Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

Abstract

In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping.

Riesz Basis Property and Exponential Stability for One-Dimensional Thermoelastic System with Variable Coefficients

“…We therefore first transform the eigenvalue problem of (1.2) into a first order matrix differential equation and find out the explicit asymptotic expression of the matrix fundamental solutions by introducing an invertible matrix function and using the asymptotic technique for the first order matrix differential equation, referred to as the matrix operator pencil method (see, e.g., [22,23]). This method has been used to spectral analysis for system of coupled partial differential equations in [13,26]. However, there is a remarkable difference between [13,26] and the present one in the process of investigation, that is, the invertible function matrix defined in [26] either associates with the eigenvalues only or the spatial variables only [13], while in present paper, the invertible matrix function is associated with both eigenvalues and the spatial variable.…”

“…This method has been used to spectral analysis for system of coupled partial differential equations in [13,26]. However, there is a remarkable difference between [13,26] and the present one in the process of investigation, that is, the invertible function matrix defined in [26] either associates with the eigenvalues only or the spatial variables only [13], while in present paper, the invertible matrix function is associated with both eigenvalues and the spatial variable. This is caused by the fact that the eigenvalues in those systems discussed in [13,26] are of the same order, which is not true for thermoelastic systems (1.1) and (1.2).…”

Exponential stability for a thermoelastic Bresse system: Theoretical and numerical study

“…Theorem 1. Let U 0 ∈ ; then, there exists a unique solution U ∈ 𝐶 (R + , ) of problem ( 12)- (13).…”

“…𝜑(x, 0) = 𝜑 0 (x), 𝜓(x, 0) = 𝜓 0 (x), w(x, 0) = w 0 (x), 𝜃(x, 0) = 𝜃 0 (x), 𝜑 t (x, 0) = 𝜑 1 (x), 𝜓 t (x, 0) = 𝜓 1 (x), w t (x, 0) = w 1 (x), 𝜃 t (x, 0) = 𝜃 1 (x), ∀x ∈ (0, 1). 𝜑(0, t) = 𝜑(1, t) = 𝜓 x (0, t) = 𝜓 x (1, t) = 0, w x (0, t) = w x (1, t) = 𝜃(0, t) = 𝜃(1, t) = 0, ∀t ≥ 0, (13) where…”

Decay rates for elastic-thermoelastic star-shaped networks

“…In recent years the problems of control and stabilisation for thermoelastic systems have been studied intensively. We refer for instance to [22], [42] in the context of controllability, and to [14], [18], [23], [24], [35] for stabilization, among others.…”