This paper is devoted to the chemotaxis system
$\left\{\begin{aligned}&u_t=Δ u-χ\nabla·(u\nabla v), &x∈Ω,\,t>0,\\& τ v_t=Δ v-v+w, &x∈Ω,\,t>0,\\&w_t=Δ w-ξ\nabla·(w\nabla z), &x∈Ω,\,t>0,\\& τ z_t=Δ z-z+u, &x∈Ω,\,t>0,\end{aligned}\right.$
which models the interaction between two species in presence of two chemicals, where
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