Boundedness in a quasilinear fully parabolic two-species chemotaxis system of higher dimension

Abstract This paper considers the following coupled chemotaxis system: { u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − χ 1 ∇ ⋅ ( u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , v t = ∇ ⋅ ( ψ ( v ) ∇ v ) − χ 2 ∇ ⋅ ( v ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , w t = Δ w − γ w + α u + β v , $$\textstyle\begin{cases} u_{t}=\nabla\cdot(\phi(u)\nabla u)-\chi_{1} \nabla\cdot(u\nabla w)+\mu_{1} u(1-u-a_{1} v), \\ v_{t}=\nabla\cdot(\psi(v)\nabla v)-\chi_{2} \nabla\cdot(v\nabla w)+\mu_{2} v(1-a_{2}u-v), \\ w_{t}=\Delta w-\gamma w+\alpha u+\beta v, \end{cases} $$ with homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R N $\Omega\subset\mathbb{R}^{N}$ ( N ≥ 3 $N\ge3$ ) with smooth boundaries, where χ 1 $\chi_{1}$ , χ 2 $\chi_{2}$ , μ 1 $\mu_{1}$ , μ 2 $\mu_{2}$ , a 1 $a_{1}$ , a 2 $a_{2}$ , α, β and γ are positive. Based on the maximal Sobolev regularity, the existence of a unique global bounded classical solution of the problem is established under the assumption that both μ 1 $\mu_{1}$ and μ 2 $\mu_{2}$ are sufficiently large