Convergence analysis from the indirect signal production to the direct one.

This paper aims to provide the rigorous convergence analysis of the chemotaxis system { ∂ n ϵ ∂ t = Δ n ϵ − ∇ ⋅ (n ϵ ∇ c ϵ) , τ 1 ∂ c ϵ ∂ t = Δ c ϵ − c ϵ + w ϵ , τ 2 ∂ w ϵ ∂ t = − w ϵ + n ϵ in a smoothly bounded planar domain Ω as τ 1 → 0 and (or) τ 2 → 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 τ 1 = τ 2 = ϵ ↘ 0 and τ 1 = 1 , τ 2 = ϵ ↘ 0 for ∫ Ω n ϵ (⋅ , 0) < 4 π , which correspond to the classical Keller-Segel systems with direct signal production, and of τ 1 = ϵ ↘ 0 , τ 2 = 1 for suitable small initial data. As a byproduct, we show that for any p ≥ 2 , the L p -norm of n ϵ blows up if ∫ Ω n ϵ (⋅ , 0) ∈ (4 π , ∞) ∖ 4 π N , which improves the existing results that the solution is unbounded in the sense of L ∞ -norm of n ϵ. [ABSTRACT FROM AUTHOR]