Global Existence and Boundedness of a Two-Competing-Species Chemotaxis Model

In this paper, we consider the following fully parabolic two-competing-species chemotaxis modelu1t= ∆u1−�∇·(u1∇v1) +�1u1(1−u1−e1u2); x∈Ω; t >0;u2t= ∆u2−�∇·(u2∇v2) +�2u2(1−e2u1−u2); x∈Ω; t >0;v1t= ∆v1+u1−v1;x∈Ω; t >0;v2t= ∆v2+u2−v2;x∈Ω; t >0under homogeneous Neumann boundary conditions, where Ω⊂Rn(n≥3) is a convex boundeddomain with smooth boundary. Relying on a comparison principle, we show that the problempossesses a unique global bounded solution if�1and�2are large enough.