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

Abstract

This paper is concerned with a class of Kirchhoff models with memory effects \documentclass[12pt]{minimal}\begin{document}$u_{tt}\break + \alpha \Delta ^2 u - \mbox{div}(\vert \nabla u \vert ^{p-2}\nabla u) - \int _{0}^{\infty } \mu (s) \Delta ^{2}u(t-s)ds - \Delta u_t + f(u) = h,$\end{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 \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^N$\end{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.