Well-posedness and ill-posedness of a multidimensional chemotaxis system in the critical Besov spaces

Abstract We study the Cauchy problem for a multidimensional chemotaxis system in critical Besov spaces B p , 1 d p − 2 ( R d ) × ( B p , 1 d p − 1 ( R d ) ) d . For 1 ≤ p 2 d , we prove locally well-posedness for large initial data and globally well-posedness for small initial data of this system. And more importantly, we show the ill-posedness in the sense that a “norm inflation” phenomenon occurs for p > 2 d . More precisely, we construct a specific initial data which can be arbitrarily small in the Besov spaces. Meanwhile, the corresponding solution u can be arbitrarily large after an arbitrarily short time.