Global existence and uniform boundedness in a chemotaxis model with signal-dependent motility

Global existence is established for classical solutions to a chemotaxis model with signal-dependent motility for a general class of motility functions $\gamma$ which may in particular decay in an arbitrary way at infinity. Assuming further that $\gamma$ is non-increasing and decays sufficiently slowly at infinity, in the sense that $\gamma (s)\sim s^{-k}$ as $s\to \mathrm{\infty }$ for some $k\in (0,N\mathrm{/}(N-2{)}_{+})$, it is also shown that global solutions are uniformly bounded with respect to time. The admissible decay of $\gamma$ at infinity here is higher than in previous works.