Polynomial stabilization of non-smooth direct/indirect elastic/viscoelastic damping problem involving Bresse system

Abstract

We consider an elastic/viscoelastic problem for the Bresse system with fully Dirichlet or Dirichlet–Neumann–Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin–Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results at the end). Our results generalize and improve many earlier ones in the literature (see El Arwadi and Youssef, in: On the stabilization of the Bresse beam with Kelvin–Voigt damping, Applied Mathematics and Optimization, pp. 1–27, 2019) and in particular some studies done on the Timoshenko system with Kelvin–Voigt damping (see for instance Ghader and Wehbe in A transmission problem for the Timoshenko system with one local Kelvin–Voigt damping and non-smooth coefficient at the interface, 2020. arXiv:2005.12756; Tian and Zhang in Z Angew Math Phys 68:20, 2017; Zhao et al. Acta Math Sin Engl Ser 21:655–666, 2005).