Smooth attractor for a nonlinear thermoelastic diffusion thin plate based on Gurtin–Pipkin’s model

Article type: Research Article

Authors: Aouadi, Moncef^{a; *} | Miranville, Alain^b

Affiliations: [a] Ecole Nationale d’Ingénieurs de Bizerte, Université de Carthage, Menzel Abderrahmane, Tunisia. E-mail: moncef_aouadi@yahoo.fr | [b] Laboratoire de Mathématiques et Applications, Université de Poitiers, Chasseneuil Futuroscope Cedex, France. E-mail: miranv@mathlabo.univ-poitiers.fr

Correspondence: [*] Corresponding author. E-mail: moncef_aouadi@yahoo.fr.

Abstract: A nonlinear system of sixth-order evolution equations which takes into account the hereditary effects via Gurtin–Pipkin’s model and the rotational inertia is considered. The system describes the behavior of thermoelastic diffusion thin plates, recently derived by Aouadi [Applied Mathematics and Mechanics (English Edition) 36 (2015), 619–632], where the heat and diffusion fluxes depend on the past history of the temperature and diffusion gradients through memory kernels, respectively. We prove the existence and uniqueness of global solutions as well as the exponential stability of the linear system at a rate proportional to the rotational inertia parameter. The existence of a global attractor whose fractal dimension is finite is proved. Finally, a smoothness property of the attractor is established with respect to the rotational inertia parameter.

Keywords: thermoelastic diffusion plate, Gurtin–Pipkin’s model, exponential stability, global attractor

DOI: 10.3233/ASY-151330

Journal: Asymptotic Analysis, vol. 95, no. 1-2, pp. 129-160, 2015