  We consider time-independent solutions of hyperbolic equations such as
$\d_{tt}u -\Delta u= f(x,u)$ 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 $\d_t u -\Delta u= f(x,u)$. Then we
treat several examples under very sharp conditions, including equations with
potential terms and equations with supercritical nonlinearities.
