Boundedness And Finite Time Blow-up Of Solutions To A Two-species Chemotaxis System With Two Chemicals

In this paper we study the global boundedness and finite time blow-up of solutions to the following two-species chemotaxis system with two chemicalsin a smooth bounded domainΩ(?)N,where(?)represents the derivative with respect to the outer normal of(?)Ω,(|x||ξ|)=(1,1)and τ ∈ {0,1}.Denote m1:= ∫Ωuo(x)dx and m2:＝∫Ωω0(x)dx If τ = 1,in terms of the repulsion mechanism with ξ=-1,we establish the global boundedness of solutions for N ≤ 3;while when(X,ξ)=(1,1),it is shown in the two dimensional case that the solutions are globally bounded for ml and m2 small enough.Ifτ = 0,for the case of N = 2 and(X,ξ)=(1,1),we prove that there exist finite time blow-up solutions provided that m1m2-2π(m1 + m2)>0.This improves the current result for finite time blow-up with min{m1,m2}>4π in[Y.Tao,M.Winkler,Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals,Discrete Contin.Dyn.Syst.Ser.B 20(2015)3165-3183].