Critical Mass For A Two-species Chemotaxis Model With Two Chemicals In R~2

This paper mainly studies the critical curve of Keller-Segel models with two-species and two-chemicals for different potentials,including the Bessel potential and the New-tonian potential.The mainly purpose of this paper is to find a initial critical curve to distinguish the global existence and the blow-up of solutions to the corresponding model.The results will provide important theoretical guidance for the production practice.For the chemotactic model with the Bessel potential,we show that the initial critical curve of the model is 4??m₁+m₂?-m₁m₂=0,where m₁,m₂are the initial mass of the two species respectively.That is,when initial mass satisfies 4??m₁+m₂?-m₁m₂>0,combining with Moser-Trudinger inequality for the vector form,we prove the global exis-tence of solutions.On the contrary,when 4??m₁+m₂?-m₁m₂<0,using the properties of the second moment,the finite time blow-up of solutions is obtained.Compared with the previous results,the initial critical curve of the two-species and two-chemicals model is firstly given on the whole space R².For the chemotactic model with the Newtonian potential,the initial critical curve is still the 4??m₁+m₂?-m₁m₂=0.Noticing that the singularity of the Newtonian potential is stronger than that of the Bessel potential,it is more easily to get the blow-up of the solutions when 4??m₁+m₂?-m₁m₂<0 by using the second moment proper-ties.Compared to the case of the Bessel potential model,the difficulty is to prove the global existence of solutions.Here we use another key functional inequality–the Hardy-Littlewood-Sobolev inequality for vector form to give the global existence of the solutions when 4??m₁+m₂?-m₁m₂>0.