Let $y\left(h\right)(t,x)$ be one solution to$${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0\<t\<T,x\in \Omega$$with a non-homogeneous term $h$, and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining $n(n+1)/2$ unknown functions $a_{ij}$ by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0\<\theta \<T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making special choices for $h_{\ell }$, $1\le \ell \le {\ell }_0$. The proof is...

Let y(h)(t,x) be one solution to $${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0<t<T,x\in \Omega$$ with a non-homogeneous term h, and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0<\theta <T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making special choices for...