A Keller–Segel type system in higher dimensions

Abstract

We analyze an equation that is gradient flow of a functional related to Hardy–Littlewood–Sobolev inequality in whole Euclidean space ${\mathbb{R}}^d$, $d\ge 3$. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of “free-energy solutions”, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a critical value. Actually, there is a critical value of a parameter in the equation below which there is a global-in-time energy solution and above which there exist blowing-up energy solutions.