Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations by boundary data. Part I: Carleman estimates

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 $\varphi(x,t)={\rm e}^{\lambda\psi(x,t)}$, $\psi(x,t)=\left|x-x_{0}\right|^{2}-\beta\left(t-t_0\right)^{2}+\beta t_0^{2}$ for $x$ in a bounded domain in $\mathbb{R}^{n}$ with $C^{3}$-boundary and $t\in(0, T)$, where $t_0=T/2$. We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding Part II paper \cite{part II}.