[Paper Review] Robust Exponential Attractors for Coleman--Gurtin Equations with Dynamic Boundary Conditions Possessing Memory

Joseph L. Shomberg|arXiv (Cornell University)|Feb 3, 2016

Stability and Controllability of Differential Equations20 references3 citations

TL;DR

This paper establishes the existence of robust, Hölder continuous exponential attractors for a Coleman–Gurtin heat equation with memory and dynamic boundary conditions. By analyzing a singularly perturbed memory kernel, the authors prove exponential attraction to these finite-dimensional sets across the entire phase space, ensuring the existence of global attractors and robustness under perturbation.

ABSTRACT

The well-posedness of a generalized Coleman--Gurtin equation 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 robust/Hölder 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.

Motivation & Objective

• To analyze the asymptotic behavior of a Coleman–Gurtin equation with memory and dynamic boundary conditions.
• To establish the existence of exponential attractors that are robust with respect to a perturbation parameter in the memory kernel.
• To prove that the basin of attraction of these exponential attractors is the entire phase space, implying the existence of global attractors.
• To ensure Hölder continuity of the exponential attractors as the perturbation parameter varies in [0,1].
• To extend the theory of dissipative systems to models with memory on both the interior and boundary, under matching memory responses.

Proposed method

• The model is formulated as a parabolic-type PDE with memory terms in the bulk and on the boundary, governed by a kernel $ k(s) $ satisfying $ \int_0^\infty k(s)ds = 1 $.
• A singular perturbation is introduced via $ k_\varepsilon(s) = \frac{1}{\varepsilon}k(\frac{s}{\varepsilon}) $, which approaches the Dirac delta as $ \varepsilon \to 0 $, recovering the standard Coleman–Gurtin system.
• The analysis is conducted in a Hilbert phase space $ \mathcal{H}^0_\varepsilon $, incorporating the state variables and memory history.
• A priori estimates and energy methods are used to derive uniform bounds in the phase space and memory space, ensuring dissipativity.
• The transitivity property of exponential attraction is applied to construct exponential attractors via a Lyapunov-type functional and Gronwall-type inequalities.
• Robustness is proven by estimating the difference between solutions for different $ \varepsilon $, showing Hölder continuity of the attractors in the Hausdorff metric.

Experimental results

Research questions

• RQ1 Does the Coleman–Gurtin equation with memory and dynamic boundary conditions admit a finite-dimensional exponential attractor?
• RQ2 Can such an exponential attractor be shown to be robust under perturbation of the memory kernel parameter $ \varepsilon \in (0,1] $?
• RQ3 Is the basin of attraction of the exponential attractor the entire phase space, implying the existence of a global attractor?
• RQ4 How does the memory response in the interior match the boundary response to ensure well-posedness and long-time dynamics?
• RQ5 What is the regularity and continuity of the exponential attractors as $ \varepsilon \to 0 $, particularly Hölder continuity?

Key findings

• A family of exponential attractors $ (\mathbb{M}_\varepsilon)_{\varepsilon \in [0,1]} $ exists for the semigroup generated by the solutions of the perturbed system.
• The exponential attractors are robust, meaning they depend Hölder continuously on the perturbation parameter $ \varepsilon \in [0,1] $, ensuring stability under small changes in the memory kernel.
• The basin of attraction of the exponential attractors is the entire phase space $ \mathcal{H}^0_\varepsilon $, which implies the existence of a global attractor.
• The global attractor is finite-dimensional, as it is contained within the finite-dimensional exponential attractor.
• The difference between solutions for $ \varepsilon_1 $ and $ \varepsilon_2 $ is bounded by $ C \frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_2} Q(R_1) $, which decays as $ \varepsilon_1 \to \varepsilon_2 $, confirming robustness.
• The results are obtained under standard dissipation assumptions on the nonlinearities $ f(u) $ and $ g(u) $, and a crucial matching condition between interior and boundary memory responses.