The Patlak-Keller-Segel model of chemotaxis on ℝ² with singular drift and mortality rate

We extend the classical Patlak-Keller-Segel model of chemotaxis on ℝ² in the following two directions: the presence of a logarithmically singular prescribed drift term, which aggregates mass at a singular point, and the effect of a linear logistic term (mortality rate) which induces an overall exponential decay of the mass. The results characterize conditional global existence (existence of solutions at all time for selected initial data) as well as unconditional blow-up at a finite time, with respect to the total mass and second moment of the initial data.