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