Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Let be a generator of an exponentially stable operator semigroup in a Banach space, and let be a linear bounded variable operator. Assuming that is sufficiently small in a certain sense for the equation , we derive exponential stability conditions. Besides, we do not require that for each , the “frozen” autonomous equation is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations.