Instability of Steady States for nonlinear parabolic and damped hyperbolic equations

Details Instability of Steady States for nonlinear parabolic and damped hyperbolic equations Instability of Steady States for nonlinear parabolic and damped hyperbolic equations

2011-10-28

arxiv.org/abs/1110.6240v2

Mathematics

Analysis of PDEs

18 pages

Scientific paper

We consider steady solutions of semi-linear hyperbolic and parabolic equations of the form $\partial_{tt}u + a(t) \partial_t u + Lu = f(x, u)$ and $a(t) \partial_t u + Lu = f(x, u)$ with a damping coefficient $a(t)$ that is possibly sign-changing and determine precise conditions for which linear instability implies nonlinear instability. More specifically, we prove that linear instability with a positive eigenfunction gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including equations with supercritical and exponential nonlinearities.

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