Time global solutions to a parabolic-elliptic system modelling chemotaxis

This paper is concerning with asymptotic behavior of infinite time blowup solutions to a parabolic-elliptic system to chemotaxis. In this paper, we show that the solutions form a delta function singularity at each blowup point in infinite time and that the weight of each delta function singularity is equal to 8pi and 4pi if the blowup point is in the domain and on the boundary, respectively. We refer to the former and the latter phenomenon as chemotactic collapse and quantization, respectively. Next, we investigate the behavior of some norms of the solutions. Finally, we show that the location of the collapse moves continuously if total mass of the solution is between 4pi and 8pi.