Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents

We study the asymptotic behavior of a non-autonomous multivalued Cauchy problem of the form∂u∂t(t)−div(D(t)|∇u(t)|p(x)−2∇u(t))+|u(t)|p(x)−2u(t)+F(t,u(t))∋0 on a bounded smooth domain Ω in RnRn, n≥1n≥1 with a homogeneous Neumann boundary condition, where the exponent p(⋅)∈C(Ω‾) satisfies p−p− := min⁡p(x)>2min⁡p(x)>2. We prove the existence of a pullback attractor and study the asymptotic upper semicontinuity of the elements of the pullback attractor A={A(t):t∈R}A={A(t):t∈R} as t→∞t→∞ for the non-autonomous evolution inclusion in a Hilbert space H under the assumptions, amongst others, that F   is a measurable multifunction and D∈L∞([τ,T]×Ω)D∈L∞([τ,T]×Ω) is bounded above and below and is monotonically nonincreasing in time. The global existence of solutions is obtained through results of Papageorgiou and Papalini.