Ill-posedness of a multidimensional chemotaxis system in the critical Besov spaces

Critical value can be used to distinguish will-posedness and ill-posedness, and there are a lot of research about it. But it still remains a lot of problems about critical value to be solved. In this paper, we prove that a multidimensional chemotaxis system is ill-posedness in <(B) over dot>(d/p-2)(p,sigma) x (<(B) over dot>(d/p-1)(p,sigma))(d) when p = 2d and sigma > 2. Nie and Yuan prove the system is well-posedness when p < 2d and is ill-posedness when p > 2d in [14]. Later Nie and Yuan also prove the system is ill-posed when p = 2d, sigma = 1 in [15]. Indeed, we enrich the text of dichotomy index when p = 2d. (C) 2022 Elsevier Inc. All rights reserved.