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 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 $\e>0$ 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 the solutions behave like self-similar solutions, while in presence of the degradation term global solutions behave like the heat kernel.

References in corpus (2)

• Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis
• Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model

Cited by in corpus (11)

• On a generalized doubly parabolic Keller-Segel system in one spatial dimension
• Suppression of blow-up in Parabolic-Parabolic Patlak-Keller-Segel via strictly monotone shear flows
• Periodic Solutions of the parabolic-elliptic Keller-Segel system on whole spaces
• Large global solutions of the parabolic-parabolic Keller-Segel system in higher dimensions
• Constructive proofs for some semilinear PDEs on $H^2(e^{|x{|}^2/4},{\mathbb{R}}^d)$
• Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains
• Multi-species Patlak-Keller-Segel system
• Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation
• Exact traveling wave solutions of 1D parabolic-parabolic models of chemotaxis
• Large self-similar solutions of the parabolic-elliptic Keller--Segel model
• On pseudo almost Periodic Solutions of the parabolic-elliptic Keller-Segel systems