Well-posedness and ill-posedness of a multidimensional chemotaxis system in the critical Besov spaces. (July 2020)

• Record Type: Journal Article
• Authors: Nie, Yao; Yuan, Jia
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: 35G55 -- 35Q92 -- 92C17; Chemotaxis model -- Besov spaces -- Well-posedness -- Ill-posedness -- Norm inflation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111782
• Languages: English
• ISSNs: 0362-546X

Abstract

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.