Superstability and Finite Time Extinction For C_0-Semigroups

A new approach to superstability and finite time extinction of strongly continuous semigroups is presented, unifying known results and providing new criteria for these conditions to hold analogous to the well-known Pazy condition for stability. That finite time extinction implies superstability which is in turn equivalent to several (both known and new) conditions follow from this new approach in a consistent fashion. Examples showing that the converse statements fail are constructed, in particular, an answer to a question of Balakrishnan on superstable systems not exhibiting finite time extinction.