Global existence and finite-time blow-up in a 2D no-flux-Dirichlet Keller-Segel system with matrix-valued sensitivity

This paper is concerned with a system that can be used to model the diffusion-advection of myosin molecules, which change direction by a certain angle when binding to actin gel. More precisely, we shall consider the parabolic-elliptic Keller-Segel system with rotation [Formula: see text]; [Formula: see text] in smoothly bounded planar domains, where [Formula: see text] is a matrix attaining value in [Formula: see text] as [Formula: see text] with [Formula: see text]. We shall investigate the system accompanied by boundary conditions of no-flux type for [Formula: see text] and of Dirichlet type for [Formula: see text]. It is shown that the associated initial-boundary value problem possesses a critical value of the initial mass [Formula: see text] to distinguish global existence and finite-time blow-up. Furthermore, we study the blowup mechanism, and prove that the number of blowup points is finite and does not exceed [Formula: see text]. Our results present a precise characterization of the effect of rotational trajectory on the properties of Keller-Segel system in the form of rotational angle.