Boundedness and Stabilization in a Two-Species and Two-Stimuli Chemotaxis System with Signaling Loop

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop {[ u_t=Δ u-χ _1∇· (u∇ v)+μ _1 u(1-u-a_1w), 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_2u), x∈Ω , t>0,; z_t=Δ z-z+u, x∈Ω , t>0 ] . in a bounded domain Ω⊂ℝ^n with n≥ 1 , where χ _1,χ _2,χ _3>0 , a_1,a_2>0 and μ _1,μ _2>0 . The system models the communication between macrophages and breast tumor cells. It will be proved that if n≤ 2 , then for all appropriately regular nonnegative initial data u_0, v_0, w_0 and z_0 , the solution to the corresponding Neumann initial-boundary value problem is global and bounded. Moreover, the asymptotic stabilization of arbitrary global bounded solutions for any n≥ 1 under some explicit conditions will be investigated.