Speaker
Description
We consider the asymptotic problems of the Keller-Segel system with the nonlinear diffusion of porous medium type and logistic sensitivity. We classify three possible distinguished limits and reveal the relations between the Keller-Segel system and the limit systems. We prove that our model converges to three various limits as the parameters are chosen adequately: a porous medium equation when the chemotactic sensitivity is small and the chemical diffuses slowly; a hyperbolic Keller-Segel system when the cell diffusion vanishes; a surface-tension-driven free boundary problem when the chemical diffuses slowly and attracts the cells strongly. Unlike previous works, the strong convergence of the density is necessary due to the nonlinearity of the sensitivity. The mathematical methods employed vary for the three different cases, including the energy formulation, the entropy equality, the kinetic formulation, and $L^1$ compactness.
