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 utt+αδ2u - div(|δu|p-2δu) -∫∞0 μ(s)δ2u(t - s)ds - δut + f (u) = h defined in a bounded domain of 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. © 2013 American Institute of Physics.