any branching node $b_j$, $j = 1, \dots, p_{bran}$ and for all $k \in I_j$, we denote $\partial_{x,k}u(b_j)$ as the derivative of $u$ at the point $b_j$, seen as the boundary of the branch $\Lambda_k$:

$$ \partial_{x,k}u(b_j) = \lim_{y \to b_j} \partial_{x,k}u(y), $$

where $\partial_{x,k}$ represents the tangential derivative along $\Lambda_k$.

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Schematic representation of the 1D/3D domains. Branching nodes are annotated with the red discs and the coupling domains at the Purkinje terminal points are shown with the small circles.</figcaption></figure>
2.1. Mathematical model for the coupling Purkinje-myocardium. Our starting point is the model introduced in [1] and studied in [15, 16]. The depolarization wave in the myocardium is governed by a reaction-diffusion equation coupled to an ODE system, modelling the gating variables $W = (w_1, ..., w_k)$ and the ionic concentrations $Z = (z_1, ..., z_m)$:

$$ egin{cases} A(C\partial_t V + I_{ion}(V, W, Z)) + \sum_{i=2}^{N_{ter}} s_i = \operatorname{div}(\sigma \nabla V) + AI_{app}, & \text{in } \Omega \times [0, T], \\ \partial_t W + g(V, W) = 0, & \text{in } \Omega \times [0, T], \\ \partial_t Z + f(V, W, Z) = 0, & \text{in } \Omega \times [0, T]. \end{cases} \quad (2.1) $$

In the myocardium, the constant $A$ represents the surface area of the membrane per unit of volume, and $C$ is the capacitance of the cell membrane. The transmembrane potential is represented by $V$, while $I_{app}$ describes the applied current. The total ionic current is denoted by $I_{ion}$. The gating variable is represented by $W$, and $Z$ indicates the ionic intracellular concentration variables. The electrical conductivity in the myocardium is given by $\sigma$. On the heart boundary $\partial\Omega$, $\mathbf{n}$ stands for the outward unit normal. We suppose that the heart is isolated, so a Neumann boundary condition is imposed:

$$ \sigma \nabla V \cdot \mathbf{n} = 0, \quad \text{on } \partial\Omega \times [0, T]. \quad (2.2) $$

Here $\sigma := \sigma_i(\sigma_i + \sigma_e)^{-1}\sigma_e$ is the bulk conductivity in the myocardium, where $\sigma_i$ and $\sigma_e$ are the intra- and extracellular conductivity tensors.

Similarly, the electrical wave in the Purkinje system is governed by:

$$ egin{cases} A_p(C\partial_t V_p + I_{ion,p}(V_p, W_p, Z_p)) = \operatorname{div}(\sigma_p \nabla V_p) + A_p I_{app,p}, & \text{in } \Lambda \times [0, T], \\ \partial_t W_p + g_p(V_p, W_p) = 0, & \text{in } \Lambda \times [0, T], \\ \partial_t Z_p + f_p(V_p, W_p, Z_p) = 0, & \text{in } \Lambda \times [0, T]. \end{cases} \quad (2.3) $$

In the Purkinje network, the constant $A_p$ represents the surface area of the membrane per unit of volume. The capacitance of the cell membrane is $C_p$, and the transmembrane potential is represented by $V_p$. The applied current is denoted by $I_{app,p}$, and the total ionic current by $I_{ion,p}$. The gating variable is $W_p$, and $Z_p$ indicates the ionic intracellular concentration variables specific to the Purkinje fibers. The electrical conductivity in the Purkinje network is given by $\sigma_p$.

The rapid conduction system is intended to be isolated at the point $x_1$, which corresponds to the location of the atrioventricular node. Here, we have:

$$ \sigma_p \partial_x V_p(x_1, \cdot) = 0, \quad \text{on } [0, T]. \quad (2.4) $$

The variables $k$ and $m$ correspond to the number of gating variables $W$, $W_p$ in $\mathbb{R}^k$ and the concentration variables $Z$, $Z_p$ in $\mathbb{R}^m$.

Also, we have Kirchhoff's law holding at branching nodes $(b_j)_{j=1,...,p^{bran}}$:

$$ \sum_{k \in I_j} \sigma_p \partial_{x,k} V_p(b_j, \cdot) = 0, \quad \forall j = 1, \dots, p^{bran}, \quad \text{on } [0, T]. \quad (2.5) $$

Finally, initial conditions for both systems (2.1) and (2.3) are defined as follows:

$$ egin{cases} V(\cdot, 0) = V_0, & W(\cdot, 0) = W_0, & Z(\cdot, 0) = Z_0, & \text{in } \Omega, \\ V_p(\cdot, 0) = V_{p,0}, & W_p(\cdot, 0) = W_{p,0}, & Z_p(\cdot, 0) = Z_{p,0}, & \text{in } \Lambda. \end{cases} \quad (2.6) $$
Following [1], $\sum_{i=2}^{N_{ter}} s_i$ is the source current flowing from the Purkinje system to the myocardium. It translates the electrical effect of the Purkinje on the myocardium. For $i = 2, \dots, N_{ter}$, the opposite effect is given by a Robin boundary condition as follows:

$$ \sigma_p(x_i) \hat{c}_x V_p(x_i) = \frac{c_p}{S_p} (\langle V \rangle_i - V_p(x_i)), \quad s_i(x) = \begin{cases} \frac{S_p}{|\Omega_i|} \sigma_p(x_i) \hat{c}_x V_p(x_i) & \text{if } x \in \Omega_i, \\ 0 & \text{else,} \end{cases} \qquad (2.7) $$

where $c_p$ is the conductance of the PMJ, $S_p$ is the surface of membrane Purkinje cells in $\Omega_i$ and

$$ \langle V \rangle_i = \frac{1}{|\Omega_i|} \int_{\Omega_i} V, \quad \text{for } i = 1, \dots, N_{ter}. $$

**Remarks 2.1.** Each equation in the system (2.3) is defined along the edges and is composed of $N_{bran}$ equations.

**Hypothesis 2.2.** We assume that the conductivities $\sigma \in C^1(\bar{\Omega}, \mathbb{R}^{3 \times 3})$ and $\sigma_p \in C^1(\bar{\Lambda}_i, \mathbb{R})$, for each $i$, $\sigma$ is symmetric and uniformly positive definite, and there exist positive constants $\bar{\sigma}$, $\underline{\sigma}$, $\bar{\sigma}_p$ and $\underline{\sigma}_p$ such that:

$$ \underline{\sigma} \|\varepsilon\|_{I_2}^2 \le \varepsilon^T \sigma(x) \varepsilon \le \bar{\sigma} \|\varepsilon\|_{I_2}^2 \quad \forall \varepsilon \in \mathbb{R}^3, \forall x \in \Omega; \quad \underline{\sigma_p} \le \sigma_p(x) \le \bar{\sigma_p} \quad \forall x \in \Lambda. \qquad (2.8) $$

2.2. General structure of cardiac cellular membrane models. We are considering the following models for both the myocardium and the Purkinje network. We assume, as in [13, 14] and their references, that the ionic current

$$ I_{\text{ion}} : \mathbb{R} \times \mathbb{R}^k \times (0, +\infty)^m \to \mathbb{R}, \quad (V, W, Z) \mapsto I_{\text{ion}}(V, W, Z) $$

has the following general form

$$ I_{\text{ion}}(V, W, Z) = \sum_{j=1}^{m} J_j(V, W, \log z_j) + \tilde{H}(V, W, Z), \qquad (2.9) $$

where $W = (w_i)_{i=1,\dots,k}$ and $Z = (z_j)_{j=1,\dots,m}$ represent respectively the gating and the ionic concentration variables.

The dynamics of the ionic concentrations is described by the following systems:

$$ \frac{\partial z_i}{\partial t} = f_i(V, W, z_i) = -J_i(V, W, \log z_i) + H_i(V, W, Z), \quad i = 1, \dots, m. \qquad (2.10) $$

And the dynamics of the gating variables are described by [13, 14]:

$$ \frac{\partial w_j}{\partial t} = g_j(V, w_j), \quad j = 1, \dots, k. \qquad (2.11) $$

Above, $g_j : \mathbb{R}^2 \to \mathbb{R}$ is given by

$$ g_j(V, w_j) = \alpha_j(V)(1 - w_j) - \beta_j(V)w_j, \quad j = 1, \dots, k. \qquad (2.12) $$

$\alpha_j, \beta_j$ are positive rational functions,

$$ H, \tilde{H} \in C^2(\mathbb{R} \times \mathbb{R}^k \times (0, +\infty)^m) \cap Lip(\mathbb{R} \times [0, 1]^k \times (0, +\infty)^m), \quad J_j \in C^2(\mathbb{R} \times \mathbb{R}^k \times \mathbb{R}), \quad \forall j = 1, \dots, m, \qquad (2.13) $$

and there exist three functions $\underline{G}$, $\bar{G}$, $L_V$ belonging to $C^1(\mathbb{R}^k, \mathbb{R}^+)$ such that:

$$ 0 < \underline{G}(W) \le \frac{\partial}{\partial \xi} J_j(V, W, \xi) \le \bar{G}(W), \quad \forall W \in \mathbb{R}^k; \quad \left| \frac{\partial}{\partial V} J_j(V, W, 0) \right| \le L_V(W). \qquad (2.14) $$
In summary, from (2.1)–(2.7), the problem we are interested in can be formally stated as follows: Find V, W, Z, $V_p$, $W_p$ and $Z_p$ in the appropriate function spaces to be defined later such that:

$$
\left\{
\begin{array}{ll}
A(C\partial_t V + I_{\text{ion}}(V, W, Z)) + \sum_{i=2}^{N_{\text{ter}}} s_i(x) = \text{div}(\boldsymbol{\sigma}\nabla V) + AI_{\text{app}}, & \text{in } \Omega \times [0, T], \\
A_p(C\partial_t V_p + I_{\text{ion},p}(V_p, W_p, Z_p)) = \text{div}(\boldsymbol{\sigma}_p\nabla V_p) + A_p I_{\text{app},p}, & \text{in } \Lambda \times [0, T], \\
\boldsymbol{\sigma}\nabla V \cdot \mathbf{n} = 0, & \text{on } \partial\Omega \times [0, T], \\
\sum_{k \in I_j} \boldsymbol{\sigma}_p \partial_{x,k} V_p(b_j) = 0, \quad \forall j = 1, \dots, p^{\text{bran}}, & \\
\boldsymbol{\sigma}_p \partial_x V_p(x) = 0, & \text{on } \{x_1\} \times [0, T], \\
\boldsymbol{\sigma}_p(x_i) \partial_x V_p(x_i) = \frac{c_p}{S_p} (\langle V \rangle_i - V_p(x_i)), & \text{for } i = 2, \dots, N_{\text{ter}}, \\
\partial_t W + g(V, W) = 0, & \text{in } \Omega \times [0, T], \\
\partial_t W_p + g_p(V_p, W_p) = 0, & \text{in } \Lambda \times [0, T], \\
\partial_t Z + f(V, W, Z) = 0, & \text{in } \Omega \times [0, T], \\
\partial_t Z_p + f_p(V_p, W_p, Z_p) = 0, & \text{in } \Lambda \times [0, T], \\
V(\cdot, 0) = V_0, \quad W(\cdot, 0) = W_0, \quad Z(\cdot, 0) = Z_0, & \text{in } \Omega, \\
V_p(\cdot, 0) = V_{p,0}, \quad W_p(\cdot, 0) = W_{p,0}, \quad Z_p(\cdot, 0) = Z_{p,0}, & \text{in } \Lambda.
\end{array}
\right.
\quad (2.15)
$$

### 3. NOTATIONS AND PRELIMINARY RESULTS

In this section, we introduce some definitions of appropriate functional spaces and we present some preliminary results which allow us to prove our main results.

#### 3.1. Functional spaces on the network.
We identify the regular curve $\Lambda_i$ with the interval $(0, l_i)$, where $l_i$ is a strictly positive real number for all $i = 1, \dots, N_{\text{ter}}$. For a function $V_p$ defined on $\Lambda$, we denote by $V_p^i$ the restriction of $V_p$ to $\Lambda_i$. The functional spaces are then treated as Hilbert products and the unknown $V_p$ is presented as a vector of $\mathbb{R}^{N_{\text{bran}}}$: $V_p = (V_p^1, V_p^2, \dots, V_p^{N_{\text{bran}}})$. The continuity condition at the connection nodes is expressed as:

$$V_p^i(b_j) = V_p^k(b_j), \quad \forall j \in \{1, \dots, p^{\text{bran}}\}, \forall i, k \in \{1, \dots, N_{\text{bran}}\}.$$

Kirchoff's law is equivalent to the Neumann condition at the external node.

We introduce, for $p \ge 1$, the Banach spaces $L^p(\Lambda) = \prod_{i=1}^{N_{\text{bran}}} L^p(\Lambda_i)$, equipped with the norm

$$\|u\|_{L^p(\Lambda)} = \begin{cases} \left( \sum_{i=1}^{N_{\text{bran}}} \int_{\Lambda_i} |u(x)|^p dx \right)^{\frac{1}{p}}, & \text{if } p \ne +\infty, \\ \sup_{1 \le i \le N_{\text{bran}}} \sup_{x \in \Lambda_i} |u(x)|, & \text{if } p = +\infty. \end{cases} \quad (3.1)$$

We also introduce the Hilbert space

$$H^1(\Lambda) = \left\{ u = (u^1, \dots, u^{N_{\text{bran}}}) \in \prod_{i=1}^{N_{\text{bran}}} H^1(\Lambda_i), \ u^i(b_l) = u^j(b_l), \ \forall l \in \{1, \dots, p^{\text{bran}}\}, \forall i, j \in \{1, \dots, N_{\text{bran}}\} \right\}, \quad (3.2)$$

equipped with its natural norm and inner product

$$\|u\|_{H^1(\Lambda)} = \sum_{i=1}^{N_{\text{bran}}} \|u^i\|_{H^1(\Lambda_i)}; \quad \langle f, g \rangle_{1,\Lambda} = \sum_{i=1}^{N_{\text{bran}}} (\langle f, g \rangle_{L^2(\Lambda)} + \langle \partial_x f, \partial_x g \rangle_{L^2(\Lambda)}). \quad (3.3)$$

Any regular curve of length $l_i$ is homeomorphic to the interval $[0, l_i]$, which itself is homeomorphic to $[0, 1]$ for each strictly positive real number $l_i$ [21].
We recall that the embedding of $H^1(\Lambda)$ into $L^2(\Lambda)$ is dense and compact [9]. We also introduce the space

$$ H^2(\Lambda) = \{u = (u^1, u^2, \dots, u^{N_{\text{bran}}}) \in H^1(\Lambda), u_i \in H^2(\Lambda_i), \forall i = 1, \dots, N_{\text{bran}}\}, \quad (3.4) $$

endowed with the inner product:

$$ \langle f, g \rangle_{2, \Lambda} = \sum_{i=1}^{N_{\text{bran}}} \left( \langle f, g \rangle_{L^2(\Lambda)} + \langle \partial_x f, \partial_x g \rangle_{L^2(\Lambda)} + \langle \partial_x^2 f, \partial_x^2 g \rangle_{L^2(\Lambda)} \right). $$

As the embedding of $H^2(0, l_j)$ into $C^1([0, l_j])$ is continuous for each $j$, then $H_2(\Lambda)$ is a Hilbert space.

Let us recall some results on interpolation within Banach spaces ([7], [[13], page 16]). Let $(X, Y)$ be a real interpolation couple of Banach spaces.

* Let $p \in [1, +\infty]$. If $u \in L^p(0, T; X)$ and $\frac{du}{dt} \in L^p(0, T; Y)$, then there exists a continuous extension of the function $u$ in $C^0([0, T]; (X, Y)_{1-\frac{1}{p}, p})$ such that

$$ \|u(t)\|_{(X,Y)_{1-\frac{1}{p},p}} \le \|u\|_{L^p(0,T;X)} + \left\| \frac{du}{dt} \right\|_{L^p(0,T;Y)}, \quad \forall t \in [0,T]. \quad (3.5) $$

* For $0 < \beta < 1$, $1 \le p, q < +\infty$ and $m \in \mathbb{N}$, we have

$$ (L^p(\Omega), W^{m,p}(\Omega))_{\beta,q} = B_{p,q}^{m\beta}(\Omega), \quad (3.6) $$

where $B_{p,q}^{m\beta}(\Omega)$ is the Besov space (real interpolation between Sobolev spaces).

* If $1 \le p < \infty$, $\Omega \subset \mathbb{R}^n$ is bounded with a Lipschitz boundary, we have

$$ B_{p,q}^{s}(\Omega) \subset C_{B}^{0}(\Omega), \quad \text{if } s > \frac{n}{p}, \text{ with continuous injection,} \quad (3.7) $$

where $s = m\beta$ and $C_B^0(\Omega)$ is the set of continuous and bounded functions on $\Omega$.

Now, let

$$ \mathbf{H}_1 = (L^2(\Omega), H^2(\Omega))_{1-\frac{1}{p},p} = B_{2,p}^{2(1-\frac{1}{p})}(\Omega), $$

$$ \mathbf{H}_{2,i} = (L^2(\Lambda_i), H^2(\Lambda_i))_{1-\frac{1}{p},p} = B_{2,p}^{2(1-\frac{1}{p})}(\Lambda_i) \quad \forall i, \quad \text{and} \quad \mathbf{H}_2 = \prod_{i=1}^{N_{\text{bran}}} \mathbf{H}_{2,i}. $$

By the properties (3.6) and (3.7), we have

$$ \mathbf{H}_1 \subset C^0(\Omega) \quad \text{if} \quad 2 - \frac{2}{p} > \frac{3}{2} \quad \text{i.e} \quad p > 4, \quad \text{and} \quad \mathbf{H}_{2,i} \subset C^0(\bar{\Lambda}_i) \quad \text{if} \quad 2 - \frac{2}{p} > \frac{1}{2} \quad \text{i.e} \quad p > 1. \quad (3.8) $$

### 3.2. Preliminary results.
Our first step is to recall from [[13], Proposition 2.1] that for any given $\varphi = (V, V_p)$ in $H^1(0, T; L^2(\Omega) \times L^2(\Lambda))$, there exist unique $\psi = (W, W_p)$ and $\theta = (Z, Z_p)$ that solve the respective following systems

$$ \left\{ \begin{array}{ll} \displaystyle \frac{\partial W}{\partial t} + g(V, W) = 0, & \text{in } \Omega \times [0, T], \\ \displaystyle \frac{\partial W_p}{\partial t} + g_p(V_p, W_p) = 0, & \text{in } \Lambda \times [0, T], \\ W(\cdot, 0) = W_0 & \text{on } \Omega, \quad W_p(\cdot, 0) = W_{p,0} \text{ on } \Lambda \end{array} \right. \quad (3.9) $$

and

$$ \left\{ \begin{array}{ll} \displaystyle \frac{\partial Z}{\partial t} + f(V, W, Z) = 0, & \text{in } \Omega \times [0, T], \\ \displaystyle \frac{\partial Z_p}{\partial t} + f_p(V_p, W_p, Z_p) = 0, & \text{in } \Lambda \times [0, T], \\ Z(\cdot, 0) = Z_0 & \text{on } \Omega, \quad Z_p(\cdot, 0) = Z_{p,0} \text{ on } \Lambda. \end{array} \right. \quad (3.10) $$

We denote by

$$ \log \mathbf{Z} = (\log Z_1, \log Z_2, \dots, \log Z_m) \quad \text{and} \quad \log \mathbf{Z}_p = (\log Z_{p,1}, \log Z_{p,2}, \dots, \log Z_{p,m}). $$

In the following proposition, we give results on the regularity of the gate variables where we assume that the system (2.15) is decoupled.
**Proposition 3.1.** Let $\varphi = (V, V_p) \in H^1(0, T; L^2(\Omega) \times L^2(\Lambda))$, $W_0 : \Omega \to [0, 1]^k$, $W_{p,0} : \Lambda \to [0, 1]^k$ be measurable functions and $Z_0 : \Omega \to (0, +\infty)^m$ and $Z_{p,0} : \Lambda \to (0, +\infty)^m$ be such that:

$$ (Z_0, \log \mathbf{Z}_0) \in (L^2(\Omega))^{2m} \quad \text{and} \quad (Z_{p,0}, \log \mathbf{Z}_{p,0}) \in (L^2(\Lambda))^{2m}. $$

i) There exist unique measurable functions $W : \Omega \times [0, T] \to [0, 1]^k$ and $W_p : \Lambda \times [0, T] \to [0, 1]^k$ that solve respective gating equations in system (3.9) with $g$ and $g_p$ given by (2.12). Moreover, for almost every $y_1 \in \Omega$ and $y_2 \in \Lambda$, we have $(W(y_1, \cdot), W_p(y_2, \cdot)) \in (C^1(0, T))^{2k}$ with the universal bounds for each $j$:

$$ \begin{aligned} 0 \le w_j \le 1 & \quad \text{a.e. in } \partial\Omega \times ]0, T[, \\ 0 \le w_{p,j} \le 1 & \quad \text{a.e. in } \{b_1, b_2, \dots, b_{p_{\text{bran}}}\} \times ]0, T[. \end{aligned} \quad (3.11) $$

ii) Then, there exist a unique measurable function $Z : \Omega \times [0, T] \to (0, +\infty)^m$ and a unique measurable function $Z_p : \Lambda \times [0, T] \to (0, +\infty)^m$ that solve the respective concentration equations (3.10), such that for almost $y_1 \in \Omega$ and $y_2 \in \Lambda$, we have: $(Z(y_1, \cdot), Z_p(y_2, \cdot)) \in (C^1(0, T))^{2m}$, $Z, \log \mathbf{Z}, \frac{\partial Z}{\partial t}$ belong to $(L^2(\Omega \times [0, T]))^m$, whereas, $Z_p, \log \mathbf{Z}_p, \frac{\partial Z_p}{\partial t}$ belong to $(L^2(\Lambda \times [0, T]))^m$. For all $t \in [0, T]$, there exist two positive constants $C_1$ and $C_2$, independent of $V, W, Z_0, V_p, W_p$, and $Z_{p,0}$, such that the following estimates hold:

$$ |Z(y_1, t)| \le C_1 (1 + |Z_0(y_1)| + \|V(y_1, \cdot)\|_{L^2(0,t)}), \quad (3.12) $$

$$ |Z_p(y_2, t)| \le C_2 (1 + |Z_{p,0}(y_2)| + \|V_p(y_2, \cdot)\|_{L^2(0,t)}), \quad (3.13) $$

$$ |\log \mathbf{Z}(y_1, t)| + \left| \frac{\partial Z}{\partial t}(y_1, t) \right| \le C_1 (1 + |Z_0(y_1)| + \|V(y_1, \cdot)\|_{C^0(0,t)}), \quad (3.14) $$

$$ |\log \mathbf{Z}_p(y_2, t)| + \left| \frac{\partial Z_p}{\partial t}(y_2, t) \right| \le C_2 (1 + |Z_{p,0}(y_2)| + \|V_p(y_2, \cdot)\|_{C^0(0,t)}). \quad (3.15) $$

For the proof, we follow the same reasoning used in [[13], Proposition 2.2].

Now, we will establish some preliminary estimates on the ionic currents. Let $W, W_p, Z, Z_p$ supposed be known functions satisfying the hypothesis of Proposition 3.1. Let $(\bar{V}, \bar{V}_p)$ given in $H^1(0, T; L^2(\Omega)) \times H^1(0, T; L^2(\Lambda))$ fixed transmembrane potentials.

**Proposition 3.2.** We put

$$ \bar{I}_{\text{ion}}(y_1, t) = I_{\text{ion}}(\bar{V}((y_1, t)), W(y_1, t), Z(y_1, t)) $$

and

$$ \bar{I}_{\text{ion},p}(y_2, t) = I_{\text{ion},p}(\bar{V}_p(y_2, t), W_p(y_2, t), Z_p(y_2, t)). $$

Then for each $p \in (1, +\infty)$

$$ \bar{I}_{\text{ion}} \in L^p(0, T; L^2(\Omega)) \quad \text{and} \quad \bar{I}_{\text{ion},p} \in L^p(0, T; L^2(\Lambda)). $$

Moreover

$$ \|\bar{I}_{\text{ion}}\|_{L^p(0,t;L^2(\Omega))} \le C_1 (1 + \|\bar{V}\|_{L^p(\Omega;H^1(0,T))}^2) \quad (3.16) $$

and

$$ \|\bar{I}_{\text{ion},p}\|_{L^p(0,t;L^2(\Lambda))} \le C_2 (1 + \|\bar{V}_p\|_{L^p(\Lambda;H^1(0,T))}^2). \quad (3.17) $$

Here $C_1$ (resp. $C_2$) is a constant independent of $\bar{V}$, $W$ and $Z$ (resp. $\bar{V}_p$, $W_p$ and $Z_p$).

**Proof.** Set

$$ \begin{aligned} \bar{I}_{\text{ion}}(y_1, t) &= I_{\text{ion}}(\bar{V}(y_1, t), W(y_1, t), Z(y_1, t)), \\ \bar{I}_{\text{ion},p}(y_2, t) &= I_{\text{ion},p}(\bar{V}_p(y_2, t), W_p(y_2, t), Z(y_2, t)). \end{aligned} \quad (3.18) $$

Using estimates (2.14), we obtain

$$ \begin{aligned} |J(\bar{V}, W, \log \mathbf{Z})| &\le |J(0, W, 0)| + L_v(\bar{V}) + \bar{G}|\log \mathbf{Z}| \\ &\le C(1 + |\bar{V}| + |\log \mathbf{Z}|), \end{aligned} \quad (3.19) $$
and

$$
\begin{align}
|J(\bar{V}_p, W_p, \mathbf{log} \mathbf{Z}_p)| & \le |J(0, W_p, 0)| + L_v(\bar{V}_p) + \bar{G}|\mathbf{log} \mathbf{Z}_p| \tag{3.20} \\
& \le C(1 + |\bar{V}_p| + |\mathbf{log} \mathbf{Z}_p|). \nonumber
\end{align}
$$

Moreover, based on the definition (2.9) and by applying estimates (3.14), (3.15), we get $\bar{I}_{ion} \in L^2(\Omega \times [0, T])$, $\bar{I}_{ion,p} \in L^2(\Lambda \times [0, T])$ and for all $t \in [0, T]$:

$$
\| \bar{I}_{ion} \|_{L^2(0,t;L^2(\Omega))} \le C(1 + \| \bar{V} \|_{L^2(0,t;L^2(\Omega))}^2), \| \bar{I}_{ion,p} \|_{L^2(0,t;L^2(\Lambda))} \le C(1 + \| \bar{V}_p \|_{L^2(0,t;L^2(\Lambda))}^2). \quad (3.21)
$$

Further, according to the estimates (3.14) and (3.15), we obtain:

$$
|\bar{I}_{\text{ion}}(y_1, t)| \le C(1 + \|\bar{V}(y_1, \cdot)\|_{H^1(0,T)}), \quad \text{a.e in } \Omega \times [0, T], \qquad (3.22)
$$

and

$$
|\bar{I}_{ion,p}(y_2, t)| \le C(1 + \|\bar{V}_p(y_2, \cdot)\|_{H^1(0,T)}), \quad \text{a.e in } \Lambda \times [0, T]. \quad (3.23)
$$

So, for each $p \in (1, +\infty)$, we have $\bar{I}_{ion} \in L^p(0, T; L^2(\Omega))$ and $\bar{I}_{ion,p} \in L^p(0, T; L^2(\Lambda))$ with

$$
\| \bar{I}_{\text{ion}} \|_{L^p(0,t;L^2(\Omega))} \le C_1 (1 + \| \bar{V} \|_{L^p(\Omega;H^1(0,T))}^2) \quad (3.24)
$$

$$
\| \bar{I}_{\text{ion},p} \|_{L^p(0,t;L^2(\Lambda))} \le C_2 (1 + \| \bar{V}_p \|_{L^p(\Lambda;H^1(0,T))}^2). \quad (3.25)
$$

for some constant $C_1$ (resp. $C_2$) not depending on $\bar{V}$, $W$ and $Z$ (resp. $\bar{V}_p$, $W_p$ and $Z_p$). $\square$

The following proposition is a classical result regarding the existence, uniqueness and maximal regularity estimates in $L^p - L^q$ for the heat equation with inhomogeneous data and Neumann boundary conditions [8, 10, 20].

**Proposition 3.3.** Assume that $\partial\Omega$ is of class $C^2$ and that $\sigma$ satisfies (2.8). Let $p \in [2, +\infty[$. Let $V_0 \in H^2(\Omega)$ satisfy the compatibility condition:

$$
\sigma \nabla V_{0,n} = 0 \quad \text{on} \quad \partial \Omega \times [0, T].
$$

Let

$$
\bar{I}_{\text{ion}} \in L^p(0, T; L^2(\Omega)), \quad I_{\text{app}} \in L^p(0, T; L^2(\Omega)) \quad \text{and} \quad \bar{s}_i \in L^p(0, T; L^2(\Omega_i)), \quad \text{for } i=2, \dots, N_{\text{ter}}.
$$

There exists a unique $V \in W^{1,p}(0, T; L^2(\Omega)) \cap L^p(0, T; H^2(\Omega))$ which satisfies:

$$
\left\{
\begin{array}{ll}
A(C \partial_t V + \bar{I}_{\text{ion}}(V, W, Z)) + \sum_{i=2}^{N_{\text{ter}}} \bar{s}_i = \text{div}(\sigma \nabla V) + AI_{\text{app}}, & \text{in } \Omega \times [0, T], \\
\sigma \nabla V \cdot \mathbf{n} = 0, & \text{on } \partial \Omega \times [0, T], \\
V(\cdot, 0) = V_0, & \text{in } \Omega,
\end{array}
\right.
\qquad (3.26)
$$

and the following a priori estimates:

$$
\|V\|_{L^p(0,T;H^2(\Omega))} + \|V\|_{W^{1,p}(0,T;L^2(\Omega))} \le C \left( \|V_0\|_{H^2(\Omega)} + \|\bar{I}_{\text{ion}}\|_{L^p(0,T;L^2(\Omega))} \right. \\
\left. \qquad + \|I_{\text{app}}\|_{L^p(0,T;L^2(\Omega))} + \sum_{i=2}^{N_{\text{ter}}} \|\bar{s}_i\|_{L^p(0,T;L^2(\Omega_i))} \right). \tag{3.27}
$$

In addition, if $V^{(1)}$ and $V^{(2)}$ are solutions of (3.26) corresponding to data $\bar{I}_{ion}^{(1)}, \bar{I}_{ion}^{(2)}, \bar{s}_i^{(1)}, \bar{s}_i^{(2)}$, it holds for any $t \in [0, T]$:

$$
C \|V^{(2)}(t) - V^{(1)}(t)\|_{L^2(\Omega)}^2 \le \|\bar{I}_{ion}^{(2)} - \bar{I}_{ion}^{(1)}\|_{L^2(0,t;L^2(\Omega))}^2 + \sum_{i=2}^{N_{\text{ter}}} \|\bar{s}_i^{(2)} - \bar{s}_i^{(1)}\|_{L^2(0,t;L^2(\Omega_i))}^2. \quad (3.28)
$$

The heat equation on a network has been discussed in the literature [4, 6], among others. A specific question pertains to the definition of functional spaces. The continuity conditions and Kirchoff's law play a fundamental role in the proof of existence and regularity. We have the following result on existence and regularity [[6], Section 2].
**Proposition 3.4.** Assume that $\sigma_p$ verifies (2.8). Let $p \in [2, +\infty[$. Let $V_{p,0} \in H_2(\Lambda)$, satisfying the compatibility conditions $\sigma_p \partial_x V_{p,0}(x_1) = 0$ and

$$ \sigma_p(x_i) \partial_{x_i} V_{p,0}(x_i) + \frac{c_p}{S_p} V_{p,0}(x_i) = \frac{c_p}{S_p} \langle V \rangle_i, \quad \text{for } i = 2, \dots, N_{\text{ter}}. $$

Let $\bar{V} \in L^p(0, T; L^2(\Omega))$, $I_{app,p} \in L^p(0, T; L^2(\Lambda))$ and $\bar{I}_{ion,p} \in L^p(0, T; L^2(\Lambda))$. There exists a unique $V_p \in W^{1,p}(0, T; L^2(\Lambda)) \cap L^p(0, T; H_2(\Lambda))$, which satisfies:

$$ \left\{ \begin{array}{ll} A_p(C\partial_t V_p + \bar{I}_{ion,p}(V_p, W_p, Z_p)) = \text{div}(\sigma_p \nabla V_p) + A_p I_{app,p}, & \text{in } \Lambda \times [0, T], \\ \sigma_p \partial_x V_p(x) = 0, & \text{for } x = x_1, \text{ on } [0, T], \\ \sigma_p \partial_x V_p(b_j) = \sum_{k \in I_j} \sigma_p \partial_{x,k} V_p(b_j), & \text{for } j = 1, \dots, p^{\text{bran}}, \\ \sigma_p(x_i) \partial_x V_p(x_i) = \frac{c_p}{S_p} (\langle \bar{V} \rangle_i - V_p(x_i)), & \text{for } i = 2, \dots, N_{\text{ter}}, \\ V_p(\cdot, 0) = V_{p,0} & \text{in } \Lambda, \end{array} \right. \quad (3.29) $$

and the following a priori estimates:

$$ \|V_p\|_{L^p(0,T;H_2(\Lambda))} + \|V_p\|_{W^{1,p}(0,T;L^2(\Lambda))} \le C \left( \|V_{p,0}\|_{H_2(\Lambda)} + \|\bar{I}_{ion,p}\|_{L^p(0,T;L^2(\Lambda))} \right. \\ \left. + \|I_{app,p}\|_{L^p(0,T;L^2(\Lambda))} + \|\bar{V}\|_{L^p(0,T;L^2(\Lambda))} \right), \quad (3.30) $$

In addition, if $V_p^{(1)}$ and $V_p^{(2)}$ are the solutions corresponding to data $\bar{I}_{ion,p}^{(1)}$ and $\bar{I}_{ion,p}^{(2)}$, it holds for all $t \in [0, T]$

$$ C' \|V_p^{(2)}(t) - V_p^{(1)}(t)\|_{L^2(\Lambda)}^2 \le \|\bar{I}_{ion,p}^{(2)} - \bar{I}_{ion,p}^{(1)}\|_{L^2(0,t;L^2(\Lambda))}^2 + \|\bar{V}^{(2)} - \bar{V}^{(1)}\|_{L^2(0,t;L^2(\Lambda))}^2. \quad (3.31) $$

## 4. MAIN RESULTS: EXISTENCE AND UNIQUENESS

We present here our main results regarding the existence of a variational solution for the system (2.15).

**Theorem 4.1.** We assume that $\Omega$ is of class $C^2$ and that $\sigma$ and $\sigma_p$ verify the hypothesis 2.2. Let be given the data $(V_0, V_{p,0}) \in H^2(\Omega) \times H_2(\Lambda)$,

$$ \begin{array}{ll} W_0 : \Omega \mapsto [0, 1]^k & \text{measurable, } W_{p,0} : \Lambda_i \mapsto [0, 1]^k \text{ measurable,} \\ (Z_0, Z_{p,0}) \in L^2(\Omega)^m \times L^2(\Lambda)^m & \text{with } (\log \mathbf{Z}_0, \log \mathbf{Z}_{p,0}) \in L^2(\Omega)^m \times L^2(\Lambda)^m, \\ (I_{app}, I_{app,p}) \in L^p(0, T; L^2(\Omega) \times L^2(\Lambda)), & \text{for } p > 4. \end{array} $$

Let be given the ionic currents $I_{ion}$ and $I_{ion,p}$ satisfying (2.9) with regularity (2.13)-(2.14), the dynamic of the ionic concentrations satisfying (2.10), and the dynamics of the gating variables satisfying (2.12).

Then, there exist a unique solution $(V, V_p, W, W_p, Z, Z_p)$ of the system (2.15) satisfying:

$$ \begin{array}{l} (V, V_p) \in W^{1,p}(0, T; L^2(\Omega) \times L^2(\Lambda)) \cap L^p(0, T; H^2(\Omega) \times H_2(\Lambda)) \cap C^0([0, T]; C^0(\Omega) \times C^0(\Lambda)), \\ W : \Omega \times [0, T] \longrightarrow [0, 1]^k, \quad Z : \Omega \times [0, T] \longrightarrow (0, +\infty)^m \text{ measurable,} \\ W_p : \Lambda \times [0, T] \longrightarrow [0, 1]^k, \quad Z_p : \Lambda \times [0, T] \longrightarrow (0, +\infty)^m \text{ measurable,} \end{array} \quad (4.1) $$

with

$$ \begin{array}{ll} (W(x, \cdot), W_p(y, \cdot)) \in (C^1(0, T) \cap C^0([0, T]))^{2k} & \text{for a.e. } x \in \Omega, y \in \Lambda \\ (Z(x, \cdot), Z_p(y, \cdot)) \in (C^1(0, T) \cap C^0([0, T]))^{2m} & \text{for a.e. } x \in \Omega, y \in \Lambda \\ Z \in H^1(0, T; (L^2(\Omega))^m) \cap (L^\infty(\Omega \times [0, T]))^m, & \text{log } \mathbf{Z} \in (L^\infty(\Omega))^m, \\ Z_p \in H^1(0, T; (L^2(\Lambda))^m) \cap (L^\infty(\Lambda \times [0, T]))^m, & \text{log } \mathbf{Z}_p \in (L^\infty(\Lambda))^m. \end{array} \quad (4.2) $$
The proof of Theorem 4.1 is based on Banach's fixed point theorem. It will be done in several steps. For a Hilbert space $H$ and for $\alpha \in \mathbb{R}_+^*$, we introduce the weighted norm on $L^2(0, T; H)$

$$ |||\varphi|||_{\alpha,H} = \left( \int_0^T \exp^{-\alpha t} \|\varphi(t)\|_H^2 dt \right)^{\frac{1}{2}}. $$

It is clear that $|||\cdot|||_{\alpha,H}$ is a norm equivalent to the natural norm of $L^2(0, T; H)$.

Since we use Banach's fixed point theorem, we need to introduce the following closed subset $K$ of the Banach space $H_T := L^2(0, T; H^2(\Omega) \times H^2(\Lambda)) \cap H^1(0, T; L^2(\Omega) \times L^2(\Lambda)) \cap L^\infty((0, T) \times \Omega) \times L^\infty((0, T) \times \Lambda)$ for $\varphi_0 \in H^2(\Omega) \times H^2(\Lambda)$, $\mathfrak{J}_{app} = (I_{app}, I_{app,p}) \in L^p(0, T; L^2(\Omega) \times L^2(\Lambda))$, $p > 4$ and for some constants $M_0, M_1$ and $M_\infty$,

$$ K = \left\{ \varphi = (V, V_p) \in H_T, \quad \varphi(x, 0) = \varphi_0, \quad |||\varphi|||_{\alpha, L^2(\Omega) \times L^2(\Lambda)} \le M_0, \quad |||\hat{\partial}_t \varphi|||_{\alpha, L^2(\Omega) \times L^2(\Lambda)} \le M_1, \quad |||\varphi|||_{\alpha, H^2(\Omega) \times H_2(\Lambda)} \le M_1, \quad |||\varphi|||_{L^\infty(\mathcal{Q})} \le M_\infty \right\}. \quad (4.3) $$

A specific choice of $M_0, M_1, M_\infty$ is important in the proof of Theorem 4.1. We retain that, for $\varphi = (V, V_p) \in H_T, V_p^j$ is continuous on $[0, T] \times [0, l_j]$ for all $j$.

Let us denote by $\mathfrak{F}$ the operator that maps a function $\bar{\varphi} = (\bar{V}, \bar{V}_p)$ to the solutions of (3.26) and (3.29). We shall now prove that the operator $\mathfrak{F}$ satisfies the following two properties: (P1) $\mathfrak{F}(K) \subset K$

(P2) $\mathfrak{F}$ is a contraction with respect to a norm inducing the $L^2([0; T] \times \Omega) \times L^2([0; T] \times \Lambda)$ topology: $|||\cdot|||_{\alpha, L^2(\Omega) \times L^2(\Lambda)}$. Thus, Banach's fixed point theorem provides existence and uniqueness for $(V, W, Z)$ and $(V_p, W_p, Z_p)$ solution of Problem (2.15).

The proof of the first property (P1) will be done in the following three lemmas.

**Lemma 4.2.** Let $\bar{\varphi} = (\bar{V}, \bar{V}_p) \in H_T$ be a given function, let $\bar{M}_\infty$ such that

$$ |\bar{V}_p(x_i, t)| \le \bar{M}_\infty, \quad \forall i = 1, \dots, N_{\text{ter}}, \quad \forall t \in [0, T]. $$

Let $\psi = (W, W_p)$, $\theta = (Z, Z_p)$ be the unique solutions of systems (3.9) and (3.10) respectively, as given by Proposition 3.1. Let $\bar{\mathfrak{J}}_{\text{ion}} = (\bar{I}_{\text{ion}}, \bar{I}_{\text{ion},p})$ be given as in (3.18) satisfying (3.16) and (3.17). Let $\varphi = (V, V_p)$ be the solution of (3.26)-(3.29). There exists $\alpha_0 > 0$ such that for all $\alpha \ge \alpha_0$:

$$ |||\varphi|||_{\alpha, L^2(\Omega) \times L^2(\Lambda)}^2 \le \max \left\{ 1, |||\bar{\varphi}|||_{\alpha, L^2(\Omega) \times L^2(\Lambda)}^2 \right\}. \quad (4.4) $$

*Proof.* Since the embedding of $H^1(0, T; L^2(\Omega) \times L^2(\Lambda))$ into $L^\infty(0, T; L^2(\Omega) \times L^2(\Lambda))$ is continuous, there exists $C_1 > 0$ such that:

$$ \|\varphi\|_{L^\infty(0, T; L^2(\Omega) \times L^2(\Lambda))} \le C_1 \|\varphi\|_{H^1(0, T; L^2(\Omega) \times L^2(\Lambda))}, $$

and then,

$$ \|\varphi(t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le C_1 \|\varphi\|_{H^1(0, T; L^2(\Omega) \times L^2(\Lambda))} \quad \text{for all } t \in [0, T]. $$

Combining this inequality with estimates (3.27) and (3.30) for $p = 2$, we obtain:

$$ \|\varphi(t)\|_{L^2(\Omega) \times L^2(\Lambda)}^2 \le C_2 \left( \|\varphi_0\|_{H^2(\Omega) \times H^2(\Lambda)}^2 + \|\bar{\mathfrak{J}}_{\text{ion}}\|_{L^2(0,t;L^2(\Omega) \times L^2(\Lambda))}^2 \right. \\ \qquad \left. + \|\mathfrak{J}_{\text{app}}\|_{L^2(0,t;L^2(\Omega) \times L^2(\Lambda))}^2 + \sum_{i=2}^{N_{\text{ter}}} \|\bar{s}_i\|_{L^2(0,t;L^2(\Omega_i))}^2 \right. \\ \qquad \left. + \|\bar{V}\|_{L^2(0,t;L^2(\Omega))}^2 \right). \quad (4.5) $$

Let

$$ \varphi(t) = \|\varphi(t)\|_{L^2(\Omega) \times L^2(\Lambda)}^2 \quad \text{and} \quad \bar{\varphi}(t) = \|\bar{\varphi}(t)\|_{L^2(\Omega) \times L^2(\Lambda)}^2. $$
Combining estimates (3.21) and (4.5) and by using the definition of $\bar{s}_i$, we obtain:

$$
\begin{align}
\phi(t) &\le C_3 + C_3 \int_0^t \bar{\phi}(s) ds + C_3 \sum_{i=2}^{N_{\text{ter}}} \int_0^t \int_{\Omega_i} |\langle \bar{V} \rangle_i - \bar{V}_p(x_i, s)|^2 dx ds \nonumber \\
&\le C_3 + C_3 \int_0^t \bar{\phi}(s) ds + C_3 \sum_{i=2}^{N_{\text{ter}}} \int_0^t \int_{\Omega_i} (|\langle \bar{V} \rangle_i|^2 + |\bar{V}_p(x_i, s)|^2) dx ds \tag{4.6} \\
&\le C_3 + C_4 \int_0^t \bar{\phi}(s) ds + C_4 \sum_{i=2}^{N_{\text{ter}}} \int_0^t |\bar{V}_p(x_i, s)|^2 ds, \nonumber
\end{align}
$$

where $C_3 = C_3(T)$ depends on $T$, $\|\phi_0\|_{H^2(\Omega)\times H^2(\Lambda)}^2$ and $\|\mathcal{J}_{app}\|_{L^2(0,T;L^2(\Omega)\times L^2(\Lambda))}^2$.

By multiplying by $\exp^{-\alpha t}$ with $\alpha > 0$ and integrating between 0 and $T$, we get:

$$
\int_0^T \exp(-\alpha t) \phi(t) dt \le C_3 \int_0^T \exp(-\alpha t) dt + C_4 \int_0^T e^{-\alpha t} \left( \int_0^t \bar{\phi}(s) ds \right) dt \\
+ C_4 \sum_{i=2}^{N_{\text{ter}}} \int_0^T e^{-\alpha t} \left( \int_0^t \bar{V}_p(x_i, s)^2 ds \right) dt.
$$

Integrating by parts, we obtain:

$$
\begin{align}
\int_0^T \exp(-\alpha t) \phi(t) dt &\le \frac{1}{\alpha} \Biggl[ C_3(1-\exp(-\alpha T)) + C_4 \int_0^T e^{-\alpha t} \bar{\phi}(t) dt - C_4 e^{-\alpha T} \int_0^T \bar{\phi}(t) dt \nonumber \\
&\phantom{{}\le \frac{1}{\alpha} \Biggl[} + C_4 \sum_{i=2}^{N_{\text{ter}}} \int_0^T e^{-\alpha t} \bar{V}_p(x_i,t)^2 dt - C_4 e^{-\alpha T} \int_0^T \sum_{i=2}^{N_{\text{ter}}} \bar{V}_p(x_i,t)^2 dt \Biggr] \tag{4.7} \\
&\le \frac{1}{\alpha} \Biggl[ C_3 + C_4 \int_0^T e^{-\alpha t} \bar{\phi}(t) dt + C_4 \sum_{i=2}^{N_{\text{ter}}} \int_0^T e^{-\alpha t} \bar{V}_p(x_i,t)^2 dt \Biggr] \nonumber
\end{align}
$$

And then we get

$$
\int_0^T e^{-\alpha t} \phi(t) dt \le \frac{C_3 + C_4}{\alpha} \max \left( 1, \int_0^T e^{-\alpha t} \bar{\phi}(t) dt, \sum_{i=2}^{N_{\text{ter}}} \int_0^T e^{-\alpha t} \bar{V}_p(x_i, t)^2 dt \right).
$$

Now, as $\sum_{i=2}^{N_{\text{ter}}} \bar{V}_p(x_i, t)^2 \le N_{\text{ter}} \bar{M}_{\infty}^2$ and

$$
\sum_{i=2}^{N_{\text{ter}}} \int_0^T \exp(-\alpha t) \bar{V}_p(x_i, t)^2 dt \le \frac{N_{\text{ter}} \bar{M}_{\infty}^2}{\alpha}, \quad (4.8)
$$

we can choose $\alpha_0 = \max(N_{\text{ter}}\bar{M}_{\infty}^2, C_3 + C_4)$ and for $\alpha \ge \alpha_0$ we get

$$
\begin{align*}
\|\varphi\|^2_{\alpha, L^2(\Omega) \times L^2(\Lambda)} &= \int_0^T e^{-\alpha t} \phi(t) dt \le \max \left\{ 1, \int_0^T e^{-\alpha t} \bar{\phi}(t) dt \right\} \\
&= \max \left\{ 1, \|\bar{\varphi}\|^2_{\alpha, L^2(\Omega) \times L^2(\Lambda)} \right\}.
\end{align*}
$$

□

Combining the result of Lemma 4.2 with estimates (3.27), (3.30) for $p = 2$, and the fact that $|||\varphi|||_{\alpha,H} \le ||\varphi||_H$ for each Hilbert space, we immediately have the following corollary.

**Corollary 4.3.** Let $M_0 \ge 1$ and $\bar{M}_\infty > 0$. Let $\bar{\phi}, \psi, \theta$ be as in the statement of Lemma 4.2 such that

$$
\|\bar{\phi}\|^2_{\alpha, L^2(\Omega) \times L^2(\Lambda)} \le M_0; \quad |\bar{V}_p(x_i, t)| \le \bar{M}_\infty, \forall i = 1, \dots, N_{\text{ter}}, t \in [0, T]. \quad (4.9)
$$

Then there exist $M_1 > 0$ depending only on $M_0$, $\bar{M}_\infty$ and the data of the problem, such that:

$$
\|\varphi\|^2_{\alpha, H^2(\Omega) \times H^2(\Lambda)} \le M_1 \quad \text{and} \quad \|\partial_t \varphi\|^2_{\alpha, L^2(\Omega) \times L^2(\Lambda)} \le M_1.
$$
Proof. For $p = 2$, the estimates (3.27) and (3.30) imply that:

$$ \begin{aligned} \|\varphi(t)\|_{H^2(\Omega) \times H^2(\Lambda)}^2 &\le C_2 (\|\varphi_0\|_{H^2(\Omega) \times H^2(\Lambda)}^2 + \|\bar{\mathcal{J}}_{\text{ion}}\|_{L^2(0,t;L^2(\Omega) \times L^2(\Lambda))}^2 \\ &\quad + \|\mathcal{J}_{\text{app}}\|_{L^2(0,t;L^2(\Omega) \times L^2(\Lambda))}^2 + \sum_{i=2}^{N_{\text{ter}}} \|\bar{s}_i\|_{L^2(0,t;L^2(\Omega_i))}^2) \\ &\quad + \|\bar{V}\|_{L^2(0,t;L^2(\Omega))}^2), \quad \text{a.e for } t \in [0,T]. \end{aligned} \quad (4.10) $$

By noting:

$$ \begin{aligned} \phi_{H^2}(t) &= \|\varphi(t)\|_{H^2(\Omega) \times H^2(\Lambda)}^2 \\ \bar{\phi}_{H^2}(t) &= \|\bar{\varphi}(t)\|_{H^2(\Omega) \times H^2(\Lambda)}^2 \end{aligned} $$

and using estimates (3.21) and (4.10), we obtain:

$$ \phi(t)_{H^2} \le C_3 + C_4 \int_0^t \bar{\phi}(s)ds, \quad \text{a.e in } t. $$

Here $C_3$ and $C_4$ are the constants defined in (4.6).

Following the approach of the previous proof, we get:

$$ \|\varphi\|_{\alpha, H^2(\Omega) \times H^2(\Lambda)}^2 \le \max\{1, \|\bar{\varphi}\|_{\alpha, L^2(\Omega) \times L^2(\Lambda)}^2\} = \max\{1, M_0\} \le M_1. $$

for $M_1 > 1$ depending only on $M_0$ and data of the problem. On the other hand, by estimates (3.27) and (3.30), we get:

$$ \begin{aligned} \|\partial_t \varphi(t)\|_{L^2(\Omega) \times L^2(\Lambda)}^2 &\le C_2 (\|\varphi_0\|_{H^2(\Omega) \times H^2(\Lambda)}^2 + \|\bar{\mathcal{J}}_{\text{ion}}\|_{L^2(0,t;L^2(\Omega) \times L^2(\Lambda))}^2 + \|\mathcal{J}_{\text{app}}\|_{L^2(0,t;L^2(\Omega) \times L^2(\Lambda))}^2 \\ &\quad + \sum_{i=2}^{N_{\text{ter}}} \|\bar{s}_i\|_{L^2(0,t;L^2(\Omega_i))}^2 + \|\bar{V}\|_{L^2(0,t;L^2(\Omega))}^2). \end{aligned} $$

Repeating the same steps, we obtain:

$$ \|\partial_t \varphi\|_{\alpha, L^2(\Omega) \times L^2(\Lambda)}^2 \le M_1. $$

□

The second lemma is as follows:

**Lemma 4.4.** Let $M_0, M_1$ and $\bar{M}_\infty$, be as in Corollary 4.3, $\bar{\mathcal{J}}_{\text{ion}}$ be as given in (3.18) and $\bar{\varphi} \in H_T$ such that

$$ \|\bar{\varphi}\|_{\alpha, L^2(\Omega) \times L^2(\Lambda)}^2 \le M_0, \quad \|\partial_t \bar{\varphi}\|_{\alpha, L^2(\Omega) \times L^2(\Lambda)}^2 \le M_1, \quad |\bar{V}_p(x_i, t)| \le \bar{M}_\infty, \quad i = 1, \dots, N_{\text{ter}}, \quad t \in [0, T]. $$

Let $\varphi_0 \in H^2(\Omega) \times H_2(\Lambda)$ and for some $p > 4$, let $\bar{\mathcal{J}}_{\text{app}} \in L^p(0, T; L^2(\Omega) \times L^2(\Lambda))$. Then, there exists $M_\infty > \bar{M}_\infty$, depending only on $M_0, \bar{M}_\infty, p$ and the data of the problem, such that:

$$ \sup \left\{ |\varphi| = (|V(y_1, t)|, |V_p(y_2, t)|) : (y_1, y_2) \in \Omega \times \Lambda, \ t \in [0, T] \right\} \le M_\infty. \quad (4.11) $$

Proof. Let $\mathbf{H}_1$ and $\mathbf{H}_2$ be the Besov spaces introduced in Section 3.1 and let $\mathbf{S}_p = \mathbf{H}_1 \times \mathbf{H}_2$. Using (3.27) and (3.30), we have:

$$ \begin{aligned} \|\varphi\|_{\mathbf{S}_p} \le C (\|\varphi_0\|_{H^2(\Omega) \times H_2(\Lambda)} + \|\bar{\mathcal{J}}_{\text{ion}}\|_{L^p(0,t;L^2(\Omega) \times L^2(\Lambda))} + \|\mathcal{J}_{\text{app}}\|_{L^p(0,t;L^2(\Omega) \times L^2(\Lambda))} \\ + \sum_{i=2}^{N_{\text{ter}}} \|\bar{s}_i\|_{L^p(0,t;L^2(\Omega_i))} + \|\bar{V}\|_{L^p(0,t;L^2(\Omega))}). \end{aligned} \quad (4.12) $$

and using the property (3.5), we have:

$$ \varphi \in C^0([0, T]; \mathbf{S}_p), \quad \forall p \ge 1 $$

By the estimates (3.16) and (3.17), there exist a constant $C_5 > 0$ such that:

$$ \begin{aligned} \|\bar{\mathcal{J}}_{\text{ion}}\|_{L^p(0,t;L^2(\Omega) \times L^2(\Lambda))} &\le C_5 (1 + \|\bar{\varphi}\|_{L^2(\Omega; H^1(0,T)) \times L^2(\Lambda; H^1(0,T))}^2) \\ &\le C_6 (1 + M_0), \quad \forall p \in (1, +\infty). \end{aligned} \quad (4.13) $$

Now, by the property (3.8), we have

$$ \mathbf{S}_p \subset C^0(\Omega) \times C^0(\Lambda), \quad \forall p > 4, $$

this allows us to say

$$ \varphi \in C^0([0, T]; C^0(\Omega) \times C^0(\Lambda)). $$
Finally, taking into account the inequalities (4.12) and (4.13) there is a constant $M_\infty > 0$ depending on $M_0, M_1, \bar{M}_\infty$, $p$ and the parameters of the problem such that

$$ \sup \left\{ |\varphi| = (|V(y_1,t)|, |V_p(y_2,t)|) : (y_1,y_2) \in \Omega \times \Lambda, \ t \in [0,T] \right\} \le M_\infty. $$

□

This lemma thus shows that property (P1) holds.

**Final step: end of the proof of Theorem 4.1.** Now, we prove the second property (P2): $\mathcal{F}(K)$ is a contraction in $L^2(0,T;L^2(\Omega) \times L^2(\Lambda))$ endowed with the norm $||| \cdot |||_{\alpha,L^2(\Omega) \times L^2(\Lambda)}$. Let $p > 4$, $\varphi_0 = (V_0, V_p) \in H^2(\Omega) \times H^2(\Lambda)$, $(W_0, W_{p,0}) : \Omega \times \Lambda \to [0,1]^{2k}$ and $(Z_0, Z_{p,0}) : \Omega \times \Lambda \to (0, +\infty)^{2m}$ such that:

$$ (Z_0, \log \mathbf{Z}_0) \in (L^2(\Omega))^{2m} \text{ and } (Z_{p,0}, \log \mathbf{Z}_{p,0}) \in (L^2(\Lambda))^{2m}. $$

Let $\bar{\varphi}_i = (\bar{V}_i, \bar{V}_{p,i})$, $i = 1,2$ and $\psi_i = (W_i, W_{p,i})$, $\theta_i = (Z_i, Z_{p,i})$ be the solutions of systems (3.9) and (3.10) respectively as in the hypothesis of Proposition 3.1 corresponding to $\bar{\varphi}_i$.

By estimates (3.11), (3.14), (3.15) and (4.11), there exists a compact set $\mathcal{C} = \mathcal{C}(T) \subseteq (\mathbb{R} \times \mathbb{R}^k \times \mathbb{R}^m)^2$ such that:

$$ (\bar{V}_i(y_1,t), W_i(y_1,t), \log \mathbf{Z}_i(y_1,t), \bar{V}_{p,i}(y_2,t), W_{p,i}(y_2,t), \log \mathbf{Z}_{p,i}(y_2,t)) \in \mathcal{C} $$

$$ \forall (y_1,t) \in \Omega \times [0,T], \forall (y_2,t) \in \Lambda \times [0,T]. $$

Therefore, there exists a compact set $\tilde{\mathcal{C}} = \tilde{\mathcal{C}}(T) \subseteq (\mathbb{R} \times \mathbb{R}^k \times (0, +\infty)^m)^2$ such that:

$$ (\bar{V}_i(y_1,t), W_i(y_1,t), Z_i(y_1,t), \bar{V}_{p,i}(y_2,t), W_{p,i}(y_2,t), Z_{p,i}(y_2,t)) \in \tilde{\mathcal{C}} $$

$$ \forall (y_1,y_2) \in \Omega \times \Lambda, \forall t \in [0,T]. $$

On another hand, $g$ and $g_p$ being locally Lipschitz continuous, there exists $L_1, L_2 > 0$ depending on $\tilde{\mathcal{C}}$ such that:

$$ |W_1(y_1,t) - W_2(y_1,t)| \le L_1 \int_0^T (|\bar{V}_1(y_1,s) - \bar{V}_2(y_1,s)| + |W_1(y_1,s) - W_2(y_1,s)|) ds \quad (4.14) $$

and

$$ |W_{p,1}(y_2,t) - W_{p,2}(y_2,t)| \le L_2 \int_0^T (|\bar{V}_{p,1}(y_2,s) - \bar{V}_{p,2}(y_2,s)| + |W_{p,1}(y_2,s) - W_{p,2}(y_2,s)|) ds. \quad (4.15) $$

Then, by summing up the squares of (4.14 and (4.15) we have:

$$ |\psi_1(y,t) - \psi_2(y,t)| \le L \int_0^T |\bar{\varphi}_1(y,s) - \bar{\varphi}_2(y,s)| + |\psi_1(y,s) - \psi_2(y,s)| ds, \quad (4.16) $$

$\forall y = (y_1, y_2) \in \Omega \times \Lambda$ and $t \in [0, T]$, where $L$ is a positive constant depending on $L_1$ and $L_2$.

Now, using Jensen's inequality and integrating on $\Omega$ and on $\Lambda$, we obtain:

$$ \|\psi_1(y,t) - \psi_2(y,t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le 2L \int_0^T \left( \|\bar{\varphi}_1(y,s) - \bar{\varphi}_2(y,s)\|_{L^2(\Omega) \times L^2(\Lambda)} \right. \\ \left. \qquad + \|\psi_1(y,s) - \psi_2(y,s)\|_{L^2(\Omega) \times L^2(\Lambda)} \right) ds. \quad (4.17) $$

Thanks to the Gronwall lemma, (4.17) writes

$$ \|\psi_1(y,t) - \psi_2(y,t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le 2Le^{2LT} \int_0^T \|\bar{\varphi}_1(y,s) - \bar{\varphi}_2(y,s)\|_{L^2(\Omega) \times L^2(\Lambda)} ds, \quad (4.18) $$

By hypothesis (2.13), we have $J_i \in C^2(\mathbb{R} \times \mathbb{R}^k \times \mathbb{R})$, thus

$$ J_i(\dots, \log(\dots)) \in C^2(\mathbb{R} \times \mathbb{R}^k \times (0, +\infty)). $$

So, $F = (f, f_p)$ is locally Lipschitz continuous on $(\mathbb{R} \times \mathbb{R}^k \times (0, +\infty))^2$ and therefore there exists $L_3, L_4 > 0$ depending on $\tilde{\mathcal{C}}$ such that, $\forall y = (y_1, y_2) \in \Omega \times \Lambda$ and $t \in [0, T]$,

$$ |Z_1(y_1,t) - Z_2(y_1,t)| \le L_3 \int_0^T (|\bar{V}_1(y_1,s) - \bar{V}_2(y_1,s)| + |W_1(y_1,s) - W_2(y_1,s)| \\ + |Z_1(y_1,s) - Z_2(y_1,s)|) ds $$
and

$$|Z_{p,1}(y_2,t) - Z_{p,2}(y_2,t)| \le L_4 \int_0^T (|\bar{V}_{p,1}(y_2,s) - \bar{V}_{p,2}(y_2,s)| \\ + |W_{p,1}(y_2,s) - W_{p,2}(y_2,s)| + |Z_{p,1}(y_2,s) - Z_{p,2}(y_2,s)|)ds.$$

Then, we have:

$$|\theta_1(y,t) - \theta_2(y,t)| \le L' \int_0^T (|\bar{\varphi}_1(y,s) - \bar{\varphi}_2(y,s)| + |\psi_1(y,s) - \psi_2(y,s)| + |\theta_1(y,s) - \theta_2(y,s)|)ds.$$

Thus, by using Jensen's inequality, Gronwall's lemma and (4.18), we find a constant $L_5 = L_5(L, L', T)$ such that:

$$\|\theta_1(y,t) - \theta_2(y,t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le L_5 \int_0^T \|\bar{\varphi}_1(y,s) - \bar{\varphi}_2(y,s)\|_{L^2(\Omega) \times L^2(\Lambda)} ds. \quad (4.19)$$

By estimates (3.28) and (3.31) we obtain for $C' > 0$:

$$C' \|\varphi_1(t) - \varphi_2(t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le \|\bar{\mathfrak{J}}_{ion,1} - \bar{\mathfrak{J}}_{ion,2}\|_{L^2(0,T;L^2(\Omega) \times L^2(\Lambda))} + \|\bar{\varphi}_1(s) - \bar{\varphi}_2(s)\|_{L^2(\Omega) \times L^2(\Lambda)} \quad (4.20)$$

and since $\bar{\mathfrak{J}}_{ion}$ is locally Lipschitz continuous, we can find $L_6 > 0$ depending on $\mathcal{C}$ such that for all $t \in [0, T]$, it holds:

$$\|\varphi_1(t) - \varphi_2(t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le L_6 \int_0^t (\|\bar{\varphi}_1(s) - \bar{\varphi}_2(s)\|_{L^2(\Omega) \times L^2(\Lambda)} \\ + \|\psi_1(s) - \psi_2(s)\|_{L^2(\Omega) \times L^2(\Lambda)} + \|\theta_1(s) - \theta_2(s)\|_{L^2(\Omega) \times L^2(\Lambda)}) ds. \quad (4.21)$$

And by using the inequalities (4.18) and (4.19), we can find $C = C(L, L', L_5, L_6, T)$ such that $\forall t \in [0, T]$ it holds:

$$\|\varphi_1(t) - \varphi_2(t)\|_{L^2(\Omega) \times L^2(\Lambda)} \le C \int_0^t \|\bar{\varphi}_1(s) - \bar{\varphi}_2(s)\|_{L^2(\Omega) \times L^2(\Lambda)} ds. \quad (4.22)$$

Now, we note:

$$\phi(t) = \|\varphi_1(t) - \varphi_2(t)\|_{L^2(\Omega) \times L^2(\Lambda)},$$

$$\overline{\phi(t)} = \|\bar{\varphi}_1(t) - \bar{\varphi}_2(t)\|_{L^2(\Omega) \times L^2(\Lambda)}.$$

Hence, (4.22) writes:

$$\phi(t) \le L \int_0^t \overline{\phi(s)} ds, \quad \forall t \in [0, T].$$

We multiply by $e^{-\alpha t}$, $\alpha > 0$, integrate between 0 and T and get

$$\int_0^T e^{-\alpha t} \phi(t) \le L \int_0^T e^{-\alpha t} \int_0^t \overline{\phi(s)} ds dt, \quad \forall t \in [0, T]. \quad (4.23)$$

Integrating by parts, (4.23) writes:

$$\int_0^T e^{-\alpha t} \phi(t) \le \frac{L}{\alpha} \left( \int_0^T e^{-\alpha t} \overline{\phi(t)} dt - C_4 \exp(-\alpha T) \int_0^T \overline{\phi(t)} dt \right) \quad (4.24) \\ \le \frac{L}{\alpha} \int_0^T e^{-\alpha t} \overline{\phi(t)} dt.$$

We conclude that:

$$\|\mathfrak{F}(\bar{\varphi}_1) - \mathfrak{F}(\bar{\varphi}_2)\|_{\alpha, L^2(\Omega) \times L^2(\Lambda)} \le \frac{L}{\alpha} \|\bar{\varphi}_1 - \bar{\varphi}_2\|_{\alpha, L^2(\Omega)}.$$

So, for $\alpha > L$, the operator $\mathfrak{F}$ is a contraction and the existence and uniqueness of a solution for Problem (2.15) is established.

## 5. CONCLUSION

In this paper, we have demonstrated the existence and uniqueness of a solution for the 1D/3D coupled problem. This coupling models the interaction between the Purkinje conduction system and myocardial electrical activity. Our proof relies on the Banach Fixed Point theorem and involves selecting appropriate functional spaces.
ACKNOWLEDGMENT

This work has been supported by SPICY cooperative research program, funded by INRIA international laboratory LIRIMA. The University of Tunis El Manar researcher's work is supported on a regular basis by the Tunisian Ministry of Higher Education, Scientific Research and Technology. This work was also supported by the Agence Nationale de la Recherche (Grant IHU LIRYC ANR-10-IAHU-04). We also thank Prof Moncef Mahjoub for the fruitful discussions.

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<footnote n="1">¹ UNIVERSITY OF TUNIS EL MANAR, FST, CAMPUS UNIVERSITAIRE EL-MANAR, 2092 EL MANAR TUNIS, TUNISIA</footnote>

<footnote n="2">² INRIA CENTRE DE L'UNIVERSITÉ DE BORDEAUX, CARMEN PROJECT TEAM. 200 AVENUE DE LA VIELLE TOUR 33405 TALENCE CEDEX, FRANCE</footnote>

<footnote n="2">² IHU-LIRYC, ELECTROPHYSIOLOGY AND HEART MODELING INSTITUTE, FOUNDATION BORDEAUX UNIVERSITY F-33600 PESSAC-BORDEAUX, FRANCE</footnote>

<footnote>Email address: khouloud.kordoghli@fst.utm.tn, saloua.mani@fst.utm.tn, nejib.zemzemi@inria.fr</footnote>