Attractors for a Class of Kirchhoff Models with $p$-Laplacian and Time Delay

Abstract

This paper is concerned with a class of Kirchhoff models with time delay and perturbation of $p$-Laplacian type \[ u_{tt}(x,t) + \Delta^2 u(x,t) - \Delta_p u(x,t) - a_0 \Delta u_t(x,t) + a_1 u_t(x,t-\tau) + f(u(x,t)) = g(x), \] where $\Delta_p u = \operatorname{div}(|\nabla u|^{p-2} \nabla u)$ is the usual $p$-Laplacian operator. Many researchers have studied well-posedness and decay rates of energy for these equations without delay effects. But, there are not many studies on attractors for other delayed systems. Thus we establish the existence of global attractors and the finite dimensionality of the attractors by establishing some functionals which are related to the norm of the phase space to our problem.