Global dynamics in a chemotaxis system involving nonlinear indirect signal secretion and logistic source.

This paper is concerned with a quasilinear parabolic–parabolic–elliptic chemotaxis system u t = ∇ · (φ (u) ∇ u - ψ (u) ∇ v) + a u - b u γ , x ∈ Ω , t > 0 , v t = Δ v - v + w γ 1 , x ∈ Ω , t > 0 , 0 = Δ w - w + u γ 2 , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n (n ≥ 1) , where a , b , γ 1 , γ 2 > 0 , γ > 1 , φ and ψ are nonlinear functions satisfying φ (s) ≥ a 0 (s + 1) α and | ψ (s) | ≤ b 0 s (1 + s) β - 1 for all s ≥ 0 with a 0 , b 0 > 0 and α , β ∈ R. When β + γ 1 γ 2 < max { n + 2 n + α , γ } , then the system has a classical solution which is globally bounded in time. Moreover, when β + γ 1 γ 2 = max { n + 2 n + α , γ } , it has been shown that the existence of global bounded classical solution depends on the size of coefficient b and initial data u 0. Furthermore, we consider a specific system with γ 1 = 1 , γ 2 = κ and γ = κ + 1 for κ > 0. If b > 0 is sufficiently large, the global classical solution(u, v, w) exponentially converges to the steady state ((a b) 1 κ , a b , a b) in L ∞ norm as t → ∞ , where convergence rate is explicitly expressed in terms of the system parameters. [ABSTRACT FROM AUTHOR]