Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models

TL;DR

This paper investigates the qualitative dynamics of Keller-Segel models with flux-saturated porous-media diffusion, focusing on support regularization, blow-up behavior, and traveling wave solutions.

Contribution

It provides new insights into the regularization, support dynamics, and traveling waves in flux-saturated Keller-Segel systems, highlighting the role of key system constants.

Findings

Support regularization inside the solution domain.

Conditions for finite-time blow-up and Dirac mass formation.

Characterization of traveling wave solutions and their dependence on system parameters.

Abstract

This paper deals with the analysis of qualitative properties involved in the dynamics of Keller-Segel type systems in which the diffusion mechanisms of the cells are driven by porous-media flux-saturated phenomena. We study the regularization inside the support of a solution with jump discontinuity at the boundary of the support. We analyze the behavior of the size of the support and blow--up of the solution, and the possible convergence in finite time towards a Dirac mass in terms of the three constants of the system: the mass, the flux--saturated characteristic speed, and the chemoattractant sensitivity constant. These constants of motion also characterize the dynamics of regular and singular traveling waves.