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

TL;DR

This paper establishes an explicit lower bound for the blow-up time in a fully parabolic chemotaxis system with nonlinear terms, extending previous results from 3D to higher dimensions and analyzing energy function blow-up times.

Contribution

It provides the first explicit lower bound estimate for blow-up time in a fully parabolic chemotaxis system with nonlinearities in dimensions greater than 3.

Findings

Proved energy blow-up time equals classical blow-up time for large energy levels.

Derived explicit lower bound for blow-up time in higher dimensions.

Extended blow-up analysis from parabolic-elliptic to fully parabolic systems.

Abstract

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.