After an overall review and description over the current researches on and the research approaches for certain infinite-dimensional dynamic systems

the authors have established an equation for elastic beams under axial loads

while taking into account of the viscous effect of beam materials.By employing Galerkin approach

the authors have proved the existence and uniqueness of the global solutions to the equation hereinabove under nonlinear boundary condition

and proved the continuous dependence of the solutions on initial values.