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

Abstract

Abstract 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 8π, whereas there exist solutions blowing up in finite time under the hypotheses of either M 0 > 8 π with any integrable initial data or M 0 < 8 π < m 0 accompanied with large initial data. Moreover, m 0 < M 0 = 8 π 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 , λ ( r ) = 8 λ ( r 2 + λ ) 2 with λ > 0. Subsequently we prove that if the initial data is strictly below m 0 M 0 U s , λ ( r ) for some λ > 0, then the solution vanishes in L l o c 1 ( R 2 ) as t → ∞ . If the initial data is strictly above m 0 M 0 U s , λ ( r ) for some λ > 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.