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 domain?(?)N,where(?)represents the derivative with respect to the outer normal of(?)?,(|x||?|)=(1,1)and ? ? {0,1}.Denote m1:= ??uo(x)dx and m2:????0(x)dx If ? = 1,in terms of the repulsion mechanism with ?=-1,we establish the global boundedness of solutions for N ? 3;while when(X,?)=(1,1),it is shown in the two dimensional case that the solutions are globally bounded for ml and m2 small enough.If? = 0,for the case of N = 2 and(X,?)=(1,1),we prove that there exist finite time blow-up solutions provided that m1m2-2?(m1 + m2)>0.This improves the current result for finite time blow-up with min{m1,m2}>4? 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].