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 general. Additionally, we also
apply the aforementioned 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.