  This paper is concerned with global existence as well as infinite-time
blowups of classical solutions to the following fully parabolic kinetic system
\begin{equation} \begin{cases} u_t=\Delta (\gamma (v)u) v_t-\Delta v+v=u
\end{cases} \qquad(0.1)\end{equation} in a smooth bounded domain
$\Omega\subset\mathbb{R}^n$, $n\geq1$ with no-flux boundary conditions. This
model was recently proposed in [8,20] to describe the process of stripe pattern
formations via the so-called self-trapping mechanism. The system features a
signal-dependent motility function $\gamma(\cdot)$, which is decreasing in $v$
and will vanish as $v$ tends to infinity. The major difficulty in analysis
comes from the possible degeneracy as $v\nearrow+\infty.$ In this work we
develop a new comparison method different from the conventional energy method
in literature, which reveals a striking fact that there is no finite-time
degeneracy in this system. More precisely, we use comparison principles for
elliptic and parabolic equations to prove that degeneracy cannot take place in
finite time in any spatial dimensions for all smooth motility functions
satisfying $\gamma(s)>0$, $\gamma'(s)\leq0$ when $s\geq0$ and
$\lim\limits_{s\rightarrow+\infty}\gamma(s)=0.$ Then we investigate global
existence of classical solutions to (0.1) when $n\leq3$ and discuss the
uniform-in-time boundedness under certain growth conditions on $1/\gamma.$ In
particular, we consider system (0.1) with $\gamma(v)=e^{-v}$. We prove that
classical solution always exists globally, which must be uniformly-in-time
bounded with arbitrary initial data of sub-critical mass. On the contrary, with
certain initial data of super-critical mass, the solution will become unbounded
at time infinity which differs from the finite-time blowup behavior of the
Keller--Segel model.
