  We consider a parabolic equation in a bounded domain $\OOO$ over a time
interval $(0,T)$ with the homogeneous Neumann boundary condition. We
arbitrarily choose a subboundary $\Gamma \subset \ppp\OOO$. Then, we discuss an
inverse problem of determining a zeroth-order spatially varying coefficient by
extra data of solution $u$: $u\vert_{\Gamma \times (0,T)}$ and $u(\cdot,t_0)$
in $\OOO$ with $t_0=0$ or $t=T$. First we establish a conditional Lipschitz
stability estimate as well as the uniqueness for the case $t_0=T.$ Second,
under additional condition for $\Gamma$, we prove the uniqueness for the case
$t_0=0$. The second result adjusts the uniqueness by M.V. Klibanov (Inverse
Problems {\bf 8} (1992) 575-596) to the inverse problem in a bounded domain
$\OOO$. We modify his method which reduces the inverse parabolic problem to an
inverse hyperbolic problem, and so even for the inverse parabolic problem, we
have to assume conditions for the uniqueness for the corresponding inverse
hyperbolic problem. Moreover we prove the uniqueness for some inverse source
problem for a parabolic equation for $t_0=0$ without boundary condition on the
whole $\ppp\OOO$.
