On the Initial-boundary-value Problem for the Extensible Beam with Attached Load
Marié Grobbelaar‐Van Dalsen · Mathematical Methods in the Applied Sciences · 1996
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.