A boundary condition with memory for Kirchhoff plates equations

Uniform decay for a von Karman plate equation with a boundary memory condition

“…Cavalcanti [1] proved the global existence of solution for the linear Euler-Bernoulli viscoelastic equation and then investigated the asymptotic behaviour by using the perturbed energy method. Santos and Junior [5] investigated the stability of solutions for Kirchho plate equations (1)-(2) with memory condition working at the boundary when = 0. The feature which distinguishes their paper from other related works is the fact that the boundary condition is of the memory type.…”

“…The feature which distinguishes their paper from other related works is the fact that the boundary condition is of the memory type. By employing the same idea of Reference [5], in this paper we prove the existence of solution to von Karman plate equations (1)-(5) and then investigate uniform decay rate of the energy…”

Energy decay of dissipative plate equations with memory-type boundary conditions

“…This result can be proved, for initial data in suitable function spaces, using standard arguments such as the Galerkin method (see [40]). …”

“…The same results were obtained by Alabau-Boussouira et al [3] for a more general abstract equation. For boundary viscoelastic damping, if k i is the resolvent kernel of −g i g i (0) for i = 1, 2, Santos and Junior [40] showed that the energy decays exponentially (polynomially), provided the resolvent kernels also decay exponentially (polynomially). In Rivera et al [33,34] investigated a class of abstract viscoelastic systems of the form…”

Stability for the beam equation with memory in non‐cylindrical domains

“…In a ÿxed domain, it is well-known that, the relaxation function g decays to zero implies that the energy of the system also decays to zero; See References [9][10][11][12]. But in a moving boundary setting, the transverse de ection u(x; t) of a beam charges its conÿguration at each instant of time, increasing its deformation and hence increasing its tension.…”