We consider reaction-diffusion  equations  on a  bounded domain
under Dirichlet boundary   condition.   We discuss   two properties
of the associated  semiflows: convergence of  bounded  solutions to
a  singleequilibrium and transversality of intersections of stable and
unstable manifolds of hyperbolic equilibria.  While these properties
are always satisfied   in one space  dimension, in   higher dimension
there exist counterexamples. We show the latter using the method
of realization of ODEs.