The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the vertical displacement

Abstract

In this paper, we study the energy decay for two one-dimensional thermoelastic Bresse-type systems in a bounded open interval under mixed homogeneous Dirichlet–Neumann boundary conditions and with two different kinds of dissipation working only on the vertical displacement and given by heat conduction of types I and III. The two systems are consisting of three wave equations (Bresse-type system) coupled, in a certain manner, with one heat equation (type I) or with one wave equation (type III). We prove that, independently of the values of the coefficients, these systems are not exponentially stable. Moreover, we show the polynomial stability for each system with a decay rate depending on the smoothness of the initial data. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach. Our results complete our study [Guesmia A. Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement. Nonauton Dyn Syst. 2017;4:78–97] for the case of a dissipation generated by an infinite memory.

COMMUNICATED BY:

• D. Motreanu

Keywords:

• Bresse system
• heat conduction
• asymptotic behavior
• energy method
• frequency domain approach

2010 Mathematics Subject Classifications:

• 35B40
• 35L45
• 74H40
• 93D20
• 93D15

Acknowledgments

The author would like to express his gratitude to the anonymous referee for very careful reading and punctual suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).