Finite-time blow-up and blow-up rates for the Gierer–Meinhardt system

Abstract

In this paper, we consider the Gierer–Meinhardt system (1.1), shown below, on a bounded smooth domain () with a homogeneous Neumann boundary condition. Under suitable conditions on the exponents , , , and , we establish sufficient conditions for finite-time blow-up and obtain blow-up rates for blow-up solutions. This work is a continuation of our earlier result in this direction. The Gierer–Meinhardt system was introduced in to model activator–inhibitor systems in pattern formation in ecological systems.

Keywords:

• finite-time blow-up
• blow-up rate
• Gierer–Meinhardt system
• reaction–diffusion system

AMS Subject Classifications:

• Primary 35K51
• 35K57
• 35K58

Notes

1 Bold face letters are used to denote vectors and all relations involving vectors are understood in the component-wise sense.