Existence and upper semicontinuity of pullback attractors for non-autonomous p-Laplacian parabolic problems

We study the asymptotic behavior of parabolic p-Laplacian problems of the form∂uλ∂t(t)−div(Dλ(t)|∇uλ(t)|p−2∇uλ(t))+|uλ(t)|p−2uλ(t)=B(t,uλ(t)) in a bounded smooth domain Ω   in RnRn, where n⩾1n⩾1, p>2p>2, Dλ∈L∞([τ,T]×Ω)Dλ∈L∞([τ,T]×Ω) with 0<β⩽Dλ(t,x)⩽M0<β⩽Dλ(t,x)⩽M a.e. in [τ,T]×Ω[τ,T]×Ω, λ∈[0,∞)λ∈[0,∞) and for each λ∈[0,∞)λ∈[0,∞) we have |Dλ(s,x)−Dλ(t,x)|⩽Cλ|s−t|θλ|Dλ(s,x)−Dλ(t,x)|⩽Cλ|s−t|θλ for all x∈Ωx∈Ω, s,t∈[τ,T]s,t∈[τ,T] for some positive constants θλθλ and CλCλ. Moreover, Dλ→Dλ1Dλ→Dλ1 in L∞([τ,T]×Ω)L∞([τ,T]×Ω) as λ→λ1λ→λ1. We prove that for each λ∈[0,∞)λ∈[0,∞) the evolution process of this problem has a pullback attractor and we show that the family of pullback attractors behaves upper semicontinuously at λ1λ1.