  We give a new stability estimate for the problem of determining the
time-dependent zero order coefficient in a parabolic equation from a partial
parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary
to the previous works, we do not need any measurement on the final time. We
also show how this result can be used to establish a stability estimate for the
problem of determining the nonlinear term in a semilinear parabolic equation
from the corresponding "linearized" Dirichlet-to-Neumann map. The key
ingredient in our analysis is a parabolic version of an elliptic Carleman
inequality due to Bukhgeim and Uhlmann. This parabolic Carleman inequality
enters in an essential way in the construction of complex geometric optics
solutions that vanish at a part of the lateral boundary.
