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

In this paper, we give a new construction of $u_0\in B^\sigma_{p,\infty}$ such that the corresponding solution to hyperbolic Keller-Segel model starting from $u_0$ is discontinuous at $t = 0$ in metric $B^\sigma_{p,\infty}(\R^d)$ with $d\geq1$ and $1\leq p\leq\infty$, which implies ill-posedness for equation $B^\sigma_{p,\infty}$. Our result generalizes recent work \cite{Zhang01} (J. Differ. Equ. 334 (2022)) where case $d=1$ $p=2$ was considered.