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

Abstract

In the present article we consider several issues concerning the doubly parabolic Keller–Segel system (1.1)–(1.2) in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that ε>0 is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term (α=0) the solutions behave like self-similar solutions, while in the presence of the degradation term (α>0) the global solutions behave like the heat kernel.