In this talk, we would like to report our recent work on a Keller-Segel system of chemotaxis, featuring a density-suppressed motility. This model was originally proposed by Keller and Segel in their seminal work in 1971, which models the cellular movements due to a local sensing chemotaxis. An extended model was also developed in some recent works of Biophysics to study the process of pattern formation, involving a density-suppressed motility, which stands for a repressive effect of the signal on cell motility. From a mathematical point of view, the model features a signal-dependent motility, which may vanish as the concentration becomes unbounded, leading to a possible degenerate problem. Conventional energy methods can only deal with some special cases and the existence of classical solutions with generic motility functions is a long standing open problem. Recently, we develop a new comparison method based on the nonlinear structure which provides us an explicit point-wise upper bound estimate for the concentration. Then, we study the global existence of classical solutions and discuss their boundedness in any dimensions. In particular, a critical mass phenomenon as well as an infinite-time blowup was verified in the two-dimensional case. The talk is based on recent joint works with Kentarou Fujie (Tohoku University), Philippe Laurençot (University of Toulouse and CNRS), and Yanyan Zhang (ECNU).

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