Long-time dynamics of N-dimensional structure equations with thermal memory

This paper is concerned with the long-time behavior for a class of N-dimensional thermoelastic coupled structure equations with structural damping and past history thermal memory u(tt) + Delta(2)u + nu Delta theta + Delta(2)u(t) - [sigma(integral(Omega)(del u)(2) dx) + phi (integral(Omega) del u del u(t) dx)Delta u + f(1)(u) = q(1)(x), in Omega x R+, theta(t) - iota Delta theta - (1 - iota) integral(infinity)(0) k(s)Delta theta(t - s)ds - nu Delta u(t) + f(2)(theta) = q(2)(x), with 0 <= iota < 1. This system arises from a model of the nonlinear thermoelastic coupled vibration structure with the clamped ends for simultaneously considering the medium damping, the viscous effect and the nonlinear constitutive relation and thermoelasticity based on a theory of non-Fourier heat flux laws. By considering the case where the internal (structural) damping is present, for 0 <= iota < 1, we show that the thermal source term f(2)(theta) is crucial to stabilizing the system and guarantees the existence of a global attractor for the above mentioned system in the present method.