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 ut=d1Δu−a1u+upvq+δ1(x,t),x∈Ω,t>0vt=d2Δv−a2v+urvs+δ2(x,t),x∈Ω,t>0∂u∂η=∂v∂η=0(oru=v=0),x∈∂Ω,t>0u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω.Here p>1, s>−1, d1,d2,q,r>0 and a1,a2≥0 are constants, while δ1(x,t) and δ2(x,t) are nonnegative continuous functions.