Some remarks on the fully parabolic Keller-Segel system in the plane

We will describe recent results on the doubly parabolic Keller-Segel system in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. We analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. We are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term the solutions behave like self-similar solutions, while in presence of the degradation term global solutions behave like the heat kernel