2022/05/18 by Bian, Shen, Wang, Quan
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.08761
We investigate the global existence and blow-up of solutions to the Keller-Segel model with nonlocal reaction term u(M0-∫\R2 u dx) in dimension two. By introducing a transformation in terms of the total mass of the populations to deal with the lack of mass conservation, we exhibit that the qualitative behavior of solutions is decided by a critical value 8π for the growth parameter M0 and the initial mass m0. For general solutions, if both m0 and M0 are less than 8π, solutions exist globally in time using the energy inequality, whereas there are finite time blow-up solutions for M0>8π (It involves the case m0<8π) with any initial data and M0<8π0, then all the radially symmetric solutions are vanishing in Lloc1(\R2) as t → ∞. If the initial data u0(r)>(m0)/(M0) (8 λ)/((r2+λ)2) for some λ>0, then there could exist a radially symmetric solution satisfying a mass concentration at the origin as t → ∞.