Asymptotics of Operator Semigroups

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Many physical, biological and economic systems which are evolving in time can be modelled by differential equations whose solutions can be studied via operator semigroups. Typical examples include models of diffusion of heat in a body, wave motions of many kinds and populations of biological systems.Such equations usually cannot be solved explicitly. Nevertheless one can often find some information about the solutions, such as their long-time behaviour. For example, one may be able to show that the quantity under examination decays (or grows, or oscillates) by analysis of the equations, and to estimate the rate of decay (or growth). In recent years there has been much research into the rates of decay of the energy of damped waves, and operator semigroups have been used to give precise estimates in that case.This research will build on the previous work in this area. It is hoped to be able to provide general results determining more precisely when decay occurs and to improve the estimates for the rates of decay when quantities other than energy are being examined. Moreover we hope to gain a better understanding of mixed systems where the question whether the solutions decay or grow depends on the initial data.