Critical blow-up curve in a two-species chemotaxis system with two chemicals involving flux-limitation

We investigate the following two-species chemotaxis system with two chemicals involving flux-limitation ★ { u t = Δ u − ∇ · ( u ( 1 + | ∇ v | 2 ) − p 2 ∇ v ) , x ∈ Ω , t > 0 , 0 = Δ v − μ w + w , μ w = f Ω w , x ∈ Ω , t > 0 , w t = Δ w − ∇ · ( w ( 1 + | ∇ z | 2 ) − q 2 ∇ z ) , x ∈ Ω , t > 0 , 0 = Δ z − μ u + u , μ u = f Ω u , x ∈ Ω , t > 0 , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = ∂ z ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ Ω , where p , q ∈ R and Ω ⊂ R n is a smooth bounded domain. In this paper, we identify a critical blow-up curve for system (⋆) with n ≥ 2. If p < n − 2 n − 1 and q < n − 2 n − 1 , and Ω = B R ( 0 ) ⊂ R n with n ≥ 2, there exist radially symmetric initial data such that the corresponding solution blows up in finite time; if either p > n − 2 n − 1 or q > n − 2 n − 1 with n ≥ 2, then solutions exist globally and remain bounded.