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Instability of steady states for nonlinear wave and heat equations

Paschalis Karageorgis·Walter A. Strauss
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Abstract

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

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Paschalis Karageorgis, Walter A. Strauss. 2006-11-18. Instability of steady states for nonlinear wave and heat equations. https://arxiv.org/abs/math/0611559

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