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

Abstract

This paper is concerned with the global boundedness and blow-up 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 ∗ > 0 m_*>0 , such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e., L 1 L^1 -norm of the initial value of cell density) is less than m ∗ m_* , while the solution may blow up if the initial cell mass is greater than m ∗ m_* .