Speaker
Description
We begin with a brief overview of the rapidly developing research area of active matter, a.k.a. active materials. These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We present a free boundary PDE model a cytoskeleton of a moving cell. The key mathematical features of our model are the nonlocal boundary conditions, nonlinear diffusion, and the Keller-Segel cross-diffusion term. We present an overview of three recent works on that model. We begin from the 2D model with linear diffusion in which we derive an explicit formula for the stability determining eigenvalue for the linearized non-self-adjoint operator. Next, we present a recent result on the nonlinear stability of stationary and traveling wave solutions in 1D model. Here we focus on non-self-adjointness of the linearized problem, which plays a key role in the spectral stability analysis. Finally, we consider 2D model with nonlinear diffusion and prove this nonlinearity results in the change of the bifurcation from supercritical to subcritical, leading to two drastically different scenarios of the onset of the cell motion. Here we derive an explicit formula that governs the change of the bifurcation type in terms of measurable physical parameters and therefore can be used for both qualitative and quantitative biological predictions. Finally, we discuss how our results lead to an open question of bistability.