We discuss the existence of the blow-up solution for some nonlinear parabolic system called attractive drift-diffusion equation in two space dimensions. We show that if the initial data satisfies a threshold condition, the corresponding solution blows up in a finite time. This is a system case for the blow-up result of the chemotactic equation proved by Nagai  and Nagai-Senba-Suzuki  and gravitational interaction of particles by Biler-Nadzieja , .
"Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type." Differential Integral Equations 16 (4) 427 - 452, 2003. https://doi.org/10.57262/die/1356060652