Blow-up and delay for a parabolic–elliptic Keller–Segel system with a source term

In this paper, we are concerned with the parabolic–elliptic Keller–Segel system with a positive source term in a bounded domain in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{R}}^{N}$\end{document} (N=2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N=2,3$\end{document}), under homogeneous Dirichlet boundary condition, with time-dependent coefficients. Lower bounds for the blow-up time if the solutions blow up in finite time are derived under appropriate assumptions on data. Moreover, the exponential decay of the associated energies is also studied.