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Global attractors for the Signorini problem with pointwise damping

Jaime E. Muñoz Rivera, Maria Grazia Naso

TL;DR

This work develops and analyzes a hybrid PDE–ODE formulation for the Signorini problem in a Timoshenko beam with pointwise damping, showing exponential decay to zero and the existence of compact global attractors. By comparing the hybrid model to the nonhybrid Timoshenko system and employing observability inequalities, it establishes stability properties that persist as the coupling parameter tends to zero, thereby recovering the Signorini problem. The authors prove well-posedness, energy dissipation, and attractor existence for both the linear hybrid model and nonlinear variants with normal compliance, and they extend these results to the variational inequality setting via robust limit arguments. The results provide a rigorous framework for understanding long-time behavior and attractor structure in contact problems with unilateral constraints and localized damping, with potential implications for engineering applications involving beam contact and frictional interfaces.

Abstract

The existence of global attractors is investigated for the Signorini problem with pointwise dissipation. It is shown that both the semilinear Signorini problem and the elastic obstacle problem with normal compliance exhibit exponential decay to zero and admit compact global attractors. To establish these results, the original problem is approximated by a hybrid PDE-ODE system, which allows for a rigorous analysis of well-posedness and the long-time behavior of its solutions.

Global attractors for the Signorini problem with pointwise damping

TL;DR

This work develops and analyzes a hybrid PDE–ODE formulation for the Signorini problem in a Timoshenko beam with pointwise damping, showing exponential decay to zero and the existence of compact global attractors. By comparing the hybrid model to the nonhybrid Timoshenko system and employing observability inequalities, it establishes stability properties that persist as the coupling parameter tends to zero, thereby recovering the Signorini problem. The authors prove well-posedness, energy dissipation, and attractor existence for both the linear hybrid model and nonlinear variants with normal compliance, and they extend these results to the variational inequality setting via robust limit arguments. The results provide a rigorous framework for understanding long-time behavior and attractor structure in contact problems with unilateral constraints and localized damping, with potential implications for engineering applications involving beam contact and frictional interfaces.

Abstract

The existence of global attractors is investigated for the Signorini problem with pointwise dissipation. It is shown that both the semilinear Signorini problem and the elastic obstacle problem with normal compliance exhibit exponential decay to zero and admit compact global attractors. To establish these results, the original problem is approximated by a hybrid PDE-ODE system, which allows for a rigorous analysis of well-posedness and the long-time behavior of its solutions.
Paper Structure (7 sections, 16 theorems, 147 equations, 1 figure)

This paper contains 7 sections, 16 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Semigroup and General Results
  3. The Hybrid Model
  4. The $C_0$ semigroup of contractions
  5. Comparison between the hybrid and the non hybrid model
  6. Applications to Contact Problem with normal compliance
  7. Signorini problem

Key Result

Theorem 2.1

Let $\{T(t)\}_{t \geq 0}$ be a $C_0$-semigroup of contractions, exponentially stable, with infinitesimal generator $\mathbb{A}$ on a Hilbert space $\mathcal{H}$. Let $\mathcal{F} : \mathcal{H} \to \mathcal{H}$ be a locally Lipschitz continuous function satisfying conditions ff0, ff1 and ff2 . Then, which decays exponentially to zero as $t \to \infty$. Moreover if $U_0 \in D(\mathcal{A})$ then the

Figures (1)

  • Figure 1: Beam subjected to a constraint at the free end $x=\ell$.

Theorems & Definitions (32)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • ...and 22 more