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

Abstract

Abstract 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 t t + △ 2 u + ν △ θ + △ 2 u t − [ σ ( ∫ Ω ( ∇ u ) 2 d x ) + ϕ ( ∫ Ω ∇ u ∇ u t d x ) ] △ u + f 1 ( u ) = q 1 ( x ) , in Ω × R + , θ t − ι △ θ − ( 1 − ι ) ∫ 0 ∞ k ( s ) △ θ ( t − s ) d s − ν △ u t + f 2 ( θ ) = q 2 ( x ) , with 0 ≤ ι < 1 . $$\begin{gathered} u_{tt}+\triangle^{2}u+\nu \triangle\theta+\triangle^{2}u_{t}-\biggl[\sigma \biggl( \int_{\Omega}(\nabla u)^{2}\,dx\biggr)+\phi\biggl( \int_{\Omega}\nabla u\nabla u_{t}\,dx\biggr)\biggr] \triangle u+f_{1}(u) \\ \quad=q_{1}(x),\quad \mbox{in }\Omega\times R^{+}, \\ \theta_{t}-\iota\triangle\theta-(1-\iota) \int_{0}^{\infty }k(s)\triangle\theta(t-s)\,ds-\nu \triangle u_{t}+f_{2}(\theta )=q_{2}(x),\quad \mbox{with } 0\leq\iota< 1. \end{gathered} $$ 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 ≤ ι < 1 $0\leq\iota<1$ , we show that the thermal source term f 2 ( θ ) $f_{2}(\theta)$ is crucial to stabilizing the system and guarantees the existence of a global attractor for the above mentioned system in the present method.

Keywords

• thermoelastic coupled structure
• internal (structural) damping
• thermal memory
• asymptotically smooth
• global attractor