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

Logarithmic stability in determining the time-dependent zero order  coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map.  Application to the determination of a nonlinear term