Initial-Boundary Value Problem for a Nonlinear Beam Vibration Equation

We consider an initial-boundary value problem for the beam vibration equation, which is a fourth-order nonlinear equation with two independent variables. It is shown that under certain conditions on the initial data this problem can be reduced to the Cauchy problem for a countable system of quasilinear ordinary differential equations. Using the method of energy inequalities, we prove that this Cauchy problem has a solution. Based on this, we establish the existence of a local solution of the original initial-boundary value problem and construct it in closed form. A theorem on the uniqueness of a global solution is proved.