Ill-posedness issue on a multidimensional chemotaxis equations in the critical Besov spaces

TL;DR

This paper demonstrates that a multidimensional chemotaxis system is ill-posed in certain critical Besov spaces, addressing an open question and highlighting the system's mathematical limitations.

Contribution

It proves the ill-posedness of the chemotaxis system in specific Besov spaces, resolving an open problem in the mathematical analysis of such equations.

Findings

Chemotaxis system is ill-posed in $ ext{dot}B_{2d, r}^{-3/2}$ spaces.

Lack of solution continuity in the specified Besov spaces.

Addresses an open question in nonlinear PDE analysis.

Abstract

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022)) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in $\dot{B}_{2d, r}^{-\frac{3}{2}} \times\big(\dot{B}_{2d, r}^{-\frac{1}{2}}\big)^{d}$ when $1\leq r<d$ due to the lack of continuity of the solution.