Boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source

Abstract

In this paper, we deal with the following quasilinear attraction-repulsion model: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u t = ∇ • (D(u)∇u)-∇ • (S(u)χ (v)∇v) + ∇ • (F(u)ξ (w)∇w) + f (u), x ∈ Ω, t > 0, v t = v + βuαv, x ∈ Ω, t > 0, 0 = w + γ u-δw, x ∈ Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω with homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 2). Let the chemotactic sensitivity χ (v) be a positive constant, and let the chemotactic sensitivity ξ (w) be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.