New Stability Results for a Linear Thermoelastic Bresse System with Second Sound

Abstract

In this paper, we consider a linear one-dimensional thermoelastic Bresse system with second sound consisting of three hyperbolic equations and two parabolic equations coupled in a certain manner under mixed homogeneous Dirichlet-Neumann boundary conditions, where the heat conduction is given by Cattaneo's law. Only the longitudinal displacement is damped via the dissipation from the two parabolic equations, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, non exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. Our method of proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.

General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law

“…Then, multiplying by W (n) and q (n) in L 2 (0, ), we obtain (a). Analogously, using equation (3.12) and Lemma 3.3 we have 1 iβn ϑ…”

“…For a brief and most comprehensive survey on Bresse systems with thermal damping we refer the reader to the Introduction of [2]. Additionally to that survey, we can mention the recent papers [1] and [6]. In [1], the authors studied a variant of (1.10) for different mixed boundary conditions.…”

“…Additionally to that survey, we can mention the recent papers [1] and [6]. In [1], the authors studied a variant of (1.10) for different mixed boundary conditions. In such variant, comparing with the model described above, the corresponding coupling terms are different: the term θ in the first equation and the term ϕ t in the coupling of the heat equation were omitted.…”

Study on the stability of thermoelastic Bresse and Timoshenko type systems with Gurtin-Pipkin's law via the vertical displacements

“…the exponential and polynomially stability of (1.7)-(1.8) with (1.11) and (1.14) were proved in [1]. The subject of [17] was the study of the exponential and polynomially stability of (1.7) with (1.10) in both cases (1.8) and (1.9), where the thermoelastic effect is effecive on the longitudinal displacements:…”

On the exponential and polynomial stability for a linear Bresse system

“…For the last few years, many researchers studied the well‐posedness and the stability of Bresse systems . Under different types of controls F i , various stability results have been yet obtained depending on the nature and the number of controls, the regularity of initial data, and the following parameters: for this purpose, we refer the reader to previous studies in case of (local or global, linear or nonlinear) frictional dampings and De Lima Santos et al, Guesmia and Kafini, and Guesmia in case of memories. In some papers, it was proved that, when each equation of is directly damped, that is, F 1 F 2 F 3 ≠0, the stability of holds regardless to s 1 , s 2 and s 3 .…”