Explicit lower bound of blow-up time in a fully parabolic attraction-repulsion chemotaxis system with nonlinear terms

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.