  We propose a new approach to the study of (nonlinear) growth and instability
for semilinear evolution equations with compact nonlinearities. We show, in
particular, that compact nonlinear perturbations of a linear evolution equation
can be treated as linear ones as far as the growth of their solutions is
concerned. We obtain exponential lower bounds of solutions for initial values
from a dense set if, e.g., the resolvent of the generator is unbounded on a
vertical line in the right halfplane.
