Global existence of solutions of an activator-inhibitor system

We consider the generalized Gierer-Meinhardt system∂ uⱼ∂ t=dⱼΔ uⱼ- aⱼuⱼ+gⱼ(x,u) \ \ in\ \ Ω×[0,T) ,∂ uⱼ∂ν=0\ \ on\ \ ∂Ω×[0,T),uⱼ(x,0) =φⱼ(x)\ \ in\ \ Ωwhere Ω is a smooth bounded domain in Rⁿwith ν its unit outer normal, $j=1,2$, u=(u₁,u₂) andg₁(x,u) =ρ₁(x,u) u₁ᵖu₂q+σ₁(x) ,g₂(x,u) =ρ₂(x,u) u₁ʳu₂ˢ+σ₂(x).Here dⱼ, aⱼ are positive constants, ρ₁≥0, ρ₂>0,σⱼ≥0 are bounded smooth functions and $p,q,r,s$arenonnegative constants satisfying $0We show that there is a unique global solution when p-1 < r, whichimproves 1987 result of K. Masuda and K. Takahashi[10]. Asymptotic bounds of global solutions are alsoestablished which yield new a priori estimates of stationarysolutions.