Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect

Abstract

a b s t r a c tWe consider vibrations modeled by the standard linear solid model of viscoelasticity which are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. We show that the exponential stability under the Fourier law of heat conduction holds. In order to obtain the asymptotic behaviour we use multiplier techniques.

Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound

“…Let us recall that in viscoelasticity theory, springs and dashpots represent the elastic and viscous components of the materials, respectively. According to [7], [1] or [6], among others, in the one dimensional case this equation represents a linear spring connected in series with a Kelvin-Voigt system, that is, another linear spring connected in parallel with a dashpot. This is a common way of approaching viscoelastic systems using a rheological point of view.…”

Convergence to Steady-state and Boundary Layer Profiles in a Linear Chromatography System

“…Let us start with the junction of I I with I II . In that junction, the value of c(x, t) is continuous and we have 1 2…”

“…This is not an unusual situation, for example, in dissipative mechanics, where the mechanical energy is often shown to be nonincreasing, but more work remains to be done in order to show that it eventually tends to zero and that it tends exponentially. See [1], for example, as a recent reference for the situation we are describing. But we have not been able to use the so-called multiplier technique and to modify the energy with the addition of new terms of the form of crossed products, as in [1].…”

“…See [1], for example, as a recent reference for the situation we are describing. But we have not been able to use the so-called multiplier technique and to modify the energy with the addition of new terms of the form of crossed products, as in [1]. We use here a different argument, no longer local in time, but based on considering the variation of the energy along intervals of time of sufficiently long length T .…”

Stability of the vibrations of an inhomogeneous flexible structure with thermal effect

“…The coupled system is generally described by a pair of partial or ordinary differential equations. Recently Alves et al [9], has achieved exponential stability under the Fourier law of heat conduction and obtained the asymptotic behaviour by using multiplier technique, in the field of vibrations modeled by standard linear solid model coupled with heat equation. In this work, we consider a coupled system of a realistic standard linear solid model (cf.…”