Instability of Steady States for Nonlinear Wave and Heat Equations

We consider time-independent solutions of hyperbolic equations such as ∂ttu − ∆u = f(x, u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as ∂tu −∆u = f(x, u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.