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

Abstract

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations urn:x-wiley:mma:media:mma5389:mma5389-math-0001 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(∂tu) = |∂tu|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.