Quantitative decay of a one-dimensional hybrid chemotaxis model with large data

Abstract

We investigate the quantitative behaviour of one-dimensional classical solutions for a hyperbolic-parabolic system describing repulsive chemotaxis. It is shown that classical solutions to the Cauchy problem always exist globally in time for large initial perturbations around constant equilibrium states. We prove rigorously the chemo-repulsion collapse scenario and show that all solutions converge to the attractive ground states as time approaches infinity. Moreover explicit decay rates of the perturbations are computed under some mild conditions on the initial data. The proof is established via a novel Lp-based energy method. We also obtain a frequency-dependent stretched-exponential decay rate by using a new Fourier method which can be of independent interest.