Nonlinear instability of solutions in parabolic and hyperbolic diffusion

Stephen Pankavich; Petronela Radu

doi:10.3934/eect.2013.2.403

Article Contents

2013, Volume 2, Issue 2: 403-422. Doi: 10.3934/eect.2013.2.403

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Nonlinear instability of solutions in parabolic and hyperbolic diffusion

Stephen Pankavich^1, and
Petronela Radu^2,

1.
Department of Mathematics, United States Naval Academy, Annapolis, MD 21402
2.
Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588

Received: October 31, 2012

Revised: November 30, 2012

Published: May 2013

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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.

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Mathematics Subject Classification: Primary: 35B35, 35B05, 35B30; Secondary: 35L70, 35K55.

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