Asymptotic Dynamics and Singular Limit of Global Attractors for Thermoelastic Lamé Systems with Memory

Abstract

We investigate the long-time dynamics of a thermoelastic Lamé system with memory, where heat conduction is governed by a Gurtin–Pipkin type law featuring a fading memory kernel. The model incorporates a viscoelastic relaxation term characterized by a small parameter $\omega >0$ , representing the thermal relaxation time. Our main goal is to analyze the asymptotic behavior of the global attractors as $\omega \to 0$ . We first establish the well-posedness of the system and prove the existence of a finite-dimensional global attractor using the quasi-stability method. We then show that, as the memory kernel is rescaled and the relaxation parameter vanishes, the global attractors converge in the sense of upper semicontinuity to the global attractor of the limiting thermoelastic Lamé system governed by Fourier’s law. Our analysis overcomes the challenges posed by the lack of strong compactness induced by the memory term and provides uniform in $\omega$ bounds for trajectories on the attractor. These results contribute to the understanding of singular perturbation problems in thermoelasticity with memory and establish robust convergence properties of the associated attractors.