  We introduce an asymmetric classical Ginzburg-Landau model in a bounded
interval, and study its dynamical behavior when perturbed by weak
spatiotemporal noise. The Kramers escape rate from a locally stable state is
computed as a function of the interval length. An asymptotically sharp
second-order phase transition in activation behavior, with corresponding
critical behavior of the rate prefactor, occurs at a critical length l_c,
similar to what is observed in symmetric models. The weak-noise exit time
asymptotics, to both leading and subdominant orders, are analyzed at all
interval lengthscales. The divergence of the prefactor as the critical length
is approached is discussed in terms of a crossover from non-Arrhenius to
Arrhenius behavior as noise intensity decreases. More general models without
symmetry are observed to display similar behavior, suggesting that the presence
of a ``phase transition'' in escape behavior is a robust and widespread
phenomenon.
