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

TL;DR

This paper investigates the large time behavior of the dissipative Timoshenko system, revealing new growth estimates, asymptotic profiles, and establishing global existence results for certain nonlinear cases, advancing understanding of beam dynamics.

Contribution

It introduces optimal polynomial growth estimates for solutions, identifies asymptotic profiles via diffusion functions, and proves global existence for nonlinear systems with super-Fujita power.

Findings

Solutions grow polynomially with rates t^{3/4} for w and t^{1/4} for ψ.

Asymptotic profiles are characterized by diffusion plate functions.

Global existence is established for nonlinear cases with power exceeding the Fujita exponent.

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 $\psi$ 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 $\psi$ 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 $\partial_x w-\psi$. 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).