Abstract
In this paper, we consider the thermoelastic Bresse system in one-dimensional bounded interval under mixed homogeneous Dirichlet–Neumann boundary conditions and two different kinds of dissipation working only on the longitudinal displacement and given by heat conduction of types I and III. We prove that the exponential stability of the two systems is equivalent to the equality of the three speeds of the wave propagations. Moreover, when at least two speeds of the wave propagations are different, we show the polynomial stability for each system with a decay rate depending on the smoothness of the initial data. The results of this paper complete the ones of Afilal et al. [On the uniform stability for a linear thermoelastic Bresse system with second sound (submitted), 2018], where the dissipation is given by a linear frictional damping or by the heat conduction of second sound. The proof of our results is based on the semigroup theory and a combination of the energy method and the frequency domain approach.
Highlights
We study in this paper the asymptotic behavior at infinity of the solutions of two coupled systems related to the Bresse model with two different types of dissipation given by heat conduction and working only on the longitudinal displacement
We show that the polynomial stability holds in general with two decay rates corresponding to the two cases, ρ1b − ρ2k = 0 and ρ1b − ρ2k = 0
Estimates on φnx, φ∼n and λnφn First, taking the inner product of with i λnw∼n in L2 (0, 1), integrating by parts and using the boundary conditions, we have, i λnw∼n
Summary
We study in this paper the asymptotic behavior at infinity of the solutions of two coupled systems related to the Bresse model with two different types of dissipation given by heat conduction and working only on the longitudinal displacement. The authors of [9] proved that the following thermoelastic Bresse system. In (0, 1) × (0, ∞) , in (0, 1) × (0, ∞) , where g : R+ → R+ is a given function satisfying some hypotheses He provided a necessary and sufficient condition for exponential stability in terms of the structural parameters of the problem. The authors of [1] considered the Cattaneo heat conduction working only on the longitudinal displacement. 3 and 4, we prove, respectively, our exponential and polynomial stability results
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
