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:{ut=d1Δu−a1u+upvq+δ1(x),x∈Ω,t>0,vt=d2Δv−a2v+urvs+δ2(x),x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, where p>1p>1, s>−1s>−1, q,r,d1,d2,a1,a2 are positive constants, δ1,δ2,u0,v0 are nonnegative smooth functions, Ω⊂RdΩ⊂Rd (d≥1d≥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=rp−1=r and qr=(p−1)(s+1)qr=(p−1)(s+1).