  It is known that for the parabolic-elliptic Keller-Segel type system in a
smooth bounded domain in 3-dimensional space, the lower bound of a blow-up time
of unbounded solution is given. This paper extends the previous works to deal
with the fully parabolic Keller-Segel type system in any spatial dimension
larger than 3. Firstly, we prove that any blow-up time of an energy function is
also the classical blow-up time when its energy level is sufficiently large.
Secondly, we give an explicit estimation for the lower bound of blow-up time
for the fully parabolic attraction-repulsion chemotaxis system with nonlinear
terms, under homogeneous Neumann boundary conditions, in a smooth bounded
domain.
