On a Timoshenko system with thermal coupling on both the bending moment and the shear force

Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling

“…Still, the main challenge encountered in our analysis is connected to the treatment of the full Dirichlet boundary conditions (1.5) which produce some "pointwise" boundary terms in the estimates. Such terms have been handled in [2] by means of a general observability inequality recently established in [3,21], combined with some localized estimates obtained by means of appropriate cut-off functions. This method heavily relies on the regularization of the temperature variables provided by the parabolicity of the heat equation, and thus cannot be applied to (1.2)-(1.5) due to the hyperbolic character of the Gurtin-Pipkin thermal law.…”

“…we obtain the so-called Timoshenko-Fourier system with full thermal coupling, whose stability properties have been recently studied in [2]. In that paper, for a wide range of boundary conditions including (1.4) and (1.5), it is shown that the associated solution semigroup is exponentially stable independently of the values of the structural parameters of the model.…”

“…The constants ρ 1 , ρ 2 > 0 are physical parameters of the model, while S and M stand for the shear force and the bending moment, respectively. When the beam is subject to an unknown temperature distribution, one may assume that S and M satisfy the constitutive laws (see [2])…”

Effects of history and heat models on the stability of thermoelastic Timoshenko systems

“…Again, exponential stability is given if and only if the EWS condition (1.2) is satisfied. For more recent results related to (1.4) we quote [2] and the references therein. A third type of damping effect is given by a history term, similar to the memory term in (1.3), resulting in the system…”

“…where we have α = 0 (without history) or α = 1 (with history), and τ = 0 (Fourier law) or τ > 0 (Cattaneo law). For the derivation of these possible dissipative hybrid models generated by (1.10), we refer to [2,3] where physical justifications are provided on thermo-(visco-)elasticity. The remaining constants are assumed to be positive constants, 11) and further conditions on the exponentially decaying kernel g will be specified later on.…”