The explicit solution to the initial boundary value problem of Gierer-Meinhardt model

Using the solutions of an elliptic system and an ODE system, under certain conditions, we get the explicit solution to the following initial boundary value problem of Gierer-Meinhardt model {u(t) = d(1)Delta u - a(1)u + u(p)/v(q) + delta(1)(x,t), x is an element of Omega, t > 0 v(t) = d(2)Delta v - a(2)u + 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 (or u = v = 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 p > 1, s > -1, d(1), d(2), q, r > 0 and a(1), a(2) >= 0 are constants, while delta(1)(x,t) and delta(2)(x, t) are nonnegative continuous functions. (C) 2018 Elsevier Ltd. All rights reserved.