Pankavich, Stephen; Radu, Petronela Nonlinear instability of solutions in parabolic and hyperbolic diffusion. (English) Zbl 1272.35029 Evol. Equ. Control Theory 2, No. 2, 403-422 (2013). Summary: 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. Cited in 2 Documents MSC: 35B35 Stability in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35L71 Second-order semilinear hyperbolic equations 35K58 Semilinear parabolic equations Keywords:sign-changing damping; variable coefficients; steady states; semilinear evolution equations; supercritical and exponential nonlinearities PDFBibTeX XMLCite \textit{S. Pankavich} and \textit{P. Radu}, Evol. Equ. Control Theory 2, No. 2, 403--422 (2013; Zbl 1272.35029) Full Text: DOI arXiv