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This paper deals with the analysis of qualitative properties involved in thedynamics of Keller-Segel type systems in which the diffusion mechanisms of thecells are driven by porous-media flux-saturated phenomena. We study theregularization inside the support of a solution with jump discontinuity at theboundary of the support. We analyze the behavior of the size of the support andblow--up of the solution, and the possible convergence in finite time towards aDirac mass in terms of the three constants of the system: the mass, theflux--saturated characteristic speed, and the chemoattractant sensitivityconstant. These constants of motion also characterize the dynamics of regularand singular traveling waves.

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