  In this paper, we present a new Carleman estimate for the adjoint equations
associated to a class of super strong degenerate parabolic linear problems. Our
approach considers a standard geometric imposition on the control domain, which
can not be removed in the general situations. Additionally, we also apply the
aformentioned main inequality in order to investigate the null controllability
of two nonlinear parabolic systems. The first application is concerned a global
null controllability result obtained for some semilinear equations, relying on
a fixed point argument. In the second one, a local null controllability for
some equations with nonlocal terms is also achieved, by using an inverse
function theorem.
