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We propose a new approach to the study of (nonlinear) growth and instabilityfor semilinear evolution equations with compact nonlinearities. We show, inparticular, that compact nonlinear perturbations of a linear evolution equationcan be treated as linear ones as far as the growth of their solutions isconcerned. We obtain exponential lower bounds of solutions for initial valuesfrom a dense set if, e.g., the resolvent of the generator is unbounded on avertical line in the right halfplane.

On growth and instability for semilinear evolution equations: an abstract approach