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=\Delta u-\nabla\cdot(u \nabla v), \\[6pt] \displaystyle v_t=\Delta v-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 $\int_\Omega u^{\frac N2+\delta}(\cdot,t)$ 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 $\{\int_\Omega u^\frac N2(\cdot,t)|~t\in(0,T_{\max})\}$ can prevent blow up.