A concentration $u(x, t)$ obeys the differential equation

$$\frac{\partial u}{\partial t}=D u_{x x}+f(u)$$

in the domain $0 \leqslant x \leqslant L$, with boundary conditions $u(0, t)=u(L, t)=0$ and initial condition $u(x, 0)=u_{0}(x)$, and where $D$ is a positive constant. Assume $f(0)=0$ and $f^{\prime}(0)>0$. Linearising the dynamics around $u=0$, and representing $u(x, t)$ as a suitable Fourier expansion, show that the condition for the linear stability of $u=0$ can be expressed as the following condition on the domain length

$$L<\pi\left[\frac{D}{f^{\prime}(0)}\right]^{1 / 2}$$