An Attraction-Repulsion Chemotaxis System: The Roles of Nonlinear Diffusion and Productions

• Published in: Acta applicandae mathematicae 2024-04, Vol.190 (1), p.5, Article 5
• Main Authors: Jiao, Zhan, Jadlovská, Irena, Li, Tongxing
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Attraction; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Flux; Mathematics; Mathematics and Statistics; Nonlinearity; Partial Differential Equations; Probability Theory and Stochastic Processes
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-024-00641-6

This article considers the no-flux attraction-repulsion chemotaxis model { u t = ∇ ⋅ ( ( u + 1 ) m 1 − 1 ∇ u − χ u ( u + 1 ) m 2 − 2 ∇ v + ξ u ( u + 1 ) m 3 − 2 ∇ w ) , x ∈ Ω , t > 0 , 0 = Δ v + f ( u ) − β v , x ∈ Ω , t > 0 , 0 = Δ w + g ( u ) − δ w , x ∈ Ω , t > 0 defined in a smooth and bounded domain Ω ⊂ R n ( n ≥ 2 ) with m 1 , m 2 , m 3 ∈ R , χ , ξ , β , δ > 0 . The functions f ( u ) , g ( u ) extend the prototypes f ( u ) = α u s and g ( u ) = γ u r with α , γ > 0 and suitable s , r > 0 for all u ≥ 0 . Our main result exhibits that there exists M ∗ > 0 such that for all properly regular initial data, the studied model admits a unique classical solution which remains bounded if m 2 + s < m 3 + r or m 2 + s = m 3 + r and ξ γ χ α > M ∗ .