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

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