ON THE INITIAL-BOUNDARY-VALUE PROBLEM FOR THE EXTENSIBLE BEAM WITH ATTACHED LOAD

When one end of an extensible beam whose ends are a fixed distance apa rt, is hinged while to other end a load is attached, the mathematical model describing the vibrations of the beam contains a non-linearity b oth in the partial differential equation as well as in the dynamical b oundary condition. We introduce weak damping and consider the resultin g boundary-value problem within the framework of the theories of B-evo lutions and fractional powers of a closed pair of operators. This appr oach enables us to construct a unique weak solution which exhibits exp onential decay by employing Faedo-Galerkin approximations.