Asymptotics of Operator Semigroups

许多随时间演化的物理、生物和经济系统可以用微分方程来建模，微分方程的解可以通过算子半群来研究。典型的例子包括热在物体中的扩散模型、多种波动模型和生物系统的种群模型。这样的方程通常不能显式求解。然而，人们通常可以找到一些关于解决方案的信息，例如它们的长期行为。例如，人们可以通过对方程的分析来表明所研究的量在衰减（或增长，或振荡），并估计衰减（或增长）的速率。近年来，人们对阻尼波的能量衰减率进行了大量研究，在这种情况下，算子半群被用来给出精确的估计。这项研究将以这一领域以前的工作为基础。人们希望能够提供更精确地确定衰变发生时间的一般结果，并在检查能量以外的量时改进对衰变速率的估计。此外，我们希望更好地理解混合系统，其中解是否衰减或增长的问题取决于初始数据。

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.