Critical mass capacity for two-dimensional Keller-Segel model with nonlocal reaction terms

This paper deals with the parabolic-elliptic Keller-Segel system on R-2, involving a source term of logistic type defined in terms of the mass capacity M-0 and the total mass of the individuals. We exhibit that the qualitative behaviour of solutions is decided by the mass capacity M-0 and the initial mass m(0). For general solutions, the existence of a global weak solution is proved under the assumption that both M-0 and m(0) are less than 8 pi, whereas there exist solutions blowing up in finite time under the hypotheses of either M-0 > 8 pi with any integrable initial data or M-0 < 8 pi < m0 accompanied with large initial data. Moreover, m(0) < M-0 = 8 pi gives rise to a compromise that solutions exist globally and blow up as time goes to infinity. For radially symmetric solutions, we introduce a strategy of relegating the lack of mass conservation via a transformation to the density and then obtain that there are stationary solutions given by U-s,U-gimel(r) = 8 gimel(r(2)+gimel)(2) with.> 0. Subsequently we prove that if the initial data is strictly below m(0) /M-0 U-s,U-gimel(r) for some gimel > 0, then the solution vanishes in L-loc(1)(R-2) as t ->infinity. If the initial data is strictly above m(0)/M-0 Us,gimel(r) for some gimel > 0, then the solution either blows up in finite time or has a mass concentration at the origin as time goes to infinity. Finally, our results are complemented by numerical simulations that demonstrate the asymptotic behaviour of solutions.