Instability of steady states for nonlinear wave and heat equations

Abstract

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.