Global attractors for Moore–Gibson–Thompson thermoelastic extensible beams and Berger plates

Abstract

For $p\in R$, and $\alpha ,\beta ,\gamma ,\delta ,\eta >0$, we consider an abstract version of the system $\left(\begin{array}{c}{u}_{tt}+{\Delta }^{2}u-\left(p+{\Vert \nabla u\Vert }^2\right)\Delta u=\eta \Delta ({\varphi }_{tt}+\alpha {\varphi }_t)+f\\ {\varphi }_{ttt}+\alpha {\varphi }_{tt}-\beta \Delta {\varphi }_t-\gamma \Delta \varphi -\delta \Delta {\varphi }_{tt}=-\eta \Delta u_t\end{array}\right)$describing the dynamics of thermoelastic extensible beams or Berger plates, where the evolution of the temperature is ruled by a regularized Moore–Gibson–Thompson type equation. The existence of a global attractor of optimal regularity is proved.