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 ∂ u λ ∂ t ( t ) − div ( D λ ( t ) | ∇ u λ ( t ) | p − 2 ∇ u λ ( t ) ) + | u λ ( t ) | p − 2 u λ ( t ) = B ( t , u λ ( t ) ) in a bounded smooth domain Ω in R n , where n ⩾ 1 , p > 2 , D λ ∈ L ∞ ( [ τ , T ] × Ω ) with 0 < β ⩽ D λ ( t , x ) ⩽ M a.e. in [ τ , T ] × Ω , λ ∈ [ 0 , ∞ ) and for each λ ∈ [ 0 , ∞ ) we have | D λ ( s , x ) − D λ ( t , x ) | ⩽ C λ | s − t | θ λ for all x ∈ Ω , s , t ∈ [ τ , T ] for some positive constants θ λ and C λ . Moreover, D λ → D λ 1 in L ∞ ( [ τ , T ] × Ω ) as λ → λ 1 . We prove that for each 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 λ 1 .