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Uniform Decaying Property of Solutions for Anisotropic Viscoelastic Systems

Maarten V. de Hoop, Ching-Lung Lin, Gen Nakamura

TL;DR

This work addresses the uniform decay properties of solutions to an anisotropic viscoelastic system with memory (VID system) under mixed-type boundary conditions. By refining Rivera-Lapa’s approach and modifying the Russell principle for a non-time-reversible setting, the authors develop a unified energy-method framework that yields UDP in two regimes: polynomial decay $(1+t)^{-(2^m-1)}$ for polynomial relaxation and exponential decay for exponentially decaying kernels, with decay rates independent of initial data. The polynomial result requires polynomial decay assumptions on the relaxation tensor $G(x,t)$ and strong convexity, while the exponential result handles general $G(x,t)$ with exponential decay, both under mixed boundary conditions that couple a traction term with a dissipation. The findings advance exact boundary controllability analyses for viscoelastic systems and provide robust decay guarantees applicable to practical devices modeling viscoelastic behavior.

Abstract

The paper concerns about the uniform decaying property (abbreviated by UDP) of solutions for an anisotropic viscoelastic system in the form of integrodifferential system (abbreviated by VID system) with mixed type boundary condition. The mixed type condition consists of the homogeneous displacement boundary condition and a homogeneous traction boundary condition or with a dissipation. By using a dissipative structure of this system, we will prove the UDP in a unified way for the two cases, which are, when the time derivative of relaxation tensor decays with polynomial order and it decays with exponential order.

Uniform Decaying Property of Solutions for Anisotropic Viscoelastic Systems

TL;DR

This work addresses the uniform decay properties of solutions to an anisotropic viscoelastic system with memory (VID system) under mixed-type boundary conditions. By refining Rivera-Lapa’s approach and modifying the Russell principle for a non-time-reversible setting, the authors develop a unified energy-method framework that yields UDP in two regimes: polynomial decay for polynomial relaxation and exponential decay for exponentially decaying kernels, with decay rates independent of initial data. The polynomial result requires polynomial decay assumptions on the relaxation tensor and strong convexity, while the exponential result handles general with exponential decay, both under mixed boundary conditions that couple a traction term with a dissipation. The findings advance exact boundary controllability analyses for viscoelastic systems and provide robust decay guarantees applicable to practical devices modeling viscoelastic behavior.

Abstract

The paper concerns about the uniform decaying property (abbreviated by UDP) of solutions for an anisotropic viscoelastic system in the form of integrodifferential system (abbreviated by VID system) with mixed type boundary condition. The mixed type condition consists of the homogeneous displacement boundary condition and a homogeneous traction boundary condition or with a dissipation. By using a dissipative structure of this system, we will prove the UDP in a unified way for the two cases, which are, when the time derivative of relaxation tensor decays with polynomial order and it decays with exponential order.
Paper Structure (18 sections, 17 theorems, 140 equations)

This paper contains 18 sections, 17 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Prior works
  4. Novelty of the results
  5. Organization of this paper
  6. Outline of the proof of Theorem \ref{['mainresult']} and preliminaries
  7. Basic identities and inequalities for (MP)'
  8. Basic identities
  9. Basic inequalities
  10. Estimates for $\mathcal{L}(t,u)$
  11. Upper and lower bounds
  12. Estimating the time derivative from above
  13. Proof of Theorem \ref{['mainresult2']}
  14. Conclusion
  15. Examples of $G$
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any given initial data $(f_1,f_2)\in H_+\times H_+$, the solution $u\in C^2([0,\infty); H^1(\Omega))$ of mixed prob with the property $\partial_t^3 u\in C([0,T]; L^2(\Omega))$, $\partial_t^4 u\in C([0,\infty); H_-)$ converges to zero at polynomial rate in time $t$ as $t\rightarrow\infty$ and thi

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 8 more