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

Abstract

Abstract In this paper, we are concerned with the parabolic–elliptic Keller–Segel system with a positive source term in a bounded domain in RN ${\mathbb{R}}^{N}$ ( N=2,3 $N=2,3$), 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.

Keywords

• Parabolic–elliptic Keller–Segel system
• Blow-up time
• Lower bounds
• Decay