The Long-time Behavior On Radial Solutions Of Keller-Segel Chemotaxis Model With Multi-species Or Multi-chemicals

This paper mainly studies the long-time behavior of solutions to the Keller-Segel chemotactic model with Newtonian potential for multi-species and multi-chemicals.This kind of model is used to describe the collective movement behavior of biological population,which plays an important role in biology and has attracted the attention of many researchers in recent years.In this paper,the super and sub-solution method is used to prove that the radial solution tends to zero in L_loc¹(R²)norm when the initial data satisfies the subcritical condition,and the radial solution blows up in finite time or has the mass concentration phenomenon in the infinite time when the initial data is in the supercritical case.We know that there is no comparison principle for Keller-Segel equations in general,so the super and sub-solution method cannot be applied directly to the equations.We introduce the mass transformation to transform the equations into a single equation with comparison principle such that the super and sub-solution method can be used.Specifically,for the model with one species and two chemicals,the super and sub-solutions are constructed by using the steady-state solutions of a single equation,and then the long-time behavior of the solutions are given by using the comparison principle.For the model with two species and two chemicals,it is found that the transformed model is still system,so the comparison principle of the equations is proved first,and the long-time behavior is obtained.