Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate

Abstract

We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. It is a 2×2 reaction-diffusion system describing the interaction of cells and a chemical signal. We study Cauchy problem for the original system and its transformed system, which is one of hyperbolic-parabolic balance laws. Our initial data are generic perturbations of a constant ground state, i.e. the initial mass of perturbation is non-zero. In the case of non-diffusive chemical, we obtain optimal L2 time decay rates for the solution with finite initial data. In the case of diffusive chemical, optimal L2 rates are also obtained with additional assumption on the smallness of the initial amplitude but still allowing large oscillation.