On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $L^p$-space

We are concerned with the uniqueness of mild solutions in the critical Lebesgue space $L^{\frac{n}{2}}({\mathbb{R}}^n)$ for the parabolic-elliptic Keller-Segel system, $n\ge 4$. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak- $L^{\frac{n}{2}}$ space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e., $L^q((0,T);L^p)$-norms with $q\ne \mathrm{\infty }$), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical $C([0,T);L^{\frac{n}{2}})$, our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate $\gamma$ of the chemoattractant is also analyzed.