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

M Afilal (1) , A. Guesmia (2, 3) , A Soufyane (4)

1 UCA - Université Cadi Ayyad [Marrakech]
2 KFUPM - King Fahd University of Petroleum and Minerals
3 IECL - Institut Élie Cartan de Lorraine
4 UoS - University of Sharjah

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.

Mots clés: Well-posedness; Semigroup theory; Frequency domain approach; Bresse system; Asymptotic behavior; Energy method; Heat conduction

Domaines: Equations aux dérivées partielles [math.AP]

M Afilal, A. Guesmia, A Soufyane. New Stability Results for a Linear Thermoelastic Bresse System with Second Sound. Applied Mathematics and Optimization, In press, ⟨10.1007/s00245-019-09560-7⟩. ⟨hal-02891554⟩