  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 $s\to\infty$ for some
$k\in (0,N/(N-2)_+)$, 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.
