In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ ( x , t ) = e λ ⁢ ψ ⁢ ( x , t ) , ψ ⁢ ( x , t ) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 \varphi(x,t)={\mathrm{e}}^{\lambda\psi(x,t)},\quad\psi(x,t)=\lvert x-x_{0}% \rvert^{2}-\beta\lvert t-t_{0}\rvert^{2}+\beta t_{0}^{2} for x in a bounded domain in ℝ n {\mathbb{R}^{n}} with C 3 {C^{3}} -boundary and t ∈ ( 0 , T ) {t\in(0,T)} , where t 0 = T 2 {t_{0}=\frac{T}{2}} . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385].

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This paper is concerned with the determination of coefficients and source term in a strong coupled quantitative thermoacoustic system of equations. Adapting a Carleman estimate established in the part I of this series of papers, we prove stability estimates of Hölder type involving the observation of only one component: the temperature or the pressure.

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We consider a Cahn–Hilliard equation in a bounded domain Ω in {\mathbb{R}^{n}} over a time interval {(0,T)} and discuss the backward problem in time of determining intermediate data {u(x,\theta)}, {\theta\in(0,T)}, {x\in\Omega} from the measurement of the final data {u(x,T)}, {x\in\Omega}. Under suitable a priori boundness assumptions on the solutions {u(x,t)}, we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{L^{2}(\Omega)}\leq C\lVert u(\,\cdot\,,T)% \rVert_{H^{2}(\Omega)}^{\kappa_{0}},and a conditional stability estimate for the linear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{H^{\beta}(\Omega)}\leq C\lVert u(\,\cdot\,,T% )\rVert_{H^{2}(\Omega)}^{\kappa_{1}},where {\theta\in(0,T)}, {\beta\in(0,4)} and {\kappa_{0},\kappa_{1}\in(0,1)}. The proof is based on a Carleman estimate with the weight function {\mathrm{e}^{2s\mathrm{e}^{\lambda t}}} with large parameters {s,\lambda\in\mathbb{R}^{+}}.