Long-time dynamics for a class of Kirchhoff models with memory

This paper is concerned with a class of Kirchhoff models with memory effects [12pt]{minimal}{document}uₜₜ + α Δ ² u - div( ∇ u ᵖ⁻²∇ u) - ∫ ₀∞ μ (s) Δ ²u(t-s)ds - Δ uₜ + f(u) = h,{document}utt+αΔ2u−div(|∇u|p−2∇u)−∫0∞μ(s)Δ2u(t−s)ds−Δut+f(u)=h, defined in a bounded domain of [12pt]{minimal}{document}RN{document}RN. This non-autonomous equation corresponds to a viscoelastic version of Kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations. Under assumptions that the exponent p and the growth of f(u) are up to the critical range, it turns out that the model corresponds to an autonomous dynamical system in a larger phase space, by adding a component which describes the relative displacement history. Then the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.