On Keller-Segel chemotaxis models with degenerate diffusion

Chemotaxis is the mechanism by which unicellular or multicellular organisms direct their movements in response to a stimulating chemical in the environment.   Bacterial chemotaxis was discovered by T. W.  Engelmann and W. Pfeffer in 1880s, and over one century's research has illustrated its importance in many physiological processes.  In the 1970s, E. Keller and L. Segel proposed a system of two coupled partial differential equations to describe the traveling bands of \textit{E. coli} in a capillary tube filled with oxygen.  The intuitively simple Keller-Segel model possesses very rich spatial-temporal dynamics (e.g. spike steady states, finite-time blow-up, traveling wave) and has achieved great academic success over the past few decades.  

An important extension of the classical Keller-Segel model is to include a density-dependent diffusion function, assuming that the random cell motility is anti-crowding and decreases as the population thins.  In this talk, we consider a Keller-Segel model with porous medium diffusion and study the effects of chemotaxis on the existence and stability of nonconstant and particularly compactly-supported steady states.  A novelty in the quadratic diffusion structure is utilized to obtain the explicit formulas of all steady states in a one-dimensional interval.  We also give a complete hierarchy of their energies on the bifurcation diagram, and with that being said, the half-bump has the least energy while the constant steady state has the largest energy.  In the large limit of chemotaxis rate, the cell population density converges to a single or several Dirac-delta functions, which can be adopted to model the cell aggregation phenomenon.  There will be numerical simulations to demonstrate the theoretical results and other interesting phenomena such as phase transition, focusing and defocusing process within this model.  This talk is open to graduate students and some of the approaches can be understood by senior undergraduate students.