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

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

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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 Shirts string or a beam.Problems with dynamical boundary conditions containing first-order derivatives with respect to both space and time variables are not Home 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 estimatesin Sobolev spaces.As a particular example of the wave equation is considered.The exact solution of a problem with dynamical condition is obtained.