| Abstract
| - Let y(h)(t,x) be one solution to \[ \partial_t y(t,x) - \sum_{i, j=1}^{n}\partial_{j} (a_{ij}(x)\partial_i y(t,x)) = h(t,x), \thinspace 0<t<T, \thinspace x\in \Omega \] with a non-homogeneous term h, and $y\vert_{(0,T)\times\partial\Omega} = 0$, where $\Omega \subset \Bbb R^n$ is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by $\{ \partial_{u}y(h_{ell})\vert_{(0,T)\times \Gamma_0}$, $y(h_{ell})(\theta,\cdot)\}_{1\le ell\le ell_0}$ after selecting input sources $h_1, ..., h_{ell_0}$ suitably, where $\Gamma_0$ is an arbitrary subboundary, $\partial_{u}$ denotes the normal derivative, $0 < \theta < T$ and $ell_0 \in \Bbb N$. In the case of $ell_0 = (n+1)^2n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1, ..., h_{ell_0})$ from a set ${\cal H} \subset \{ C_0^{\infty} ((0,T)\times omega)\}^{ell_0}$ 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 based on a Carleman estimate.
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