We consider a parabolic-elliptic system of drift-diffusion type in the entire two-dimensional Euclidean space, modeling chemotaxis and self-attracting particles. Under the assumption that the total mass of nonnegative initial data is less than $8\pi$, we give global existence and decay estimates of nonnegative solutions to the Cauchy problem for this system.
Toshitaka Nagai. "Global existence and decay estimates of solutions to a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$." Differential Integral Equations 24 (1/2) 29 - 68, January/February 2011. https://doi.org/10.57262/die/1356019044