General Decay Rates for a Laminated Beam with Memory

Abstract

In previous work [23], Mustafa considered a viscoelastic laminated beam system with structural damping in the case of equal-speed wave propagations, and established explicit energy decay formula which gives the best decay rates. In this paper, we continue to consider the similar problems and establish the general decay result for the energy, to system with structural damping in the case of non-equal wave speeds and to system without structural damping in the case of equal wave speeds, respectively. For the first case, we use the second-order energy method to overcome the difficulty of estimating the non-equal speeds term. For the second case, we construct an appropriated perturbed functional to estimate w t 2 2 so as to overcome the absence of structural damping.

Memory-type boundary control of a laminated Timoshenko beam

“…Mustafa [15, 16] extended the result to obtain two general decay results under the assumption of equation (4). Li et al [17] and Chen et al [18] extended the result to the case of non-equal speeds. Raposo et al [19] considered the following system:…”

On a Thermoelastic Laminated Timoshenko Beam: Well Posedness and Stability

“…Complexity us, B is coercive on V × V. Consequently, using Lax-Milgram theorem, we conclude that (30) has a unique solution: 5 , and v 8 into (24)- (26) and 28, respectively, we have…”

“…By adding suitable damping effects, such as internal damping, (boundary) frictional damping, and viscoelastic damping, it was shown that if the linear damping terms are added in two of the three equations, system (4) is exponentially stable under the "equal wave speeds" assumption (ρ/I ρ ) � (G/D). But if the damping terms are added in the three equations, then the system decays exponentially without the equal wave speeds assumption, see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For thermoelastic laminated Timoshenko beam, there are few published works, we can mention the results due to Liu and Zhao [18] and Apalara [19].…”

Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term

“…Moreover, we referred to [11,14,15] for the decay estimate of degenerate wave equations with localized damping and viscoelastic damping, and system of linear equations with boundary memory dissipations. The general decay of a thermoelastic laminated beam with past history was researched in [16,30] and the general and optimal decay result for a Moore-Gibson-Thompson equation with memory was established in [28].…”