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In the present article we consider several issues concerning the doublyparabolic Keller-Segel system in the plane, when the initial data belong tocritical scaling-invariant Lebesgue spaces. More specifically, we analyze theglobal existence of integral solutions, their optimal time decay, uniquenessand positivity, together with the uniqueness of self-similar solutions. Inparticular, we prove that there exist integral solutions of any mass, providedthat $\e>0$ is sufficiently large. With those results at hand, we are then ableto study the large time behavior of global solutions and prove that in theabsence of the degradation term the solutions behave like self-similarsolutions, while in presence of the degradation term global solutions behavelike the heat kernel.

Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane