Exponential stability in thermoviscoelastic mixtures of solids

Abstract

a b s t r a c tIn this paper we investigate the asymptotic behavior of solutions to the initial boundary value problem for a one-dimensional mixture of thermoviscoelastic solids. Our main result is to establish the exponential stability of the corresponding semigroup and the lack of exponential stability of the corresponding semigroup.

Stabilization of ternary mixtures with frictional dissipation

“…Taking the product in L 2 (0, ) of the identity above with u and exploiting (6.3) we infer that 3 As before, we set σ = α(1 − γ 2 + γ 3 ).…”

“…In recent years, an increasing interest has been directed to the study of the qualitative properties of solutions related to mixtures composed by two interacting continua. In particular, existence, uniqueness, continuous dependence and stability of solutions [1][2][3]12,15,18,19]. The present paper is focused on the asymptotic behavior of the solution semigroup associated with system (1.6).…”

On the time decay of solutions in micropolar viscoelasticity

“…The author proved the slow decay of the solutions with respect to the time for elastic-porous materials when there is only one dissipation mechanism in the system: the viscoporosity. After that, a lot of works have been developed to clarify the behavior of the solutions (exponential decay, slow decay, impossibility of localization and/or analyticity) for solids with voids [4,5,11,12,15,16,18,19,20,23,22,24,26,27,28,33], for non-simple materials [10,29] or for mixtures of elastic solids [1,2,3,31,32]. Nevertheless, few attention has been paid up to now to micropolar elastic solids.…”

On the uniqueness and analyticity of solutions in micropolar thermoviscoelasticity

“…The author proved the slow decay of the solutions with respect to the time for elastic-porous materials when the only dissipation mechanism is the porous dissipation. After that, a lot of works were intended to clarify the behavior of the solutions (exponential decay, slow decay, impossibility of localization and/or analyticity) for solids with voids [4,5,9,10,13,14,16,17,18,20,19,21,23,24,29], for non-simple materials [8,25] or for mixtures of elastic solids [1,2,3,27,28]. Nevertheless, any attention has been paid up to now to micropolar elastic solids.…”