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 Ḃp,1dp−2(Rd)×(Ḃp,1dp−1(Rd))d. For 1≤p<2d, 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>2d. 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.