Boundedness in a quasilinear attraction–repulsion chemotaxis system with variable logistic source

This paper deals with a quasilinear attraction–repulsion chemotaxis system with a source term of variable logistic type uₜ = ∇·(φ(u)∇ u)-∇·(ψ(u)∇ v)+∇·(φ(u)∇ w)+g(u), τ₁vₜ = Δ v-v+u, -Δ w = -w+u in a smooth bounded domain Ω⊂ Rⁿ (n≥1), and endowed with nonnegative initial data and homogeneous Neumann boundary conditions. Moreover, the logistic source verifies g(x, s)≤η sᵏ⁽ˣ⁾-μ sᵐ⁽ˣ⁾, $ s > 0 $ with g(x, 0)≥0, x∈Ω, where η≥0, μ > 0 are constants, $ k, m $ are measurable functions fulfilling 0≤ k⁻: = x∈Ωess\ inf k(x)≤ k(x)≤ k⁺: = x∈Ωess\ k(x) < +∞ and 1 < m⁻: = x∈Ωess\ inf m(x)≤ m(x)≤ m⁺: = x∈Ωess\ m(x) < +∞, as well as φ, ψ, and φ are regular functions satisfying c₁sᵖ≤φ(s), ψ(s)≤ c₂sq, and c₃sˡ≤φ(s)≤ c₃sˡ with p, q, l∈ R, c₁, c₂, c₃, c₃ > 0 and s≥ s₀ > 1. We show that when q = m⁻ − 1 and l≤ m⁻ − 1, there exists μ^* > 0 such that if μ > μ^*, then the corresponding initial-boundary value problem possesses a unique globally bounded classical solution. Moreover, the same conclusion holds true provided that q < m⁻ − 1 and l≤ m⁻ − 1 for any μ > 0.