We are concerned with the Gierer–Meinhardt system with zero Neumann boundary condition:
where p>1, s>−1, are positive constants, are nonnegative smooth functions, Ω⊂Rd (d≥1) is a bounded smooth domain. We obtain new sufficient conditions for global existence and finite time blow-up of solutions, especially in the critical exponent cases: p−1=r and 62242c9790f7f83b22b24bcb2055776" title="Click to view the MathML source">qr=(p−1)(s+1).