Limits of Semigroups Depending on Parameters

Abstract

It is reasonable to compare dissipative semigroups with a global attractor by restricting the flows to the attractor. However, if the rate of approach to the attractor is not uniform with respect to parameters, then the transient behavior near the attractor will give more information. We introduce a concept which takes into account this transient behavior. The concept also is useful when the limit system is conservative. We give the general theory with applications to parabolic and hyperbolic PDE on thin domains as well as situations where the limit problem is conservative.