Critical mass on the Keller-Segel system with signal-dependent motility

This paper is concerned with the global boundedness and blowup of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is there is a number $m_{\ast }>0$, such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e. $L^1$-norm of the initial value of cell density) is less than $m_{\ast }$, while the solution may blow up if the initial cell mass is greater than $m_{\ast }$.