An attraction–repulsion chemotaxis with logistic source involving the exponents depending on the spatial variables An attraction–repulsion chemotaxis

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 76; no. 1
• Main Author: Ayazoglu, Rabil
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2025
• Subjects: Engineering; Mathematical Methods in Physics; Theoretical and Applied Mechanics; 35K20; 92C17; 35K59; Chemotaxis system; Global boundedness; Logistic source involving the exponents depending on the spatial variables; 35K55; Attraction–repulsion
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-024-02394-6

We study the quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source involving the exponents depending on the spatial variables: u t = Δ u - χ ∇ · u u + 1 r - 1 ∇ υ + ξ ∇ · u u + 1 r - 1 ∇ ω + a u - b u m ( x ) , 0 = Δ υ - β υ + α u , 0 = Δ ω - δ ω + γ u , where α , β , δ , γ , χ , ξ , b > 0 , a ≥ 0 , r ∈ R and m : Ω → 1 , ∞ is a measurable function, subject to the homogeneous Neumann boundary conditions in a bounded domain R N N ≥ 1 with smooth boundary. We prove that this system possesses a unique global bounded classical solution, which is an extension of known results, if the repulsion cancels the attraction in the sense that (balance) χ α = ξ γ with e s s inf x ∈ Ω m x > r + N - 2 + N , 1 , and if the attraction prevails over the repulsion in the sense that χ α > ξ γ with e s s inf x ∈ Ω m x > max r + 1 , 1 , and if the repulsion prevails over the attraction in the sense that χ α < ξ γ .