The Global Well-posedness And Blow-up Of Solutions For The Keller-Segel Model With Different Potentials

The Keller-Segel model is used to describe the phenomenon of chemotaxis in nature,which has a fundamental place in biology.This article proves the global well-posedness and blow-up of solutions for Keller-Segel chemotaxis model with different potentials.Different from the previous method of obtaining the critical mass from the free energy,in this paper,we adopt the construction of the mild solution,obtain the monotonicity formula,and then prove the existence and blow-up of solutions.In particular,for the Keller-Segel equation with the Bessel potential,let M be the initial mass,for each nonnegative initial data in the L1(R2).when M<8π/1+1/e2.the mild sulotion is global;when M>8π.the solutions blow up in finite time.Traditional way of proving is based on the second moment,free energy functional to give the basic energy estimate of solutions.However,here the monotony formula is given based on the estimates of the Bessel kernel and the heat kernel,and then the existence and blow-up of solutions are proved.For the two species and two chemicals Keller-Segel model with Newtonian potential,let m1,m2 be the initial mass of two species respectively,for each nonnegative initial data in the L1(R2),when max{m1,m2<8π,the global existence of solutions is obtained;when 8π<m1<m2;m1+m2<3m12-m22/8πor 8π<m2≤m1,m1+m2<3m22-m12/8π,the finite time blow-up of the solutions occurs.Compared with the classical proof method,the initial condition is very weak,and the proof process is also simple.The monotone formula is obtained based on the control for heat kernel,and the existence and blow-up of solutions are given by using the symmetric structure of the equation to deal with the coupling relationship.