  This paper is devoted to global existence of weak solutions to the following
degenerate kinetic model of chemotaxis
  \begin{equation}
  \begin{cases}\label{chemo0}
  u_t=\Delta (\gamma (v)u)
  \tau v_{t}=\Delta v-v+u
  \end{cases}
  \end{equation}in a smooth bounded domain with no-flux boundary conditions.
The problem features a positive signal-dependent motility function
$\gamma(\cdot)$ which may vanish as $v$ becomes unbounded. In this paper, we
first modify the comparison approach developed recently in \cite{GM,GM2} to
derive the upper bounds of $v$ under weakened assumptions on $\gamma(\cdot)$.
Then by introducing a suitable approximation scheme which is compatible with
the comparison method, we establish the global existence of weak solutions in
any spatial dimension via compactness argument. Our weak solution has higher
regularity than those obtained in previous literature.
