[PAST EVENT] Mathematics Colloquium: Xuefeng Wang, Tulane University

April 18, 2014
2pm - 3pm
Jones Hall, Room 301
200 Ukrop Way
Williamsburg, VA 23185Map this location
Abstract: Chemotaxis is the biased motion of cells under the influence of chemicals that attract the cells. The most important phenomenon about chemotaxis is cell aggregation. The mathematical model for this phenomenon, proposed by Keller and Segel 40 years ago, is a quasilinear parabolic system coupled at the leading order terms (which makes it difficult to analyze).

In this talk, I will derive the chemotaxis system, first from the macroscopical perspective, and then from the microscopical perspective (in the same fashion that the heat equation is derived via random walk). Then I will review the two approaches to model cell aggregation: blowing up of time-dependent solutions and steady states with striking features such as spikes and transition-layers (step function-like). Finally, in the case of 1D spatial domains, we present two methods to establish the existence of steady states with striking features: (i) global bifurcation theory combined with Helly's compactness theorem and Sturm oscillation theorem; (ii) singular perturbation method. We also prove local asymptotic stability and uniqueness of these steady states.

[[jxshix, Junping Shi]]