Rubrique

Inverse heat source problem for a coupled
hyperbolic-parabolic system

Bellassoued Mourad and Riahi Bochra

LASMIN, University of Tunis El Manar, National engineering school of Tunis, ENIT, Tunisia,
mourad.bellassoued@fsb.rnu.tn

Department of Mathematics, University of Carthage, Bizerte, Tunisia

riahi.bochra@gmail.com

**RÉSUMÉ.** Dans ce papier, on a prouvé une estimation de stabilité de type Höldérienne pour un problème inverse de détermination du terme source de l'équation de la chaleur à l'aide d'une inégalité de Carleman pour un système d'équations hyperbolique-parabolique couplé.

**ABSTRACT.** In this paper we consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity. Using a local Carleman estimate for a coupled hyperbolic-parabolic system, we prove the uniqueness and a Hölder stability in determining the heat source by a single measurement of solution over $\omega \times (0, T)$, where $T > 0$ is a sufficiently large time and a suitable subdomain $\omega \subset \Omega$ such that $\partial\omega \supset \partial\Omega$.

**MOTS-CLÉS** : Problème inverse, estimation de Carleman, système couplet

**KEYWORDS** : Inverse problem, Carleman estimate, coupled system
# 1. Introduction

Let $\Omega \subset \mathbb{R}^3$ an open and bounded domain with $C^\infty$ boundary $\Gamma = \partial\Omega$, and let $t$ and $x = (x_1, x_2, x_3) \in \Omega$ denote the time variable and the spatial variable respectively. Given $T > 0$, we consider a coupled hyperbolic-parabolic system

$$ egin{cases} 
\mathbf{u}_{tt} - \Delta_{\mu,\lambda}\mathbf{u} - \nabla(\lambda^*(x)\mathrm{div}\mathbf{u}_t) + \varrho_1(x)\nabla\theta = 0 & \text{in } Q \equiv \Omega \times (0, T), \\
\theta_t - \Delta\theta + \varrho_2(x)\mathrm{div}\mathbf{u}_t = g & \text{in } Q, \\
\mathbf{u}(x,t) = 0, \quad \theta(x,t) = 0 & \text{on } \Sigma \equiv \Gamma \times (0, T),
\end{cases} 
\quad (1) $$

where the $\cdot_t$ stands for the time derivative, $\nabla = (\partial_1, \partial_2, \partial_3)$, and $\Delta_{\mu,\lambda}$ is the elliptic second order linear differential operator given by

$$ \Delta_{\mu,\lambda}\mathbf{v}(x) \equiv \mu\Delta\mathbf{v}(x) + (\mu + \lambda)(\nabla(\mathrm{div}\mathbf{v}(x))) \\ + \mathrm{div}\mathbf{v}(x)\nabla\lambda(x) + (\nabla\mathbf{v} + (\nabla\mathbf{v})^T)\nabla\mu(x), \quad x \in \Omega, \quad (2) $$

for $\mathbf{v} = (v_1, v_2, v_3)^T$, where $\cdot^T$ denotes the transpose of matrices. Throughout this paper, $\mathbf{u} = (u_1, u_2, u_3)^T$ denotes the displacement at the location $x$ and the time $t$, and $\theta \equiv \theta(x, t)$, the temperature, is a scalar function, $g \in H^1(0, T; L^2(\Omega))$ is a heat source. We will assume that the Lamé parameters $\mu, \lambda \in C^2(\bar{\Omega})$, satisfy

$$ \mu(x) \geq \mu_0 > 0, \quad \lambda(x) + 2\mu(x) > 0, \quad \forall x \in \bar{\Omega}. $$

and $\lambda^* \in C^2(\bar{\Omega})$ is the consolidation coefficient which satisfy :

$$ \lambda^*(x) \geq k_0 > 0, \quad x \in \bar{\Omega}. $$

Furthermore, the coupling coefficients $\varrho_1, \varrho_2$ satisfies :

$$ (\varrho_1, \varrho_2) \in (C^2(\bar{\Omega}))^2; \quad \varrho_1(x) > \varrho_0 > 0 \quad \text{in for all } x \in \bar{\Omega}. $$

We assume that the heat source is given by

$$ g(x, t) = q(x)k(x, t), \quad (3) $$

where $k \in W^{2,\infty}(Q)$ and $q \in H^2(\Omega)$.

We can prove (e.g., [1, 5]) that the system (1) possesses a unique solution

$$ (\mathbf{u}, \theta) \equiv (\mathbf{u}_{(\lambda^*, \varrho_1, \varrho_2)}, \theta_{(\lambda^*, \varrho_1, \varrho_2)}), $$

Let $\omega \subset \Omega$ be a given arbitrarily subdomain such that $\partial\omega \supset \Gamma$, i.e. $\omega = \Omega \cap V$ where $V$ is a neighbourhood of $\Gamma$ on $\mathbb{R}^3$ and let $k$ and $t_0 \in (0, T)$ be appropriately given.

**Inverse Problem :** Determine $q(x), x \in \Omega$, by measurements

$$ \mathbf{u}|_{\omega \times (0,T)}, \quad \mathbf{u}(x, t_0), \quad \text{and} \quad \theta(x, t_0) \quad x \in \Omega. $$

The main subject of this paper is the inverse problem of determining of $g$, in the Biot consolidation model in poro-elasticity, uniquely from observed data of displacement vector $\mathbf{u}$ on a suitable subdomain $\omega \subset \Omega$ and the observation data of $\mathbf{u}$ and $\theta$ at given a
suitable time $t_0$. Such kinds of observation data are similar to those considered in (e.g. [2], [3], [4]).

The key ingredient in our argument is an $L^2$-weighted inequality of Carleman type for coupled mixed hyperbolic-parabolic system. We prove a Hölder stability estimate in our inverse problem. We note that the uniqueness in the inverse problem follows directly from the Hölder stability.

## 1.1. Statement of main result

Let $t_0 \in (0, T)$ and $x_0 \in \mathbb{R}^3 \setminus \bar{\Omega}$ such that

$$ \frac{1}{\sqrt{r_0}} \max_{x \in \bar{\Omega}} |x - x_0| < \min\{t_0, T - t_0\}, \quad (4) $$

where $r_0 \in (0, \mu_0)$.

We denote $(\mathbf{u}, \theta)$ the solution of (1) corresponding to $(\varrho_1, \varrho, \lambda^*)$.

**Theorem 1.1** Let $x_0 \in \mathbb{R}^3 \setminus \bar{\Omega}$ and $t_0 \in (0, T)$ satisfies (4).

Let $k \in W^{2,\infty}(Q)$ such that $\|k\|_{W^{2,\infty}} \le M$ and

$$ k(x, t_0) \neq 0, \quad x \in \bar{\Omega}. $$

We assume that the solution $(\mathbf{u}, \theta)$ satisfies the a priori boundedness :

$$ \|\mathbf{u}\|_{H^2(0,T;H^2(\Omega))} + \|\theta\|_{H^1(0,T;L^2(\Omega))} \le M_0, \quad (5) $$

for some given positive constant $M_0$.

Then there exist constants $C > 0$ and $\delta \in (0, 1)$ such that the following stability estimate hold

$$ \|q(x)\|_{L^2(\Omega)} \le C (\|\mathbf{u}_t\|_{H^4(\omega \times (0,T))} + \|\theta(\cdot, t_0)\|_{H^2(\Omega)} + \|\mathbf{u}_t(\cdot, t_0)\|_{H^3(\Omega)} + \|\mathbf{u}_{tt}(\cdot, t_0)\|_{H^1(\Omega)})^\delta. $$

By Theorem 1.1, we can readily derive the uniqueness in the inverse problem

**Corollary 1.2** Under the same assumptions as in Theorem 1.1, we have the uniqueness.

Let $(\mathbf{u}, \theta)$ satisfy the Biot system (1) such that

$$ \mathbf{u}(x, t) = 0, \quad (x, t) \in \omega \times (0, T) $$

and

$$ \mathbf{u}(x, t_0) = 0, \quad \theta(x, t_0) = 0, \quad x \in \Omega. $$

Then

$$ q(x) = 0 \text{ for all } x \in \Omega. $$

The remainder of the paper is organized as follows. In section 2, we give a Carleman estimate for a coupled hyperbolic-parabolic system. In section 3, we prove Theorem 1.1.
## 2. Carleman estimate

Here we present a Carleman estimate, which was proved in [3]. For formulating our Carleman estimate, we introduce some notations. Let $\psi: \Omega \times \mathbb{R} \to \mathbb{R}$ by setting

$$ \psi(x, t) = \vartheta(x) - \beta((t - t_0)^2 - M) = |x - x_0|^2 - \beta((t - t_0)^2 - M), \quad x \in \bar{\Omega}, \quad (6) $$

for $M > 0$ large, where $x_0 \in \mathbb{R}^3 \setminus \bar{\Omega}$ and $t_0 \in (0, T)$ such that

$$ \min\{t_0^2, (T - t_0)^2\} > r_0^{-1} \left( \max_{x \in \bar{\Omega}} \vartheta(x) \right). \quad (7) $$

We fix $\delta > 0$ and $\beta > 0$ satisfying

$$ \beta \min\{t_0^2, (T - t_0)^2\} > \max_{x \in \bar{\Omega}} \vartheta(x) + \delta, \quad 0 < \beta < r_0. \quad (8) $$

Therefore, by (6) and (8) the function $\psi(x, t)$ verifies the following properties

$$ \psi(x, 0) \le \beta M - \delta, \quad \psi(x, T) \le \beta M - \delta, \quad \text{for all } x \in \bar{\Omega}, \quad (9) $$

there exists $\epsilon \in (0, T/4)$ such that

$$ \max_{x \in \bar{\Omega}} \psi(x, t) \le \beta M - \frac{\delta}{2}, \quad \text{for all } t \in (0, 2\epsilon) \cup (T - 2\epsilon, T), \quad (10) $$

and

$$ \min_{x \in \bar{\Omega}} \psi(x, t_0) \ge \beta M. \quad (11) $$

We next introduce a function $\varphi: \bar{\Omega} \times \mathbb{R} \to \mathbb{R}$ by

$$ \varphi(x, t) = e^{\gamma \psi(x, t)} := \rho(x) \alpha(t), \quad \gamma > 0, \quad (12) $$

where $\gamma$ is a large parameter, $\rho(x)$ and $\alpha(t)$ are defined by

$$ \rho(x) = e^{\gamma(|x-x_0|^2+\beta M)} \ge e^{\gamma\beta M} = d_1, \forall x \in \Omega \text{ and } \alpha(t) = e^{-\beta\gamma(t-t_0)^2} \le 1, \forall t \in (0, T) \quad (13) $$

and let

$$ \sigma \equiv \sigma(x, t) = s\gamma\varphi(x, t). \quad (14) $$

We use usual function spaces, $H^k(Q)$, and

$$ H^{2,1}(Q) = H^1(0, T; L^2(\Omega)) \cap L^2(0, T; H^2(\Omega)). $$

Let $(v, y)$ a solution of the linear Biot consolidation system

$$ \left\{ \begin{array}{l} v_{tt}(x, t) - \Delta_{\mu, \lambda} v(x, t) - \nabla(\lambda^* \text{div} v_t(x, t)) + \varrho_1 \nabla y(x, t) = f(x, t) \quad \text{in } Q, \\ y_t(x, t) - \Delta y(x, t) + \varrho_2 \text{div} v_t(x, t) = h(x, t) \quad \text{in } Q, \end{array} \right. \quad (15) $$

such that

$$ \text{Supp}(v(\cdot, t)) \subset \Omega, \quad \text{Supp}(y(\cdot, t)) \subset \Omega, \quad \text{for all } t \in (0, T). $$

$$ \partial_t^j v(x, 0) = \partial_t^j v(x, T) = 0, \quad y(x, 0) = y(x, T) = 0 \quad \text{for all } x \in \Omega, j = 0, 1. \quad (16) $$

The following theorem is a weighted Carleman estimate with second large parameter for Biot's consolidation system (15) with assumption (16).
**Theorem 2.1 (Carleman estimate)** There exist two constants $\gamma_*$ > 0 and $C > 0$ such that for any $\gamma > \gamma_*$, there exists $s_* = s_*(\gamma) > 0$ such that the following estimate holds

$$ C \int_Q \left( \sigma |\nabla_{x,t} \mathbf{v}|^2 + \sigma^3 |\mathbf{v}|^2 + \sigma^4 |\mathrm{div} \mathbf{v}|^2 + \sigma^3 |\mathrm{div} \mathbf{v}_t|^2 + \sigma^2 |\nabla \mathrm{div} \mathbf{v}|^2 + \sigma |\nabla \mathrm{div} \mathbf{v}_t|^2 \right. \\ \left. + |\Delta y|^2 + \sigma^2 |\nabla y|^2 + \sigma^4 |y|^2 \right) e^{2s\varphi} dx dt \\ \le \int_Q (|\mathbf{f}|^2 + |\nabla \mathbf{f}|^2 + \gamma^{-1} \sigma |h|^2) e^{2s\varphi} dx dt \\ + \int_Q \left( \gamma^{-1} |\Delta \mathrm{div} \mathbf{v}(x, t_0)|^2 + \gamma^{-1} |\mathrm{div} \mathbf{v}_t(x, t_0)|^2 \right. \\ \left. + \sigma^4 |\mathrm{div} \mathbf{v}(x, t_0)|^2 + \sigma^2 |\nabla \mathrm{div} \mathbf{v}(x, t_0)|^2 \right) e^{2s\varphi} dx dt, \quad (17) $$

for any solution $(\mathbf{v}, y) \in H^2(Q) \times H^{2,1}(Q)$ to problem (15) which satisfy (16) and any $s \ge s_*$.

The proof is given by Bellassoued and Riahi [3].

### 3. Proof of the main result

This section is devoted to the proof of the Theorem 1.1. The idea of the proof is based on the Carleman estimate method in [3]. A usual methodology by [2, 3].

#### 3.1. Preliminaries estimate

**Lemma 3.1** Let $\omega$ be an open subdomain of $\Omega$ with regular boundary $\partial\omega \supset \Gamma$. There exists constants $\gamma_*$, $s_*$ and $C > 0$ such that for any $s \ge s_*$ and any $\gamma \ge \gamma_*$ the following estimate holds:

$$ \int_{\omega \times (0, T)} \sigma^4 |v|^2 e^{2s\varphi} dx dt \le C \int_{\omega \times (0, T)} \sigma^2 |\nabla v|^2 e^{2s\varphi} dx dt \quad (18) $$

for any $v \in H^1(\omega \times (0, T))$ such that $v(x, t) = 0$ on $\partial\omega \times (0, T)$.

**Proof** We multiply $\nabla v$ by $(\nabla\varphi)ve^{2s\varphi}$ and using the divergence theorem, we obtain

$$ \begin{aligned} \int_{\omega} \nabla v \cdot (\nabla \varphi) v e^{2s\varphi} dx &= - \int_{\omega} v \operatorname{div}((\nabla \varphi) v e^{2s\varphi}) dx \\ &= - \int_{\omega} |v|^2 \Delta \varphi e^{2s\varphi} dx - 2s \int_{\omega} |v|^2 |\nabla \varphi|^2 e^{2s\varphi} dx \\ &\quad - \int_{\omega} \nabla v \cdot (\nabla \varphi) v e^{2s\varphi} dx. \end{aligned} $$

Therefore,

$$ 2 \int_{\omega} \sigma \nabla v \cdot (\nabla \vartheta) v e^{2s\varphi} dx = -2 \int_{\omega} \sigma^2 |v|^2 |\nabla \vartheta|^2 e^{2s\varphi} dx \\ - \int_{\omega} \sigma |v|^2 \Delta \vartheta e^{2s\varphi} dx - \gamma \int_{\omega} \sigma |v|^2 |\nabla \vartheta|^2 e^{2s\varphi} dx. $$

Taking $\gamma \ge \gamma_*$ and $s \ge s_*$ sufficiently large, we obtain for any $\varepsilon > 0$

$$ C \int_{\omega} \sigma^2 |v|^2 e^{2s\varphi} dx \le C_{\varepsilon} \int_{\omega} |\nabla v|^2 e^{2s\varphi} dx + \varepsilon \int_{\omega} \sigma^2 |v|^2 e^{2s\varphi} dx. \quad (19) $$
Integrating in $(0, T)$ and taking $\epsilon$ small we obtain

$$C \int_{\omega \times (0, T)} \sigma^2 |v|^2 e^{2s\varphi} dx \le C \int_{\omega \times (0, T)} |\nabla v|^2 e^{2s\varphi} dx. \quad (20)$$

Applying the last inequality to $\sigma^2 v$ we obtain

$$\int_{\omega \times (0, T)} \sigma^4 |v|^2 e^{2s\varphi} dx \le C \int_{\omega \times (0, T)} \sigma^2 |\nabla v|^2 e^{2s\varphi} dx, \quad (21)$$

for any $\gamma \ge \gamma_*$ and $s \ge s_*$. This completes the proof. Hence, by Lemma 3.1, we obtain the following Lemma.

**Lemma 3.2** Let $(\mathbf{v}, y) \in H^2(Q) \times H^{2,1}(Q)$, satisfying

$$\left\{
\begin{array}{ll}
\mathbf{v}_{tt} - \Delta_{\mu,\lambda} \mathbf{v} - \nabla (\lambda^*(x) \mathrm{div} \mathbf{v}_t) + \varrho_1(x) \nabla y = \mathbf{f} & (x, t) \in Q, \\
y_t - \Delta y + \varrho_2(x) \mathrm{div} \mathbf{v}_t = h & (x, t) \in Q, \\
\mathbf{v} = 0, & y = 0
\end{array}
\right.
\quad (22)$$

and

$$\partial_t^j \mathbf{v}(x, 0) = \partial_t^j \mathbf{v}(x, T) = 0, \quad y(x, 0) = y(x, T) = 0 \quad \text{for all } x \in \Omega, j = 0, 1. \quad (23)$$

There exist positive constants $\gamma_*$ and $C > 0$ such that, for any $\gamma \ge \gamma_*$ we can find $s_*$ and $D$, the following inequality holds

$$\int_Q e^{2s\varphi} \left( \sigma |\nabla_{x,t} \mathbf{v}|^2 + \sigma^3 |\mathbf{v}|^2 + \sigma^4 |\mathrm{div} \mathbf{v}|^2 + \sigma^3 |\mathrm{div} \mathbf{v}_t|^2 + \sigma^2 |\nabla \mathrm{div} \mathbf{v}|^2 \right. \\
\left. + \sigma |\nabla \mathrm{div} \mathbf{v}_t|^2 + |\Delta y|^2 + \sigma^2 |\nabla y|^2 + \sigma^4 |y|^2 \right) dxdt \le C \int_Q \left( \gamma^{-1}\sigma |h|^2 \right. \\
\left. + \sigma^2 |\mathbf{f}|^2 + |\nabla \mathbf{f}|^2 \right) dxdt + C e^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 \right. \\
\left. + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 \right), \quad (24)$$

for any $s \ge s_*$.

**Proof** Let $\omega' \subset \omega$ such that $\partial\omega' \supset \Gamma$. In order to apply Carleman estimate, we introduce a cut-off function $\chi$ satisfying $0 \le \chi \le 1$, $\chi \in C^\infty(\mathbb{R}^3)$, $\chi = 1$ in $\Omega \setminus \omega'$ and $\mathrm{supp}\chi \subset \Omega$. Put

$$\hat{\mathbf{v}}(x, t) = \chi(x)\mathbf{v}(x, t), \quad \hat{y}(x, t) = \chi(x)y(x, t).$$

Noting that $(\hat{\mathbf{v}}, \hat{y}) \in H^2(Q) \times H^{2,1}(Q)$, and satisfying

$$\left\{
\begin{array}{ll}
\hat{\mathbf{v}}_{tt} - \Delta_{\mu,\lambda} \hat{\mathbf{v}} - \nabla (\lambda^* \mathrm{div} \hat{\mathbf{v}}_t) + \varrho_1 \nabla \hat{y} = \hat{\mathbf{f}} & (x, t) \in Q, \\
\hat{y}_t - \Delta \hat{y} + \varrho_2 \mathrm{div} \hat{\mathbf{v}}_t = \hat{g} & (x, t) \in Q,
\end{array}
\right.
\quad (25)$$

with

$$\mathrm{Supp}(\hat{\mathbf{v}}(\cdot, t)) \subset \Omega \quad \mathrm{Supp}(\hat{y}(\cdot, t)) \subset \Omega, \quad \forall t \in (0, T).$$

Here

$$\begin{aligned}
\hat{\mathbf{f}} &= \chi(x)\mathbf{f}(x, t) - [\Delta_{\mu,\lambda}, \chi]\mathbf{v} - \nabla(\lambda^*\nabla\chi\cdot\mathbf{v}_t) - \lambda^*(\nabla\chi)\mathrm{div}\mathbf{v}_t + \varrho_1 y\nabla\chi, \\
&\equiv \chi(x)\mathbf{f}(x, t) + P_1\mathbf{v} + Q_1\mathbf{v}_t + A_0 y
\end{aligned}$$
$$ 
\begin{aligned}
\hat{g} &= \chi(x)h(x,t) - 2\nabla\chi \cdot \nabla y - y\Delta\chi + \rho_2\nabla\chi \cdot \mathbf{v}_t \\
&\equiv \chi(x)h(x,t) + A_1y + Q_0\mathbf{v}_t
\end{aligned}
$$

and $P_1, Q_1, A_1$ are a first order partial differential operators with the coefficients are supported in $\omega'$ and $A_0, Q_0$ are zeroth order partial differential operators supported also in $\omega'$. Noting that $(\hat{\mathbf{v}}, \hat{y})$ satisfies (25), then we can apply the Carleman estimate for Biot's system (17) to $(\hat{\mathbf{v}}, \hat{y})$, we obtain

$$ 
\begin{aligned}
C \int_{(\Omega \setminus \omega') \times (0,T)} & e^{2s\varphi} (\sigma |\nabla_{x,t} \mathbf{v}|^2 + \sigma^3 |\mathbf{v}|^2 + \sigma^4 |\mathrm{div} \mathbf{v}|^2 + \sigma^3 |\mathrm{div} \mathbf{v}_t|^2 \\
& + \sigma^2 |\nabla \mathrm{div} \mathbf{v}|^2 + \sigma |\nabla \mathrm{div} \mathbf{v}_t|^2 + |\Delta y|^2 + \sigma^2 |\nabla y|^2 + \sigma^4 |y|^2) dxdt \\
&\leq \int_Q e^{2s\varphi} (\gamma^{-1}\sigma |\hat{g}|^2 + |\hat{f}|^2 + |\nabla \hat{f}|^2) dxdt + \int_Q (\gamma^{-1} |\Delta \mathrm{div} \hat{\mathbf{v}}(x, t_0)|^2 \\
& + \gamma^{-1} |\mathrm{div} \hat{\mathbf{v}}_t(x, t_0)|^2 + \sigma^4 |\mathrm{div} \hat{\mathbf{v}}(x, t_0)|^2 + \sigma^2 |\nabla \mathrm{div} \hat{\mathbf{v}}(x, t_0)|^2) e^{2s\varphi} dxdt.
\end{aligned}
\quad (26) $$

Using the first equation of system (22) after taking divergence, we obtain the following estimate

$$ 
\int_{\omega' \times (0,T)} |\Delta y|^2 e^{2s\varphi} dxdt \leq C e^{Ds} \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 \\
+ \int_{\omega' \times (0,T)} |\nabla y|^2 e^{2s\varphi} dxdt + \int_Q e^{2s\varphi} |\nabla \mathbf{f}|^2 dxdt.
$$

By the last inequality and (26), we deduce

$$ 
\begin{aligned}
C \int_{\Omega \times (0,T)} & e^{2s\varphi} (\sigma |\nabla_{x,t} \mathbf{v}|^2 + \sigma^3 |\mathbf{v}|^2 + \sigma^4 |\mathrm{div} \mathbf{v}|^2 + \sigma^3 |\mathrm{div} \mathbf{v}_t|^2 + \sigma^2 |\nabla \mathrm{div} \mathbf{v}|^2 \\
& + \sigma |\nabla \mathrm{div} \mathbf{v}_t|^2 + |\Delta y|^2 + \sigma^2 |\nabla y|^2 + \sigma^4 |y|^2) dxdt \\
\leq & C \int_Q (\gamma^{-1}\sigma |h|^2 + |\mathbf{f}|^2 + |\nabla \mathbf{f}|^2) dxdt \\
& + C \left( \int_{\omega' \times (0,T)} \sigma^4 |y|^2 e^{2s\varphi} dxdt + \int_{\omega' \times (0,T)} \sigma^2 |\nabla y|^2 e^{2s\varphi} dxdt \right) \\
& + e^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 \right).
\end{aligned}
\quad (27) $$

Let $\chi_1$ be a cut-off function satisfying $0 \le \chi_1 \le 1$, $\chi_1 \in \mathcal{C}^\infty(\mathbb{R}^3)$, $\chi_1 = 1$ in $\overline{\omega}'$ and $\mathrm{Supp}(\chi_1) \subset \omega$. Let us consider $z(x, t) = \chi_1(x)y(x, t) \in H^1(\omega \times (0, T))$ and $z(x, t) = 0$ for all $(x, t) \in \partial\omega \times (0, T)$, so that by Lemma 3.1, we have

$$ 
\begin{aligned}
\int_{\omega' \times (0,T)} \sigma^4 |y|^2 \sigma e^{2s\varphi} dxdt &\le \int_{\omega \times (0,T)} \sigma^4 |z|^2 e^{2s\varphi} dxdt \\
&\le C \int_{\omega \times (0,T)} \sigma^2 |\nabla y|^2 e^{2s\varphi} dxdt + C \int_Q \sigma^2 |y|^2 e^{2s\varphi} dxdt.
\end{aligned}
\quad (28) $$

Furthermore by the first equation of (22), we have

$$ 
\int_{\omega' \times (0,T)} \sigma^2 |\nabla y|^2 e^{2s\varphi} dxdt \le C e^{Ds} \|\mathbf{v}\|_{H^3(\omega \times (0,T))}^2
$$
$$ + \int_{\Omega \times (0,T)} \sigma^2 |\mathbf{f}|^2 e^{2s\varphi} dxdt. \qquad (29) $$

Inserting (28) and (29) in (27), we obtain (24). This completes the proof of the Lemma.

Let $(\mathbf{v}, y) \in H^2(Q) \times H^{2,1}(Q)$ satisfying

$$ \mathbf{v}_{tt} - \Delta_{\mu,\lambda} \mathbf{v} - \nabla(\lambda^* \mathrm{div} \mathbf{v}_t) + \varrho_1 \nabla y = \mathbf{f} \quad (x,t) \in Q, $$

$$ y_t - \Delta y + \varrho_2 \mathrm{div} \mathbf{v}_t = h \quad (x,t) \in Q, $$

$$ \mathbf{v} = 0, y = 0 \quad (x,t) \in \Sigma. $$

Put

$$ \tilde{\mathbf{v}}(x,t) = \eta(t)\mathbf{v}(x,t), \quad \tilde{y}(x,t) = \eta(t)y(x,t). $$

Noting that $(\tilde{\mathbf{v}}, \tilde{y}) \in H^2(Q) \times H^{2,1}(Q)$ satisfies

$$ \tilde{\mathbf{v}}_{tt} - \Delta_{\mu,\lambda} \tilde{\mathbf{v}} - \nabla(\lambda^* \mathrm{div} \tilde{\mathbf{v}}_t) + \varrho_1 \nabla \tilde{y} = \eta \mathbf{f} + \eta_{tt} \mathbf{v} + 2\eta_t \mathbf{v}_t - \eta_t \nabla(\lambda^* \mathrm{div} \mathbf{v}) \quad (x,t) \in Q, $$

$$ \tilde{y}_t - \Delta \tilde{y} + \varrho_2 \mathrm{div} \tilde{\mathbf{v}}_t = \eta h + \eta_t (y - \varrho_2 \mathrm{div} \mathbf{v}) \quad (x,t) \in Q, $$

$$ \tilde{\mathbf{v}} = 0, \quad \tilde{y} = 0 \quad (x,t) \in \Sigma, \tag{30} $$

Henceforth we fix $\gamma > 0$ sufficiently large. By $N_{s,\varphi}$ we denote the quantity

$$ N_{s,\varphi}(\mathbf{v}, y) = \int_Q e^{2s\varphi} \left( s |\nabla_{x,t} \mathbf{v}|^2 + s^3 |\mathbf{v}|^2 + s^4 |\mathrm{div} \mathbf{v}|^2 + s^3 |\mathrm{div} \mathbf{v}_t|^2 + s^2 |\nabla \mathrm{div} \mathbf{v}|^2 + s |\nabla \mathrm{div} \mathbf{v}_t|^2 + |\Delta y|^2 + s^2 |\nabla y|^2 + s^4 |y|^2 \right) dxdt. \quad (31) $$

We introduce a cut-off function $\eta$ satisfying $0 \le \eta \le 1$, $\eta \in C^\infty(\mathbb{R})$, $\eta = 1$ in $(2\varepsilon, T-2\varepsilon)$ and
$\mathrm{Supp}(\eta) \subset (\varepsilon, T-\varepsilon)$. Finally we denote

$$ \tilde{\mathbf{v}} = \eta \mathbf{v}, \quad \tilde{y} = \eta y. $$

Setting $d_0 = e^{(\beta M - \delta/2)\gamma}$, by (9), we have

$$ \max_{x \in \Omega} \varphi(x,t) \le d_0, \quad t \in (0, 2\varepsilon) \cup (T - 2\varepsilon, T). \qquad (32) $$

We have the following lemma :

**Lemma 3.3** There exist three positive constants $s_*$, $C > 0$ and $D$ such that the following inequality holds :

$$ \begin{aligned} CN_{s,\varphi}(\tilde{\mathbf{v}}, \tilde{y}) &\le \int_Q (s|h|^2 + s^2|\mathbf{f}|^2 + |\nabla \mathbf{f}|^2) e^{2s\varphi} dxdt \\ &\quad + e^{Ds} (\|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2) \\ &\quad + Cs^2 e^{2d_0s} (\|\mathbf{v}\|_{H^1(0,T;H^2(\Omega))}^2 + \|y\|_{L^2(Q)}^2), \end{aligned} $$

for any $s \ge s_*$ and any $(\tilde{\mathbf{v}}, \tilde{y}) \in H^2(Q) \times H^{2,1}(Q)$ satisfying (30).

**Proof** Applying Carleman estimate (24) to $(\tilde{\mathbf{v}}, \tilde{y})$, we obtain
$$
\begin{aligned}
CN_{s,\varphi}(\tilde{\mathbf{v}}, \tilde{y}) \le \int_Q & (s|h|^2 + s^2|\mathbf{f}|^2 + |\nabla \mathbf{f}|^2) e^{2s\varphi} dxdt \\
& + e^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 \right) \\
& + \int_Q (s^2(|\eta_{tt}|^2 + |\eta_t|^2)(|y|^2 + |\mathbf{v}|^2 + |\mathbf{v}_t|^2 + |\nabla \mathbf{v}|^2 + |\nabla \mathbf{v}_t|^2 + |\nabla (\mathrm{div} \, \mathbf{v})|^2) e^{2s\varphi} dxdt,
\end{aligned}
$$

for any $\gamma \ge \gamma_*$ and $s \ge s_*$. Since $\mathrm{Supp}(\eta_{tt})$, $\mathrm{Supp}(\eta_t) \subset (0, 2\epsilon) \cup (T - 2\epsilon, T)$, we obtain from (32)

$$
\int_Q (s^2(|\eta_{tt}|^2 + |\eta_t|^2)(|y|^2 + |\mathbf{v}|^2 + |\mathbf{v}_t|^2 + |\nabla \mathbf{v}|^2 + |\nabla \mathbf{v}_t|^2 + |\nabla (\mathrm{div} \, \mathbf{v})|^2) e^{2s\varphi} dxdt \\
\le C s^2 e^{2d_0 s} \left( \|\mathbf{v}\|_{H^1(0,T;H^2(\Omega))}^2 + \|y\|_{L^2(Q)}^2 \right).
$$

This completes the proof of the lemma.

**3.2. Completion of the proof of the main result**

Consider now the following system

$$
\begin{align}
\mathbf{u}_{tt} - \Delta_{\mu,\lambda}\mathbf{u} - \nabla(\lambda^*\mathrm{div}\mathbf{u}_t) + \varrho_1\nabla\theta &= 0 \quad (x,t) \in Q, \nonumber \\
\theta_t - \Delta\theta + \varrho_2\mathrm{div}\mathbf{u}_t &= g \quad (x,t) \in Q, \tag{33} \\
\mathbf{u} &= 0, \quad \theta = 0 \quad (x,t) \in \Sigma, \nonumber
\end{align}
$$

where the heat source term $g$ is given by

$$
g(x, t) = q(x)k(x, t).
$$

Let $\mathbf{v} = u_t$ and $y = \theta_t$. Then, we have

$$
\begin{align}
\mathbf{v}_{tt} - \Delta_{\mu,\lambda} \mathbf{v} - \nabla (\lambda^* \mathrm{div} \, \mathbf{v}_t) + \varrho_1 \nabla y &= 0 \quad (x,t) \in Q, \nonumber \\
y_t - \Delta y + \varrho_2 \mathrm{div} \, \mathbf{v}_t &= g_t \quad (x,t) \in Q, \tag{34} \\
\mathbf{v} &= 0, \quad y = 0 \quad (x,t) \in \Sigma. \nonumber
\end{align}
$$

We apply Lemma 3.3 to $(\tilde{\mathbf{v}}, \tilde{y})$ solution of the following system

$$
\begin{align}
\tilde{\mathbf{v}}_{tt} - \Delta_{\mu,\lambda} \tilde{\mathbf{v}} - \nabla(\lambda^* \mathrm{div} \tilde{\mathbf{v}}_t) + \varrho_1 \nabla \tilde{y} &= \eta_{tt} \mathbf{v} + 2\eta_t \mathbf{v}_t - \eta_t \nabla(\lambda^* \mathrm{div} \mathbf{v}) \quad (x,t) \in Q, \nonumber \\
\tilde{y}_t - \Delta \tilde{y} + \varrho_2 \mathrm{div} \tilde{\mathbf{v}}_t &= \eta g_t + \eta_t (y - \varrho_2 \mathrm{div} \mathbf{v}) \quad (x,t) \in Q, \nonumber \\
\tilde{\mathbf{v}} = 0, \quad \tilde{y} = 0 \quad (x,t) \in \Sigma, \tag{35}
\end{align}
$$

we obtain the following estimate :

$$
\begin{align}
CN_{s,\varphi}(\tilde{\mathbf{v}}, \tilde{y}) \le \int_Q s |g_t|^2 e^{2s\varphi} dxdt & \nonumber \\
& + e^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 \right) \nonumber \\
& + Cs^2 e^{2d_0 s} \left( \|\mathbf{v}\|_{H^1(0,T;H^2(\Omega))}^2 + \|y\|_{L^2(Q)}^2 \right), \tag{36}
\end{align}
$$

for sufficiently large $s > 0$.

We have the following Lemma
**Lemma 3.4** There exists a positive constant $C > 0$ such that the following estimate

$$ \int_{\Omega} |z(x, t_0)|^2 dx \leq C \int_{Q} (s|z(x, t)|^2 + s^{-1}|z_t(x, t)|^2) dxdt, $$

for any $z \in H^1(0, T; L^2(\Omega))$.

**Proof** By direct computations, we have

$$ \begin{aligned} \int_{\Omega} \eta^2(t_0) |z(x, t_0)|^2 dx &= \int_{0}^{t_0} \frac{d}{dt} \left( \int_{\Omega} \eta^2(t) |z(x, t)|^2 dx \right) dt \\ &= 2 \int_{0}^{t_0} \int_{\Omega} \eta^2(t) z(x, t) z_t(x, t) dxdt + 2 \int_{0}^{t_0} \int_{\Omega} \eta_t(t) \eta(t) |z(x, t)|^2 dxdt. \end{aligned} $$

Then, we have

$$ \int_{\Omega} |z(x, t_0)|^2 dx \leq C \int_{Q} (s|z(x, t)|^2 + s^{-1}|z_t(x, t)|^2) dxdt. $$

This complete the proof of the lemma.

Second, we apply lemma 3.4 to $y_1(x, t) = \eta(t)e^{2s\varphi(x,t)}y(x, t) = e^{2s\varphi(x,t)}\tilde{y}(x, t)$ and by the second equation of (35), we obtain

$$ \begin{aligned} \int_{\Omega} e^{2s\rho(x)} |y(x, t_0)|^2 dx &\leq \int_{Q} \sigma e^{2s\varphi} |\tilde{y}(x, t)|^2 dxdt + \int_{Q} \sigma^{-1} e^{2s\varphi} |\tilde{y}_t(x, t)|^2 dxdt \\ &\leq \int_{Q} \sigma e^{2s\varphi} |\tilde{y}(x, t)|^2 dxdt + \int_{Q} \sigma^{-1} e^{2s\varphi} |g_t(x, t)|^2 dxdt \\ &\quad + \int_{Q} \sigma^{-1} e^{2s\varphi} |\Delta\tilde{y}(x, t)|^2 dxdt + \int_{Q} \sigma^{-1} e^{2s\varphi} |\mathrm{div}\tilde{\mathbf{v}}_t|^2 dxdt \\ &\quad + \int_{Q} \sigma^{-1} e^{2s\varphi} |\eta_t|^2 (|y|^2 + |\mathrm{div}\mathbf{v}|^2) dxdt \\ &\leq \int_{Q} \sigma e^{2s\varphi} |\tilde{y}(x, t)|^2 dxdt + \int_{Q} \sigma^{-1} e^{2s\varphi} |g_t(x, t)|^2 dxdt \\ &\quad + \int_{Q} \sigma^{-1} e^{2s\varphi} |\Delta\tilde{y}(x, t)|^2 dxdt + \int_{Q} \sigma^{-1} e^{2s\varphi} |\mathrm{div}\tilde{\mathbf{v}}_t|^2 dxdt \\ &\quad + e^{2sd_0} (\|\mathbf{v}\|_{H^1(Q)}^2 + \|y\|_{L^2(Q)}^2). \end{aligned} \quad (37) $$

Then,

$$ \begin{aligned} &Cs \int_{\Omega} e^{s\rho(x)} |y(x, t_0)|^2 dx \leq \int_{Q} e^{2s\varphi} |g_t(x, t)|^2 dxdt \\ &\quad + \int_{Q} e^{2s\varphi} (|\Delta\tilde{y}|^2 + \sigma^2 |\tilde{y}|^2 + |\mathrm{div}\tilde{\mathbf{v}}_t|^2) dxdt + s e^{2sd_0} (\|\mathbf{v}\|_{H^1(Q)}^2 + \|y\|_{L^2(Q)}^2) \\ &\leq \int_{Q} e^{2s\varphi} |g_t(x, t)|^2 dxdt + CN_{s,\varphi}(\tilde{\mathbf{v}}, \tilde{y}) + se^{2sd_0} (\|\mathbf{v}\|_{H^1(Q)}^2 + \|y\|_{L^2(Q)}^2). \quad (38) \end{aligned} $$

Moreover, using (36), we get

$$ s \int_{\Omega} e^{2s\rho(x)} |y(x, t_0)|^2 dx \leq C s \int_{Q} e^{2s\varphi} |g_t(x, t)|^2 dxdt $$
$$ +e^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 \right) \\ +Cs^2 e^{2sd_0} \left( \|\mathbf{v}\|_{H^1(0,T;H^2(\Omega))}^2 + \|y\|_{L^2(Q)}^2 \right). \quad (39) $$

On the one hand, by the second equation in (33), we have

$$ y(x, t_0) = \Delta\theta(x, t_0) - \rho_2 \mathrm{div}\mathbf{v}(x, t_0) + q(x)k(x, t_0). \quad (40) $$

Moreover, we have

$$ |q(x)| \le |q(x)| |k(x, t_0)| \le |y(x, t_0)| + |\Delta\theta(x, t_0)| + |\mathrm{div}\mathbf{v}(x, t_0)|. \quad (41) $$

Then,

$$ \int_{\Omega} e^{2s\rho(x)} |q(x)|^2 dx \le \int_{\Omega} e^{2s\rho(x)} |y(x, t_0)|^2 dx \\ + \int_{\Omega} e^{2s\rho(x)} |\mathrm{div}\mathbf{v}(x, t_0)|^2 dx + \int_{\Omega} e^{2s\rho(x)} |\Delta\theta(x, t_0)|^2 dx. \quad (42) $$

We deduce that

$$ s \int_{\Omega} e^{2s\rho(x)} |q(x)|^2 dx \le s \int_{\Omega} e^{2s\rho(x)} |y(x, t_0)|^2 dx \\ +s \int_{\Omega} e^{2s\rho(x)} |\mathrm{div}\mathbf{v}(x, t_0)|^2 dx + Ce^{sD} \|\Delta\theta(x, t_0)\|_{L^2(\Omega)}^2. \quad (43) $$

Inserting (39) into (43), we get

$$ s \int_{\Omega} e^{2s\rho(x)} |q(x)|^2 dx \le Cs \int_{Q} e^{2s\varphi} |g_t(x, t)|^2 dxdt + Cs^2 e^{2sd_0} M_0^2 \\ +Ce^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 \right. \\ \left. + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 + \|\Delta\theta(x, t_0)\|_{L^2(\Omega)}^2 \right). \quad (44) $$

Selecting $\kappa \in (d_0, d_1)$ such that $Cs^2e^{2sd_0} \le e^{2\kappa s}$ for any $s$ large, we get

$$ s \int_{\Omega} e^{2s\rho(x)} |q(x)|^2 dx \le Cs \int_{Q} e^{2s\varphi} |g_t(x, t)|^2 dxdt + e^{2\kappa s} M_0^2 \\ +Ce^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(\cdot, t_0)\|_{H^3(\Omega)}^2 \right. \\ \left. + \|\mathbf{v}_t(\cdot, t_0)\|_{H^1(\Omega)}^2 + \|\theta(\cdot, t_0)\|_{H^2(\Omega)}^2 \right). \quad (45) $$

Now we return to the first integral term in $Q$ on the right-hand side term of (45).

$$ \int_{Q} e^{2s\varphi} |q(x)k_t|^2 dxdt \le \int_{\Omega} e^{2s\rho(x)} |q(x)|^2 \left( \int_{0}^{T} e^{-2s(\rho-\varphi)} \|k_t(\cdot, t)\|_{L^{\infty}(\Omega)}^2 dt \right) dx. \quad (46) $$

By the Lebesgue theorem, we obtain

$$ \int_{0}^{T} e^{-2s(\rho-\varphi)} \|k_t(\cdot, t)\|_{L^{\infty}(\Omega)}^2 dt = \int_{0}^{T} e^{-2s\rho(x)(1-\alpha(t))} \|k_t(\cdot, t)\|_{L^{\infty}(\Omega)}^2 dt $$
$$ \leq \int_{0}^{T} e^{-2s(1-\alpha(t))} \|k_{t}(., t)\|_{L^{\infty}}^{2} dt = o(1), \quad (47) $$

as $s \to \infty$. By (45), we have

$$ \begin{aligned} s \|e^{s\rho}q\|_{L^2(\Omega)}^2 &\le o(1) \int_Q se^{2s\rho} |q(x)|^2 dxdt + e^{2\kappa s} M_0^2 \\ &\quad + Ce^{Ds} \left( \|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(., t_0)\|_{H^3(\Omega)}^2 \right. \\ &\quad \left. + \|\mathbf{v}_t(., t_0)\|_{H^1(\Omega)}^2 + \|\theta(., t_0)\|_{H^2(\Omega)}^2 \right). \end{aligned} \quad (48) $$

for all $s \ge s_*$.

On the other hand, using the fact that $\rho(x) \ge d_1$ for all $x \in \Omega$. Then for sufficiently large $s > 0$, we have

$$ \begin{aligned} \|q\|_{L^2(\Omega)}^2 &\le Ce^{\gamma_1 s} (\|\mathbf{v}\|_{H^4(\omega \times (0,T))}^2 + \|\mathbf{v}(., t_0)\|_{H^3(\Omega)}^2 \\ &\quad + \|\mathbf{v}_t(., t_0)\|_{H^1(\Omega)}^2 + \|\theta(., t_0)\|_{H^2(\Omega)}^2) + Ce^{-\gamma_2 s} M_0^2. \end{aligned} \quad (49) $$

Finally, minimizing the right hand side with respect to $s$, we obtain

$$ \|q\|_{L^2(\Omega)} \le C (\|\mathbf{v}\|_{H^4(\omega \times (0,T))} + \|\theta(., t_0)\|_{H^2(\Omega)} + \|\mathbf{v}(., t_0)\|_{H^3(\Omega)} + \|\mathbf{v}_t(., t_0)\|_{H^1(\Omega)})^{\delta}. $$

Thus proof of Theorem 1.1 is completed.

# 4. Bibliographie

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[2] M. BELLASSOURED, M. YAMAMOTO, « Carleman estimates and inverse heat source problem for thermoelasticity system », *Inverse problems*, vol. 27, 2011.

[3] M. BELLASSOURED, B. RIAHI « Carleman estimate for Biot consolidation system in poro-elasticity and application to inverse problem », *Mathematical Method in the Applied Sciences*, 2016.

[4] B. WU, J. LIU, « Conditional stability and uniqueness for determining two coefficients in a hyperbolic-parabolic system », *Journal*, vol. 27, 2011.

[5] J.-L. LIONS, E. MAGENES, « Non-homogenous boundary value problems and applications », Springer-Verlag, vol. I and II, 1972.