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Nonlinear instability of solutions in parabolic and hyperbolic diffusion. (English) Zbl 1272.35029

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.

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
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