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Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam

Tamara Fastovska, Dirk Langemann, Iryna Ryzhkova

Abstract

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damped part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove well-posedness of the corresponding PDE system in energy space and establish existence of a regular global attractor under certain conditions on nonlinearities and coefficients of the damped part only. Moreover, we study singular limits of the problem when $l\to 0$ or $l\to 0$ simultaneously with $k_i\to +\infty$ and perform numerical modelling for these processes.

Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam

Abstract

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damped part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove well-posedness of the corresponding PDE system in energy space and establish existence of a regular global attractor under certain conditions on nonlinearities and coefficients of the damped part only. Moreover, we study singular limits of the problem when or simultaneously with and perform numerical modelling for these processes.
Paper Structure (13 sections, 15 theorems, 198 equations, 13 figures)

This paper contains 13 sections, 15 theorems, 198 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Abstract formulation
  3. Spaces and notations
  4. Abstract formulation
  5. Well-posedness
  6. Existence of attractors.
  7. Gradient structure
  8. Asymptotic smoothness.
  9. Existence of attractors.
  10. Singular Limits on finite time intervals
  11. Singular limit $l\rightarrow 0$
  12. Singular limit $k_i\rightarrow \infty, \;l\rightarrow 0$
  13. Discussion

Key Result

Lemma 4.1

The operator $A$ is positive and self-adjoint. Moreover, and $D(A^{1/2})=H_d\subset H_v$.

Figures (13)

  • Figure 1: Composite Bresse beam.
  • Figure 2: Transversal displacement of the beam, cross-section $x=2$.
  • Figure 3: Transversal displacement of the beam, cross-section $x=6$.
  • Figure 4: Shear angle variation of the beam, cross-section $x=2$.
  • Figure 5: Shear angle variation of the beam, cross-section $x=6$.
  • ...and 8 more figures

Theorems & Definitions (28)

  • Lemma 4.1
  • Definition 5.1
  • Definition 5.2
  • Definition 5.3
  • Theorem 5.4: Well-posedness
  • proof
  • Theorem 5.5: ChuEllLa2002
  • Lemma 5.6
  • proof
  • Remark 1
  • ...and 18 more