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

We are concerned with the following Gierer–Meinhardt model on a bounded domain Ω⊂RN(N∈{1,2,3}) with smooth boundary ∂Ω∂Ω which is a biological pattern formation model proposed by A. Gierer and H. Meinhardtequation(GM)ut=ν2Δu-u+upvq+μinΩ×R+,σvt=1ρΔv-v+urvsinΩ×R+,∂u∂n=∂v∂n=0on∂Ω×R+,where νν, μμ, σσ and ρρ 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=0σ=0 in (SS). Our framework is a functional space Xα34<α<1, where X=L2(Ω)X=L2(Ω). After we see that each of the systems (GM), (SS) and (RE) generates a global semiflow on X≔Xα⊕XαX≔Xα⊕Xα, Y≔Xα⊕R+Y≔Xα⊕R+ and XαXα, respectively, we will prove the existence of global attractors AX,σ,ρAX,σ,ρ, AY,σAY,σ and AXαAXα of (GM), (SS) and (RE), respectively, Moreover, we will prove the upper semicontinuity of AX,σ,ρAX,σ,ρ at ρ=0ρ=0 and AY,σAY,σ at σ=0σ=0.