Nonlinear instability of solutions in parabolic and hyperbolic diffusion

TL;DR

This paper establishes conditions under which linear instability in certain semilinear evolution equations leads to nonlinear instability, including exponential growth or finite-time blow-up, with applications to equations with supercritical nonlinearities.

Contribution

It provides a rigorous link between linear and nonlinear instability for a broad class of parabolic and hyperbolic equations with sign-changing damping.

Findings

Linear instability with a sign-definite eigenfunction implies nonlinear instability.

Conditions are derived for exponential growth or finite-time blow-up.

Applicable to equations with supercritical and exponential nonlinearities.

Abstract

We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign 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 evolution equations with supercritical and exponential nonlinearities.