Existence and uniqueness theorem of weak solutions to the parabolic-elliptic Keller-Segel system

Abstract

In ${ℝ}^n$ ($n\ge 3$), we first define a notion of weak solutions to the Keller-Segel system of parabolic-elliptic type in the scaling invariant class $L^s\left(\left(0,T\right);L^r\left({ℝ}^n\right)\right)$ for $2∕s+n∕r=2$ with $n∕2<r<n$. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for every initial data in $L^{n∕2}\left({ℝ}^n\right)$. We prove also their uniqueness. As for the marginal case when $r=n∕2$, we show that if $n\ge 4$, then the class $C\left(\left[0,T\right);L^{n∕2}\left({ℝ}^n\right)\right)$ enables us to obtain the only weak solution.