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

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 74; no. 6
• Main Authors: Wang, Chang-Jian, Wang, Pengyan, Zhu, Xincai
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2023; Springer Nature B.V
• Subjects: Boundary conditions; Convergence; Engineering; Mathematical Methods in Physics; Theoretical and Applied Mechanics; Keller–Segel system; Nonlinear indirect secretion; Asymptotic behavior; 35A01; 92C17; Global boundedness; Chemotaxis; 35K55; 35Q92
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-023-02126-2

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.