$ u_t = \Delta u + f(u),    t>0,  x  \in \Omega.    $

We show that there is a nonempty set G of functions $f$, that is open
in  a $C^1$  topology and  such that  for any  $f\in G$  the following
situation  occurs. The  above equation  has two  hyperbolic equilibria
$\phi$,  $\psi$   such  that  their  stable   and  unstable  manifolds
$W^s(\psi)$ and $W^u(\varphi)$ intersect nontransversally.  This shows
that  the transversality  of stable  and unstable  manifolds is  not a
generic property in the  class of spatially homogeneous equations.  In
contrast,  this property  is  generic if  nonlinearities  of the  form
$f=f(x,u)$ are considered (this  was shown by Brunovsk\'y and Pol\'a\v
cik).  In  one space dimension, the transversality  occurs always (not
only  generically),  as  shown  in  independent  works  of  Henry  and
Angenent.