We consider a reaction-diffusion equation of the type [ partial_tpsi = partial^2_xpsi + V(psi) + lambdasigma(psi)dot{W} qquadtext{on $(0,,infty)timesmathbb{T}$}, ] 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 $V:mathbb{R}tomathbb{R}$ 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 $sigma:mathbb{R}tomathbb{R}$ 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.
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