  This paper deals with the fully parabolic chemotaxis system of local sensing
in higher dimensions. Despite the striking similarity between this system and
the Keller--Segel system, we prove the absence of finite-time blow-up
phenomenon in this system even in the supercritical case. It means that for any
regular initial data, independently of the magnitude of mass, the classical
solution exists globally in time in the higher dimensional setting. Moreover,
for the exponential decaying motility case, it is established that solutions
may blow up at infinite time for any magnitude of mass. In order to prove our
theorem, we deal with some auxiliary identity as an evolution equation with a
time dependent operator. In view of this new perspective, the direct
consequence of the abstract theory is rich enough to establish global existence
of the system.
