Straughan B., The Energy Method, Stability, and Nonlinear Convection (2004) claims that (p. 4)
[linear stability analysis] can only yield linear instability. It tells us nothing whatsoever about stability. There are many equations for which nonlinear solutions will become unstable well before the linear instability analysis predicts this.
Can you give the simplest possible example of a PDE whose solution exhibits the described behavior? That is, it becomes unstable before the linear stability threshold?
Consider for example the following PDE $$ \frac{\partial u}{\partial t} = \frac{1}{\lambda}\frac{\partial^2 u}{\partial x^2} + u - u^2, $$ $$ u(0) = u(1) = 0, $$ where $\lambda$ is a parameter. Then, $\widehat{u} \equiv 0$ is clearly a solution to the above system. Linear stability analysis easily yields (by neglecting $u^2$ and plugging $u(t,x)=U(x)e^{\sigma t}$ into the equation) an eigenvalue problem $$ \sigma U = \frac{1}{\lambda} \frac{\mathrm{d}^2 U}{\mathrm{d} x^2} + U, $$ which gives a stability threshold $\lambda_{\mathrm{crit}} = \pi^2$. We can thus be certain that for $\lambda > \pi^2$ the solution $\widehat{u}$ will be unstable. But is it stable for $\lambda \leq \pi^2$?