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

TL;DR

This paper demonstrates that the hyperbolic Keller-Segel model is ill-posed in certain Besov spaces by constructing initial data leading to discontinuous solutions at initial time, extending previous results to higher dimensions and different p-values.

Contribution

The paper provides a new construction showing ill-posedness of the hyperbolic Keller-Segel model in Besov spaces, generalizing prior one-dimensional results.

Findings

Constructed initial data causing solution discontinuity at t=0

Proved ill-posedness in B^\sigma_{p,\infty} spaces for d≥1

Extended previous one-dimensional results to higher dimensions and p-values

Abstract

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