Attractors for a class of Kirchhoff models with \(p\)-Laplacian and time delay (Q2316543)

The author examines the long term behavior, in the sense of global attractors, of a class of Kirchhoff equations with time delay and a \(p\)-Laplacian type perturbation; hence, \[ u_{tt}(x,t)+\Delta^2u(x,t)-\Delta_pu(x,t)-a_0\Delta u_t(x,t)+a_1u_t(x,t-\tau)+f(u(x,t)) = g(x), \] where \(\Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the usual \(p\)-Laplacian operator. The main result is the existence of a finite dimensional global attractor. The global attractor for the associated flow and its finite-dimensionality follows from the method of quasi-stable dynamical systems in [\textit{I. Chueshov} and \textit{I. Lasiecka}, Von Karman evolution equations. Well-posedness and long-time dynamics. New York, NY: Springer (2010; Zbl 1298.35001)].