Long-Time Behavior for a Class of Semi-linear Viscoelastic Kirchhoff Beams/Plates

In this paper, we consider a transmission problem for Kirchhoff-type wave equations with boundary condition determined by the long-range memory. The wave propagation over bodies consisting of two physically different types of materials. One component is clamped, and the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. Under a wider assumption of the kernel, we establish a more general decay rate of energy, which improves earlier results established in the literature.

Section: Introductionmentioning

confidence: 99%

Memory‐type boundary stabilization of a transmission problem for Kirchhoff wave equations

Fan

Feng

2022

“…Many results have been established for the asymptotic behavior of wave equations and beams with finite memory; we refer the reader to see previous studies [11][12][13][14][15][16][17][18][19] and the references therein. The study of the Kirchhoff plates subject to a viscoelastic damping has attracted the attention of many authors, and several decay results have been established; see, for instance, previous works 20,21 and the references therein. The well-posedness and asymptotic stability of a plate equation with memory and time-varying delay has been extensively studied in literature; see, for instance, references.…”

Section: Introductionmentioning

confidence: 99%

A new stability result for a flexible satellite system with viscoelastic damping

Hamdi

Berkani

2022

In this paper, we study a viscoelastic flexible satellite under unknown distributed disturbances during attitude maneuvering. The system composed of a rigid central hub that represents the spacecraft with two symmetrical viscoelastic Euler-Bernoulli beams and subject to undesirable vibrations. The problem can be modeled by a set of partial differential equations (PDEs) taking into account therefore the dynamic boundary condition. The well-posedness of the closed-loop system is discussed. By applying a control force at the center body of the spacecraft, we shall suppress these vibrations; namely, we establish stability results of the system under appropriate assumptions imposed on the relaxation function. These new results generalize and improve many results in the literature. Our results are obtained by using the multiplier technique. Numerical simulation results show the effectiveness of the proposed control scheme.

“…The uniqueness of a weak solution and its continuous dependence on the initial data were also proved. The multi-dimensional generalizations of Equation ( 2), namely, as some of viscoelastic Kirchhoff beams/plates, have been recently studied by B. Feng 5 and Jorge Silva. 6 In Silva et al, 6 the authors considered a viscoelastic Kirchhoff beam/plate model with a viscoelastic damping…”

Section: Introductionmentioning

confidence: 99%

On a nonlinear boundary problem for thermoelastic coupled beam equations with memory term

Ngoc

Khanh

Nhân

et al. 2022

In this paper, we study a one-dimension nonlinear problem for thermoelastic coupled beam equations with memory term. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, we first prove the local existence and the uniqueness of a weak solution. Next, by establishing assumptions and constructing energy functionals suitably, we consider the global existence and general decay behavior of the solution. Finally, the blow-up property in the special case of this problem is also given. KEYWORDSthermoelastic coupled beam equations, Faedo-Galerkin method, linearization method, local existence and global existence, general decay, blow-up MSC CLASSIFICATION35A01, 35M33, 35B40, 35B44 J. A. Burns et al 9 proved the exponential decay energy for a linear homogeneous thermoelastic bar in length 𝜋 with unit reference density subject to various boundary conditions. C. M. Dafermos 10 proved on the existence, differentiability, and asymptotic stability of solutions to a system of linear thermoelasticity. For a model of linear non-Fourier thermoelastic bar, C. Giorgi and M. G. Naso 11 proved the exponential stability of c 0 -semigroup associated with the corresponding system. J. E. Muñoz Rivera 12 established the decay rate of energy in one-dimensional linear thermoelasticity obeying Fourier's law without memory effect. Afterward, J.E. Muñoz Rivera and Y. Qin 13 studied the global existence, uniqueness, and asymptotic behavior of solutions to the equations of one-dimensional nonlinear thermoelasticity with thermal memory subject to Dirichlet-Dirichlet boundary conditions at the endpoints. Slemrod 14 proved the global existence and asymptotic stability of small solutions with Neumann-Dirichlet or Dirichlet-Neumann boundary conditions.In the following, let us mention to the results in some previous works for the long-time dynamics of thermoelastic coupled beam system with thermal effects. Giorgi et al 15 studied the dissipative system { u tt + Δ 2 u + Δ𝜃 − (𝛽 + ‖u x (t)‖ 2 )Δu = 𝑓 , 𝜃 t − Δ𝜃 − Δu t = g,