Individual asymptotics of \(C_0\)-semigroups: Lower bounds and small solutions (Q1381913)

The author studies small solutions of the abstract Cauchy problem \[ \dot x(t)=Tx(t),\quad x(0)=x_0 \] for a \(C_0\)-semigroup generator \(T\). Here a ``small solution'' means the solution which tends to zero faster than any exponential. For a general \(T\) the author reveals a relation between the decay rate of a small solution \(e^{Tt}x_0\) and the growth of the local resolvent. He shows that, generally speaking, the faster \(\| e^{tT}x\|\) tends to \(0\), the slower the local resolvent \((z-T)^{-1}x_0\) grows as \(\text{Re } z\to\infty\). In a special case when \(T\) is the generator of a holomorphic semigroup of angle \(\pi/2\) the author provides a sufficient condition for an exponential lower bound of solution's decay, i.e. for absence of small solutions. On the other hand, the author gives an example of a holomorphic semigroup of angle \(\pi/2\) which has small solutions. The paper also contains some applications of the results obtained to differential operators.