Large global solutions of the parabolic-parabolic Keller–Segel system in higher dimensions

Abstract

We study the global existence of the parabolic-parabolic Keller–Segel system in Rd, d≥2. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter τ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler et al. (2015) [7] and Corrias et al. (2014) [12]. Our analysis improves earlier results and extends them to any dimension d≥3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in τ, when τ≫1: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper [4] finite time blowup for a class of large solutions.