Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system

We are concerned with the Gierer-Meinhardt system with zero Neumann boundary condition: {u(t) = d(1)Delta u - a(1)u + u(p)/v(q) + delta(1)(x), x is an element of Omega, t > 0, v(t) = d(2)Delta v - a(2)v + u(r)/v(s) + delta(2)(x), x is an element of Omega, t > 0, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), x is an element of Omega, where p > 1, s > -1, q, r, d(1), d(2), a(1), a(2) are positive constants, delta(1), delta(2), u(0), v(0) are nonnegative smooth functions, Omega subset of R-d (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 qr = (p - 1)(s +1). (C) 2016 Elsevier Inc. All rights reserved.