Ill-posedness of the hyperbolic Keller-Segel model in Besov spaces

In this paper, we give a new construction of $u_0\in B_{p,\mathrm{\infty }}^{\sigma }$ such that the corresponding solution to the hyperbolic Keller-Segel model starting from $u_0$ is discontinuous at $t=0$ in the metric of $B_{p,\mathrm{\infty }}^{\sigma }({\mathbb{R}}^d)$ with $d\ge 1$ and $1\le p\le \mathrm{\infty }$, which implies the ill-posedness for this equation in $B_{p,\mathrm{\infty }}^{\sigma }$. Our result generalizes the recent work in \cite{Zhang01} (J. Differ. Equ. 334 (2022)) where the case $d=1$ and $p=2$ was considered.