演讲人: Professor Xuefeng Wang, Center for PDE, East China

讲座时间: 2014年9月27日，3:00-4:00pm

讲座地点: 中国人民大学环境学院316房间

讲座内容:

Bifurcation theory is a great methodology for understanding the solution set of a PDE, and for revealing critical roles played by physical/biological parameters. Indeed, Crandall-Rabinowitz local bifurcation theorem and Rabinowitz global bifurcation theorem have become standard tools to achieve these goals. However,Rabinowitzglobal theoremrequires that the equation be written in the form of a compact perturbationof the identity; for complicated systems of PDEs (possibly with nonlinear boundary conditions) arising in applications, it is often cumbersome, if possible at all, to transform them into that form.

In this talk, I will introduce a relatively new global bifurcation theory for Fredholm operators, due toFitzpatric, Pejsachowicz anf Rabier,that generalizes Rabinowitz theorem. It does not require the conversion of the PDE to a compact perturbation of the identity; and as a special case of the theory,by adding a Fredholmness condition onthe nonlinear operator, the Crandall-Rabinowitz local bifurcation theorem becomesa global bifurcation theorem. I will alsointroduce the unilateral bifurcation theorem, and sufficient conditions for an elliptic system with general boundary conditions to be Fredholm with 0 index, due to Junping Shi and myself.

Finally, I will show how to apply the global bifurcation theory toa class of chemotaxis systems,obtaining steady states with striking features such as spikes and transition layers.