Global Existence of Solutions of an Activator-inhibitor System

We consider the generalized Gierer-Meinhardt system ><>: ∂uj ∂t = dj4uj − ajuj + gj (x, u) in Ω× [0, T ) , ∂uj ∂ν = 0 on ∂Ω× [0, T ) , uj (x, 0) = φj (x) in Ω where Ω is a smooth bounded domain in Rn with ν its unit outer normal, j = 1, 2, u = (u1, u2) and <: g1 (x, u) = ρ1 (x, u) u p 1 u q 2 + σ1 (x) , g2 (x, u) = ρ2 (x, u) u1 u2 + σ2 (x) . Here dj , aj are positive constants, ρ1 ≥ 0, ρ2 > 0, σj ≥ 0 are bounded smooth functions and p, q, r, s are nonnegative constants satisfying 0 < p− 1 r < q s + 1 . We show that there is a unique global solution when p−1 < r, which improves 1987 result of K. Masuda and K. Takahashi [10]. Asymptotic bounds of global solutions are also established which yield new a priori estimates of stationary solutions.