Recent results for the logarithmic Keller-Segel-Fisher/KPP System

Abstract

We consider a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. It is a 2 × 2 system describing the interaction of cells and a chemical signal. We study Cauchy problem with finite initial data, i.e., without the commonly used smallness assumption on initial perturbations around a constant ground state. We survey a sequence of recent results by the authors on the existence of global-in-time solution, long-time behavior, vanishing coefficient limit and optimal time decay rates of the solution.