  In this work we determine the second-order coefficient in a parabolic
equation from the knowledge of a single final data. Under assumptions on the
concentration of eigenvalues of the associated elliptic operator, and the
initial state, we show the uniqueness of solution, and we derive a Lipschitz
stability estimate for the inversion when the final time is large enough. The
Lipschitz stability constant grows exponentially with respect to the final
time, which makes the inversion ill-posed. The proof of the stability estimate
is based on a spectral decomposition of the solution to the parabolic equation
in terms of the eigenfunctions of the associated elliptic operator, and an ad
hoc method to solve a nonlinear stationary transport equation that is itself of
interest.
