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

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 , with homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R N ( N ≥ 3 ) with smooth boundaries, where χ 1 , χ 2 , μ 1 , μ 2 , a 1 , 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 and μ 2 are sufficiently large.