Some results on the Gierer-Meinhardt model with critical exponent p-1=r

We consider the following problem with the critical exponent p - 1 = r {u(t) = d(1)Delta u - a(1)u + u(p)/u(r)(q) + delta(1)(x, t), x is an element of Omega, t>0, v(t)= d(2)Delta v - a(2)v + u(r)/v(s) + delta(2)(x,t), x is an element of Omega, t>0, partial derivative u/partial derivative eta = partial derivative v/partial derivative eta =0, x is an element of partial derivative Omega, t>0, u(x, 0) = u(0)(x), v(x,0) = v(0)(x), x is an element of Omega. Here q,r,d(1), d(2), a(1) and a(2) are positive constants, p > 1, s > -1, delta(1), delta(2), Up and vo are nonnegative smooth functions, Omega subset of R-d (d >= 1) is a bounded smooth domain. Whether d(1) not equal d(2) or d(1) = d(2), we establish the results on the finite-time blowup and global existence of the solution, which improves those of Theorem 1.2 in Li et al. (2017). (C) 2020 Elsevier Ltd. All rights reserved.