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Global existence is established for classical solutions to a chemotaxis modelwith 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 slowlyat infinity, in the sense that $\gamma(s)\sim s^{-k}$ as $s\to\infty$ for some$k\in (0,N/(N-2)_+)$, it is also shown that global solutions are uniformlybounded with respect to time. The admissible decay of $\gamma$ at infinity hereis higher than in previous works.

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