On nonexistence of maximal asymptotics of C_0 semigroups

We develop the approach proposed in 1982 by the author and V.Ya. Shirman for analysis of asymptotic stability of a linear differential equation in Banach space. We show that the method introduced in the mentioned above work allows also to prove the nonexistence of the fastest growing solution for a wide class of linear equations, or in other words, nonexistence of maximal asymptotics of the corresponding C_0 semigroup.