| Abstract : In this talk, we will discuss various procedures for inferring asymptotic decay of solutions to partial differential equations which are under appropriate dissipation. After initially recalling the classic treatments involving Lasalle Invariance principles and the Nagy-Foias decomposition, we will proceed to consider the abstract operator theoretic result of Arendt-Batty/Lyubich-Phong, which gives a spectral criterion for strong stability. Subsequently, we will discuss a recently developed approach for establishing strong decay, based upon a resolvent strong stability criterion of Boyadzhiev-Levan/Tomilov. The benefit of this new approach is that its implementation does not require an explicit resolvent representation of the underlying semigroup generator. All this machinery will be discussed in the context of several PDE examples. |