In this paper we give some sharp qualitative and quantitative
conditions to guarantee that the exponential type of a linear
nonautonomous equation is modified by a linear perturbation. No
assumption (periodic, almost periodic, quasi periodic etc) is made on
the time behavior of the coefficients of the
equation or the perturbation.
The results are applied to the study of the asymptotic behavior, both
forwards and pullback, of nonautonomous nonlinear equations.