Steady states and their qualitative properties of several classes of Keller-Segel models

Abstract

Chemotaxis is the directed movement of bacterial organisms towards or away from stimulating chemicals in their environment. This intrinsic ability is important for species to profit from the good, avoid the harm and to survive their environment. In order to study cellular chemotactic movement, Evelyn Fox Keller and Lee Aaron Segel proposed a class of parabolic PDE systems to describe the evolution of bacterial population density and chemical concentration. Though these systems have relatively simple structures, they admit rich spatial-temporal dynamics and can well be used to model cell aggregation, one of the most important phenomena in chemotaxis, hence become one of the most extensively studied topics in PDEs over the past few decades. Cell aggregation can be modeled by the $\\delta$-profile out of finite-time blowups in the parabolic systems, and also by the steady states with striking spatial concentrating profiles such as spikes, transition layers, etc. The former approach has been surveyed by a few articles, while a review of the latter is not available, and therefore it is the goal of this paper to survey the steady states and their qualitative properties of several classes of Keller-Segel models, with a focus on the spikes and transition layers. In particular, we shall describe the classical results and recent developments of this problem, as well as inspiring mathematics developed meanwhile such as the variation method, singular perturbation, bifurcation theory. We shall also introduce some open problems concerning stationary systems.