  We consider a reaction-diffusion equation of the type \[
  \partial_t\psi = \partial^2_x\psi + V(\psi) + \lambda\sigma(\psi)\dot{W}
  \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] subject to a "nice"
initial value and periodic boundary, where $\mathbb{T}=[-1\,,1]$ and $\dot{W}$
denotes space-time white noise. The reaction term $V:\mathbb{R}\to\mathbb{R}$
belongs to a large family of functions that includes Fisher--KPP nonlinearities
[$V(x)=x(1-x)$] as well as Allen-Cahn potentials [$V(x)=x(1-x)(1+x)$], the
multiplicative nonlinearity $\sigma:\mathbb{R}\to\mathbb{R}$ is non random and
Lipschitz continuous, and $\lambda>0$ is a non-random number that measures the
strength of the effect of the noise $\dot{W}$.
  The principal finding of this paper is that: (i) When $\lambda$ is
sufficiently large, the above equation has a unique invariant measure; and (ii)
When $\lambda$ is sufficiently small, the collection of all invariant measures
is a non-trivial line segment, in particular infinite. This proves an earlier
prediction of Zimmerman et al. (2000). Our methods also say a great deal about
the structure of these invariant measures.
