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

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 ${\dot{B}}_{2d,r}^{-\frac{3}{2}}\times ({\dot{B}}_{2d,r}^{-\frac{1}{2}}{)}^d$ when $1\le r<d$ due to the lack of continuity of the solution.

17 pages, to appear Journal of Geometric Analysis