Solution
= Solution
For spatially homogeneous solutions, the <bistable cubic reaction-diffusion equation> reduces to $\dot u=f(u)=u(1-u)(u-r)$. Its equilibria are $0,r,1$, and
$$
\boxed{f'(0)=-r<0,\qquad f'(r)=r(1-r)>0,\qquad f'(1)=-(1-r)<0.}
$$
Thus $0$ and $1$ are asymptotically stable, while $r$ is unstable. Small spatial Fourier perturbations have growth rates $f'(u_*)-mk^2$, so diffusion also leaves the two outer uniform states linearly stable.