An interpolation inequality and its application in Keller-Segel model

Abstract

In this paper, we first prove an interpolation inequality of Ehrling-type, which is an improvement of a special case to the well known Gargliardo-Nirenberg inequality. Then we apply it to study the classical Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=Δu-\nabla\cdot(u \nabla v), \\[6pt] \displaystyle v_t=Δv-v+u, \end{array} \right. \end{equation} in a bounded domain $\Omega \subset {\mathbb{R}}^N$ ($N\ge 2$) with smooth boundary. It is known that for any $\delta >0$, if ${\int }_{\Omega }{u}^{\frac{N}{2}+\delta }(\cdot ,t)$ is bounded, then the solution is global and bounded. Here we show that the same conclusion holds for a weaker assumption: the equi-integrability of $\{{\int }_{\Omega }{u}^{\frac{N}{2}}(\cdot ,t)|t\in (0,T_{\max })\}$ can prevent blow up.