  This paper is concerned with the spatio-temporal dynamics of nonnegative
bounded entire solutions of some reaction-diffusion equations in R N in any
space dimension N. The solutions are assumed to be localized in the past. Under
certain conditions on the reaction term, the solutions are then proved to be
time-independent or heteroclinic connections between different steady states.
Furthermore, either they are localized uniformly in time, or they converge to a
constant steady state and spread at large time. This result is then applied to
some specific bistable-type reactions.
