Convergence analysis from the indirect signal production to the direct one

Abstract

This paper aims to provide the rigorous convergence analysis of the chemotaxis system{∂nϵ∂t=Δnϵ−∇⋅(nϵ∇cϵ),τ1∂cϵ∂t=Δcϵ−cϵ+wϵ,τ2∂wϵ∂t=−wϵ+nϵ in a smoothly bounded planar domain Ω as τ1→0 and (or) τ2→0. Precisely, we reveal that actually the indirect signal properties are strong enough to allow for a uniformly global-in-time convergence in terms of τ1=τ2=ϵ↘0 and τ1=1,τ2=ϵ↘0 for ∫Ωnϵ(⋅,0)<4π, which correspond to the classical Keller-Segel systems with direct signal production, and of τ1=ϵ↘0,τ2=1 for suitable small initial data. As a byproduct, we show that for any p≥2, the Lp-norm of nϵ blows up if ∫Ωnϵ(⋅,0)∈(4π,∞)∖4πN, which improves the existing results that the solution is unbounded in the sense of L∞-norm of nϵ.