Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations

Abstract

Abstract. The purpose of this paper is to establish strong lower energy estimates for strong solutions of nonlinearly damped Timoshenko beams, Petrowsky equations in two and three dimensions and wave-like equations for bounded one-dimensional domains or annulus domains in two or three dimensions. We also establish weak lower velocity estimates for strong solutions of the nonlinearly damped Petrowsky equation in two and three dimensions. The feedbacks in consideration have arbitrary growth close to the origin. These results improve the strong lower energy decay rates obtained in our previous papers (Alabau-Boussouira in J Differ Equ 249: 1145-1178 J Differ Equ 248:1473-1517 for strong solutions of the nonlinearly locally damped wave equation and extend to systems and to Petrowsky equation the method of Alabau-Boussouira

Discrete energy behavior of a damped Timoshenko system

“…Second, for the nonlinear damped Timoshenko system having a damping term with no growth assumption at the origin, Alabau-Boussouira [2] established a general semi-explicit formula for the decay rate of the energy at infinity in the case of equal speeds of propagation, and she proved a polynomial decay in the case of different speeds of propagation for both linear and nonlinear globally Lipschitz feedbacks. Later, in [1], Alabau-Boussouira also established a strong lower energy estimate for the strong solutions of nonlinearly damped Timoshenko beams and, as an extension of this result, for the nonlinearly damped Timoshenko system with thermoelasticity (see [6]). From the numerical point of view, the authors in [4] used a fourth-order finite difference scheme to compute the numerical solutions of the Timoshenko system with thermoelasticity with second sound (coupled with the Cattaneo Law and giving rise to a system with four equations).…”

Lie Symmetries, Optimal System and Invariant Reductions to a Nonlinear Timoshenko System

“…Case 12 a 1 = 0, a 2 = 0, a 8 = 0, a 4 = 0, a 7 = 0, a 6 = 0 : Let 1 = a 3 , 2 = 0 to makeã 3 = 0: then the conjugacy class is X 4 + αX 5 , α ∈ R. Case 13 a 1 = 0, a 2 = 0, a 8 = 0, a 4 = 0, a 7 = 0 : Let 2 = a 6 to haveã 6 = 0: the conjugacy class is αX 3 + βX 5 + X 7 , α, β ∈ R. Case 14 a 1 = 0, a 2 = 0, a 8 = 0, a 4 = 0, a 7 = 0, a 6 = 0 : Let 2 = a 5 to haveã 5 = 0: the conjugacy class is αX 3 + X 6 , α ∈ R.…”

“…Dimplekumar et al [3] discuss the mathematical modeling for the mechanics of a solid using the distribution theory of Schwartz to the beam bending differential equations; the governing differential equations of a Timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness and shear stiffness were obtained in this paper in the space of generalized functions. Mustafa and Messaoudi [4] and Fatiha [5] studied stability of the following Timoshenko system with the nonlinear frictional damping term in one equation:…”

Lower bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity

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In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau-Boussouira's energy comparison principle introduced in [3] (see also [6]). We extend to our model the nice results achieved in [6] for the case of nonlinearly damped Timoshenko system with thermoelasticity.

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“…We first investigate the stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau-Boussouira's energy comparison principle introduced in [3] (see also [6]). We extend to our model the nice results achieved in [6] for the case of nonlinearly damped Timoshenko system with thermoelasticity. The proof of our results relies on the approach in [1, 2].MSC codes: 35B35, 35B40, 35L51, 93D20.…”