主 题: Global bifurcation theory for Fredholm operators & an application to chemotaxis systems
报告人: 王学锋教授 (Center for PDE, East China Normal University, Tulane University)
时 间: 2014-09-26 10:10-11:10
地 点: 理科一号楼1418S（主持人：蒋美跃）
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, Rabinowitz global theorem requires that the equation be written in the form of a compact perturbation of 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 to Fitzpatric, 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 on the nonlinear operator, the Crandall-Rabinowitz local bifurcation theorem becomes a global bifurcation theorem. I will also introduce the unilateral bifurcation theorem, and sufficient conditions for a 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 to a class of chemotaxis systems, obtaining steady states with striking features such as spikes and transition layers.
