A note on pp-Laplacian parabolic problems in RnRn

In this work we study the asymptotic behavior of parabolic pp-Laplacian problems of the form ∂uλ∂t−div(Dλ|∇uλ|p−2∇uλ)+a|uλ|p−2uλ=B(uλ) in L2(Rn)L2(Rn), where n≥1n≥1, p>2p>2, Dλ∈L∞(Rn)Dλ∈L∞(Rn), ∞>M≥Dλ(x)≥σ>0∞>M≥Dλ(x)≥σ>0 a.e. in Rn,λ∈[0,∞)Rn,λ∈[0,∞), B:L2(Rn)→L2(Rn)B:L2(Rn)→L2(Rn) is a globally Lipschitz map and a:Rn→Ra:Rn→R is a non-negative continuous function. We prove, under suitable assumptions on aa, the existence of a global attractor in L2(Rn)L2(Rn) for each positive finite diffusion coefficient and we show that the family of attractors behaves upper semicontinuously with respect to positive finite diffusion parameters.