Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion

Abstract

We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

“…However, there are no general results in the literature for the nonlinear diffusion case ( 1.1 ), , except for the particular case of , , with W given by the fundamental solution of the Laplacian with no flux boundary conditions (the Newtonian interaction) recently studied in [ CCW+20 ]. Despite the simplicity of the setting in [ CCW+20 ], this example revealed how complicated phase transitions for nonlinear diffusion cases could be. The authors showed that infinitely many discontinuous phase transitions occur for that particular problem.…”

Asymptotic stability of diffusion waves of a quasi-linear hyperbolic-parabolic model for vasculogenesis

“…Thus the stationary solution satisfies d dt F [ρ, u, φ] = 0 which gives rise to ρu = 0 and φ t = 0 in R 3 . Since we are interested in non-constant profile for ρ, u ≡ 0 is the only (physical) stationary profile for the velocity u.…”

Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis

“…A typical form of p fulfilling (1.2) is p(ρ) = K 2 ρ 2 with K > aµ b . The stationary solutions of one dimensional (1.1) with vacuum (bump solutions) in a bounded interval with zero-flux boundary condition were constructed in [2,3,34]. The model (1.1) with p(ρ) = ρ and periodic boundary conditions in one dimension was numerically explored in [11].…”