Instability of steady states for nonlinear wave and heat equations

Details Instability of steady states for nonlinear wave and heat equations Instability of steady states for nonlinear wave and heat equations

2006-11-18

arxiv.org/abs/math/0611559v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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