  The approach to Lipschitz stability for uniformly parabolic equations
introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates,
seems hard to apply to the case of Grushin-type operators of interest to this
paper. Indeed, such estimates are still missing for parabolic operators
degenerating in the interior of the space domain. Nevertheless, we are able to
prove Lipschitz stability results for inverse source problems for such
operators, with locally distributed measurements in arbitrary space dimension.
For this purpose, we follow a mixed strategy which combines the appraoch due to
Lebeau and Robbiano, relying on Fourier decomposition, with Carleman
inequalities for heat equations with nonsmooth coefficients (solved by the
Fourier modes). As a corollary, we obtain a direct proof of the observability
of multidimensional Grushin-type parabolic equations, with locally distributed
observations, which is equivalent to null controllability with locally
distributed controls.
