Electron. J. Diff. Equ., Vol. 2016 (2016), No. 47, pp. 1-35.

Robust exponential attractors for Coleman-Gurtin equations with dynamic boundary conditions possessing memory

Joseph L. Shomberg

Well-posedness of generalized Coleman-Gurtin equations equipped with dynamic boundary conditions with memory was recently established by the author with C. G. Gal. In this article we report advances concerning the asymptotic behavior and stability of this heat transfer model. For the model under consideration, we obtain a family of exponential attractors that is robustHolder continuous with respect to a perturbation parameter occurring in a singularly perturbed memory kernel. We show that the basin of attraction of these exponential attractors is the entire phase space. The existence of (finite dimensional) global attractors follows. The results are obtained by assuming the nonlinear terms defined on the interior of the domain and on the boundary satisfy standard dissipation assumptions. Also, we work under a crucial assumption that dictates the memory response in the interior of the domain matches that on the boundary.

Submitted August 15, 2015. Published February 10, 2016.
Math Subject Classifications: 35B40, 35B41, 45K05, 35Q79.
Key Words: Coleman-Gurtin equation; dynamic boundary conditions; memory relaxation; exponential attractor; basin of attraction; global attractor; finite dimensional dynamics; robustness.

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Joseph L. Shomberg
Department of Mathematics and Computer Science
Providence College, Providence, RI 02918, USA
email: jshomber@providence.edu

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