Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III
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June 2013, 2(2): 403-422. doi: 10.3934/eect.2013.2.403
Nonlinear instability of solutions in parabolic and hyperbolic diffusion
Stephen Pankavich 1, and Petronela Radu 2,
Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, United States
Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588
Received November 2012 Revised December 2012 Published March 2013
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.
Keywords: steady states., variable coefficients, instability, Evolution equations, sign-changing damping.
Mathematics Subject Classification: Primary: 35B35, 35B05, 35B30; Secondary: 35L70, 35K5.
Citation: Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403
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Stephen Pankavich Petronela Radu