Singular patterns in Keller–Segel-type models

Abstract

The aim of this paper is to elucidate the existence of patterns for Keller–Segel-type models that are solutions of the traveling pulse form. The idea is to search for transport mechanisms that describe this type of waves with compact support, which we find in the so-called nonlinear diffusion through saturated flux mechanisms for the movement cell. At the same time, we analyze various transport operators for the chemoattractant. The techniques used combine the analysis of the phase diagram in dynamic systems together with its counterpart in the system of partial differential equations through the concept of entropic solution and the admissible jump conditions of the Rankine–Hugoniot type. We found traveling pulse waves of two types that correspond to those found experimentally.