  This paper is devoted to studying the null and approximate controllability of
two linear coupled parabolic equations posed on a smooth domain of R^N (N>1)
with coupling terms of zero and first orders and one control localized in some
arbitrary nonempty open subset of the domain. We prove the null controllability
under a new sufficient condition and we also provide the first example of a not
approximately controllable system in the case where the support of one of the
nontrivial first order coupling terms intersects the control domain.
