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Nonlinear damping effects for the 2D Mindlin-Timoshenko system

Ahmed Bchatnia, Sabrine Chebbi, Makram Hamouda

Abstract

We study in this article the asymptotic behavior of the Mindlin-Timoshenko system subject to a nonlinear dissipation acting only on the equations of the rotation angles. First, we briefly recall the existence of the solution of this system. Then, we prove that the energy associated with the Mindlin-Timoshenko system fulfills a dissipation relationship showing that the energy is decreasing. Moreover, when the wave speeds are equal, we establish an explicit and general decay result for the energy.

Nonlinear damping effects for the 2D Mindlin-Timoshenko system

Abstract

We study in this article the asymptotic behavior of the Mindlin-Timoshenko system subject to a nonlinear dissipation acting only on the equations of the rotation angles. First, we briefly recall the existence of the solution of this system. Then, we prove that the energy associated with the Mindlin-Timoshenko system fulfills a dissipation relationship showing that the energy is decreasing. Moreover, when the wave speeds are equal, we establish an explicit and general decay result for the energy.

Paper Structure

This paper contains 11 sections, 13 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. General asymptotic behavior
  4. Dissipation of the energy
  5. Assumptions
  6. Technical Lemmas
  7. Proof of Proposition \ref{['lemmado']}
  8. Proof of Theorem \ref{['th2']}
  9. Examples
  10. Example 1
  11. Example 2

Key Result

Theorem 2.2

Assume that $(H_{0})$ is satisfied. Then, the operator $\mathcal{ A}+\mathcal{B}$ generates a continuous semi-group $(\mathcal{T}(t))_{t\geq 0}$ on $\mathcal{H}$. Moreover, for all initial data $U_{0}\in \mathcal{H}$, the Cauchy problem sst has a unique solution $U\in \mathcal{C}([0,\infty );\mathca

Theorems & Definitions (28)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 18 more