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Large time asymptotic behavior for the dissipative Timoshenko system and its application

Wenhui Chen

TL;DR

The paper analyzes the dissipative Timoshenko system on $\\mathbb{R}$, focusing on large-time behavior of the transversal displacement $w$ and rotation angle $\\psi$. It introduces a Fourier-space, fourth-order reduction to capture precise growth rates, revealing $\\|w(t)\\|_{L^2} \sim t^{3/4}$ and $\\|\\psi(t)\\|_{L^2} \sim t^{1/4}$ under a nonzero $P_{w_1}$, and diffusion-plate asymptotic profiles $w^{pf},\\psi^{pf}$ that describe the asymptotics via the diffusion-plate kernel $G(t,x)$. The analysis uncovers a hidden cancellation in the shear stress $\\partial_x w - \\psi$ and provides refined small-frequency behavior, linking linear growth to explicit asymptotic structures. As an application, the authors prove global-in-time existence for the semilinear dissipative Timoshenko system with a power nonlinearity when the exponent exceeds the Fujita threshold $p>3$ in the equal-speed case, advancing understanding of nonlinear dynamics and partially answering an open question in Racke-Said-Houari (2013).

Abstract

In this paper, we study large time behavior for the dissipative Timoshenko system in the whole space $\mathbb{R}$, particularly, on the transversal displacement $w$ and the rotation angle $ψ$ of the filament for the beam. Different from decay properties of the energy term derived by Ide-Haramoto-Kawashima (2008), we discover new optimal growth $L^2$ estimates for the solutions themselves. Under the non-trivial mean condition on the initial data $w_1$, the unknowns $w$ and $ψ$ grow polynomially with the optimal rates $t^{3/4}$ and $t^{1/4}$, respectively, as large time. Furthermore, asymptotic profiles of them are introduced by the diffusion plate function, which explains a hidden cancellation mechanism in the shear stress $\partial_x w-ψ$. As an application of our results, we study the semilinear dissipative Timoshenko system with a power nonlinearity. Precisely, if the power is greater than the Fujita exponent, then the global in time existence of Sobolev solution is proved for the case of equal wave speeds, which partly gives a positive answer to the open problem in Racke-Said-Houari (2013).

Large time asymptotic behavior for the dissipative Timoshenko system and its application

TL;DR

The paper analyzes the dissipative Timoshenko system on , focusing on large-time behavior of the transversal displacement and rotation angle . It introduces a Fourier-space, fourth-order reduction to capture precise growth rates, revealing and under a nonzero , and diffusion-plate asymptotic profiles that describe the asymptotics via the diffusion-plate kernel . The analysis uncovers a hidden cancellation in the shear stress and provides refined small-frequency behavior, linking linear growth to explicit asymptotic structures. As an application, the authors prove global-in-time existence for the semilinear dissipative Timoshenko system with a power nonlinearity when the exponent exceeds the Fujita threshold in the equal-speed case, advancing understanding of nonlinear dynamics and partially answering an open question in Racke-Said-Houari (2013).

Abstract

In this paper, we study large time behavior for the dissipative Timoshenko system in the whole space , particularly, on the transversal displacement and the rotation angle of the filament for the beam. Different from decay properties of the energy term derived by Ide-Haramoto-Kawashima (2008), we discover new optimal growth estimates for the solutions themselves. Under the non-trivial mean condition on the initial data , the unknowns and grow polynomially with the optimal rates and , respectively, as large time. Furthermore, asymptotic profiles of them are introduced by the diffusion plate function, which explains a hidden cancellation mechanism in the shear stress . As an application of our results, we study the semilinear dissipative Timoshenko system with a power nonlinearity. Precisely, if the power is greater than the Fujita exponent, then the global in time existence of Sobolev solution is proved for the case of equal wave speeds, which partly gives a positive answer to the open problem in Racke-Said-Houari (2013).
Paper Structure (20 sections, 8 theorems, 149 equations)

This paper contains 20 sections, 8 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Dissipative Timoshenko system
  3. Main purposes of this manuscript
  4. Notation
  5. Main results
  6. Large time behavior for the linear dissipative Timoshenko system
  7. Global in time existence for the semilinear dissipative Timoshenko system
  8. Asymptotic behavior of solutions in the Fourier space
  9. Reduction to fourth order differential models
  10. Asymptotic expansions of characteristic roots
  11. Representation of solutions via conjugate roots in the Fourier space
  12. The transversal displacement in the Fourier space
  13. The rotation angle in the Fourier space
  14. Large time asymptotic behavior of solutions in the $L^2$ norm
  15. Preliminary on optimal growth/decay estimates
  16. ...and 5 more sections

Key Result

Theorem 2.1

Suppose that the initial data $(w_0,w_1)\in\mathcal{D}_0^0\times \mathcal{D}_0^{-1}$ and $(\psi_0,\psi_1)\in \mathcal{D}_0^{-1}\times\mathcal{D}_0^{-2}$. Then, the transversal displacement $w$ to the dissipative Timoshenko system Dissipative-Timoshenko with the wave speed $a>1/2$ satisfies the follo for large time $t\gg1$, provided that $P_{w_1}\neq0$. Furthermore, assuming $w_1\in L^{1,2}$ as wel

Theorems & Definitions (22)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.3
  • Remark 2.7
  • ...and 12 more