A quasilinear attraction–repulsion chemotaxis system with logistic source. (April 2023) - Cai, Yuanyuan; Li, Zhongping
Abstract: This paper is concerned with the quasilinear attraction–repulsion chemotaxis system with logistic source u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( Φ ( u ) ∇ v ) + ∇ ⋅ ( Ψ ( u ) ∇ w ) + f ( u ), ( x, t ) ∈ Ω × ( 0, ∞ ), 0 = Δ v + α u − β v, ( x, t ) ∈ Ω × ( 0, ∞ ), 0 = Δ w + γ u − δ w, ( x, t ) ∈ Ω × ( 0, ∞ ) under Neumann boundary conditions in a bounded domain Ω ⊂ R N ( N ≥ 1 ), where D, Φ, Ψ ∈ C 2 [ 0, ∞ ) are nonnegative functions with D ( s ) ≥ ( s + 1 ) p for s ≥ 0, Φ ( s ) ≤ χ s q, ξ s r ≤ Ψ ( s ) ≤ ζ s r for s > 1, and the smooth function f satisfies f ( s ) ≤ μ s ( 1 − s k ) for s > 0, f ( 0 ) ≥ 0 . Tian et al. (2016) proved that when q = m a x r, k and q − p ≥ 2 N, if one of the following assumptions holds: (i) q = r = k, μ > ( α χ − γ ξ ) ( 1 − 2 N ( q − p ) ) / ( 1 + 2 ( q − 1 ) N ( q − p ) ) ; (ii) q = r > k, α χ − γ ξ < 0 ; (iii) q = k > r, μ > α χ ( 1 − 2 N ( p − q ) ) / ( 1 + 2 ( q − 1 ) N ( p − q ) ), then the solution of the equations is globally bounded. The present work further shows that the same conclusion still holds for the critical cases: (a) q = r = k, μ = ( α χ − γ ξ ) ( 1 − 2 N ( q − p ) ) / ( 1 + 2 ( q − 1 ) N ( q − p ) ) ; (b) q = r > k, α χ − γ ξ = 0 with q − p < 1 N m i n 4 ( N + 1 ) N + 2, N + 2 ; (c) q = k > r, μ = α χ ( 1 − 2 N ( p − q ) ) / ( 1 + 2 ( q − 1 ) N ( p − q ) ) .