  Bistability is a key property of many systems arising in the nonlinear
sciences. For example, it appears in many partial differential equations
(PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even
has take on different names within communities such as Allee, Allen-Cahn,
Chafee-Infante, Nagumo, Ginzburg-Landau, $\Phi_4$, Schl\"ogl, Stommel, just to
name a few structurally similar bistable model names. One key mechanism, how
bistability arises under parameter variation is a pitchfork bifurcation. In
particular, taking the pitchfork bifurcation normal form for reaction-diffusion
PDEs is yet another variant within the family of PDEs mentioned above. More
generally, the study of this PDE class considering steady states and stability,
related to bifurcations due to a parameter is well-understood for the
deterministic case. For the stochastic PDE (SPDE) case, the situation is less
well-understood and has been studied recently. In this paper we generalize and
unify several recent results for SPDE bifurcations. Our generalisation is
motivated directly by applications as we introduce in the equation a spatially
heterogeneous term and relax the assumptions on the covariance operator that
defines the noise. For this spatially heterogeneous SPDE, we prove a
finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the
theory of early warning signs in our context and we explain, the role of
universal exponents between covariance operator warning signs and the lack of
finite-time Lyapunov uniformity. Our results are accompanied and
cross-validated by numerical simulations.
