A problem with dynamical boundary condition for a one-dimensional hyperbolic equation

In this paper, we consider a problem with dynamical boundary conditions for a hyperbolic equation. The dynamical boundary condition is a convenient method to take into account the presence of certain damper when fixing the end of a string or a beam. Problems with dynamical boundary conditions containing first-order derivatives with respect to both space and time variables are not self-ajoint, that complicates solution by spectral analysis. However, these difficulties can be overcome by a method proposed in the paper. The main tool to prove the existence of the unique weak solution to the problem is the priori estimates in Sobolev spaces. As a particular example of the wave equation is considered. The exact solution of a problem with dynamical condition is obtained.