Upper semicontinuity of the global attractor for the Gierer–Meinhardt model

We are concerned with the following Gierer–Meinhardt model on a bounded domain with smooth boundary ∂Ω which is a biological pattern formation model proposed by A. Gierer and H. Meinhardt where ν, μ, σ and ρ are small positive constants. We also consider the so-called shadow system (SS) of (GM) and another reduced equation (RE) which is obtained by taking σ=0 in (SS). Our framework is a functional space , where X=L2(Ω). After we see that each of the systems (GM), (SS) and (RE) generates a global semiflow on , and Xα, respectively, we will prove the existence of global attractors , and of (GM), (SS) and (RE), respectively, Moreover, we will prove the upper semicontinuity of at ρ=0 and at σ=0.