Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay

Roberto Triggiani,Jing Zhang

doi:10.3934/eect.2018008

Abstract

We consider a heat-plate interaction model where the 2-dimensional plate is subject to viscoelastic (strong) damping. Coupling occurs at the interface between the two media, where each components evolves. In this paper, we apply low, physically hinged boundary interface conditions, which involve the bending moment operator for the plate. We prove three main results: analyticity of the corresponding contraction semigroup on the natural energy space; sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point \begin{document}$\lambda = -1/ρ$\end{document} in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation.

Highlights

Fluid-structure interactions models in the physical dimensions d = 2, 3, have been the object of mathematical studies in the past several years, since the appearance of [16] which in turn followed [29, p.121]
The overall system provides a physical illustration of hyperbolic-parabolic coupling
The structure was taken at first to have static interface, a case justified to be appropriate under the assumption of small, rapid oscillations of the structure [16]; see [1],[2]-[8] [9], [10], [20], [21], [23], [26], [31] for a certainly non-exhaustive list of works

Summary

Introduction

Abstract results of the late 80s in [12, 13, 14, 15] on damped abstract equations may only provide a motivation as well as a direct suggestion of what basic dynamical property may arise in describing the overall coupled interaction model: this is, analyticity of the s.c. semigroup describing the overall coupled dynamics The abstract results of the late 80s [12, 13, 14, 15] on “strongly” damped second order abstract equations (stimulated by [11]) are not directly applicable to such interactive PDE- models with strong damping of the structure, as they concerned differential operators with homogeneous boundary conditions, while fluid/heat viscoelastic wave models are driven by highly coupled boundary conditions at the interface. A physical 2-dimensional visco-elastic plate coupled with heat at the interface

Γs ν ν

But then

Proof that λ

Let ω