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
$\{ \partial_{\nu}y(h_{\ell})\vert_{(0,T)\times \Gamma_0}$
,
$y(h_{\ell})(\theta,\cdot)\}_{1\le \ell\le \ell_0}$
after selecting input sources $h_1, ...,
h_{\ell_0}$
suitably, where $\Gamma_0$
is an arbitrary subboundary,
$\partial_{\nu}$
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.