Boundedness in a three‐dimensional two‐species and two‐stimuli chemotaxis system with chemical signalling loop

The following two‐species chemotaxis system with a chemical signalling loop under Lotka–Volterra competitive kineticsu t = Δ u − χ 1 ∇ · ( u ∇ v ) + μ 1 u ( 1 − u − a 1 w ) , x ∈ Ω , t > 0 ,v t = Δ v − v + w , x ∈ Ω , t > 0 ,w t = Δ w − χ 2 ∇ · ( w ∇ z ) − χ 3 ∇ · ( w ∇ v ) + μ 2 w ( 1 − w − a 2 u ) , x ∈ Ω , t > 0 ,z t = Δ z − z + u , x ∈ Ω , t > 0 ,is considered in a bounded domain Ω ⊂ ℝ 3with homogeneous Neumann boundary conditions, and the parameters μ i , a i >0 ( i =1,2) and χ j >0( j =1,2,3), the initial data are nonnegative and satisfy ( u 0 , v 0 , w 0 , z 0 ) ∈ C 0 ( Ω ¯ ) × C 1 ( Ω ¯ ) × C 0 ( Ω ¯ ) × C 1 ( Ω ¯ ) . Global boundedness solution of this system is proved under the conditionsμ 1 > max10 + 21 +34χ 1 2 + χ 2 2 + χ 3 2+ 1 ,2 χ 1 2 +4 + 310 + 22χ 1 2 + χ 2 2 + χ 3 2χ 1 2andμ 2 > max2 χ 3 2 +4 + 310 + 22χ 1 2 + χ 2 2 + χ 3 2χ 3 2 +10 + 21 +34χ 1 2 + χ 2 2 + χ 3 2+ 1 ,2 χ 2 2 +4 + 310 + 22χ 1 2 + χ 2 2 + χ 3 2χ 2 2.