On the Initial-boundary-value Problem for the Extensible Beam with Attached Load

Abstract

When one end of an extensible beam whose ends are a fixed distance apart, is hinged while to other end a load is attached, the mathematical model describing the vibrations of the beam contains a non‐linearity both in the partial differential equation as well as in the dynamical boundary condition. We introduce weak damping and consider the resulting boundary‐value problem within the framework of the theories of B‐evolutions and fractional powers of a closed pair of operators. This approach enables us to construct a unique weak solution which exhibits exponential decay by employing Faedo–Galerkin approximations.

Stability for variable-coefficient wave equation with local frictional damping and memory type dynamic boundary conditions

“…However, if no local reaction occurs, the Laplace-Beltrami term ∆ Γ u needs to be added to the boundary. For earlier studies on systems with dynamic boundary conditions, refer to [4,13,14].…”

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

“…Many authors are interested in such systems with dynamic boundary conditions, because dynamics influences the stability and active control of large elastic structures. In the early results dealing with the dynamic boundary conditions, the studies of Dalsen 1,2 contributed to this field. Gerbi and Said-Houari 3 studied the following problem:…”

“…Many authors are interested in such systems with dynamic boundary conditions, because dynamics influences the stability and active control of large elastic structures. In the early results dealing with the dynamic boundary conditions, the studies of Dalsen 1,2 contributed to this field. Gerbi and Said‐Houari 3 studied the following problem: They obtained local existence of solution, global existence, and the energy decay and got the blow‐up result of the solution for positive initial energy.…”