Upper semicontinuity of global attractors for a viscoelastic equations with nonlinear density and memory effects

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations | ∂ t u | ρ ∂ t t u − Δ ∂ t t u − Δ u + ∫ 0 + ∞μ ( s ) Δ u ( t − s ) d s + f ( u ) = h , in a three‐dimensional space, for f growing up to the critical exponent and dependent on ρ  ∈ [0,4), as ρ →0 + . This equation models extensional vibrations of thin rods with nonlinear material density ϱ ( ∂ t u ) = | ∂ t u | ρ and presence of memory effects. This type of problems has been extensively studied by several authors; the existence of a global attractor with optimal regularity for each ρ  ∈ [0,4) were established only recently. The proof involves the optimal regularity of the attractors combined with Hausdorff's measure.