Asymptotics of Operator Semigroups

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.

The planned research may have direct impacts in operator theory, complex analysis, partial differential equations, control theory, dynamical systems and differential geometry, which may benefit researchers in those areas. The applications to partial differential equations and control theory may provide indirect benefit to physicists, biologists and engineers. One of the most important results on stability semigroups (the ABLV theorem) has been applied in numerous such ways. Moreover results on rates of decay (of energy of waves, for example) have clear relevance to many physical problems. It is not possible to say what impacts the research may have on society in general as any such effects are unlikely to emerge for many years.

The PI, VR and RA will each alert mathematicians to this work by circulating and posting preprints, and by giving talks in seminars and conferences in the UK and around the world. The PI and VR have already given several such talks on the research on stabilty and quasi-hyperbolicity that has already been carried out. We have organised workshops on Asymptotics of Operator Semigroups in Oxford in 2009 and Luminy, France in 2011. We expect to have opportunities to explain our research to specialists in evolution equations and partial differential equations, and to discuss potential applications, at similar events during and after the work on the project.

The PI and the VR have worked together intermittently on various occasions in the last 10 years, and both have many contacts in operator theory already. The PI has already collaborated with a specialist in wave equations and PDEs (T. Duyckaerts) in work on energy decay, and we expect to expand our contacts to include others in that field. The VR has already collaborated with a specialist in complex analysis (A. Borichev). One part of the project related will require expertise in differential geometry, which we will initially seek in Oxford but the contact is likely to expand to
other geometers.