  For a given semilinear parabolic equation with polynomial nonlinearity, many
solutions blow up in finite time. For a certain large class of these equations,
we show that some of the solutions which do not blow up actually tend to
equilibria. The characterizing property of such solutions is a finite energy
constraint, which comes about from the fact that this class of equations can be
written as the $L^2$ gradient of a certain functional.
