  A nonlinear parabolic differential equation with a quadratic nonlinearity is
presented which has at least one equilibrium. The linearization about this
equilibrium is asymptotically stable, but by using a technique inspired by H.
Fujita, we show that the equilibrium is unstable in the nonlinear setting. The
perturbations used have the property that they are small in every $L^p$ norm,
yet they result in solutions which fail to be global.
