On a class of wave equations with a nonlinear dynamic boundary condition of Kirchhoff type

An initial boundary value problem for a class of nonlinear wave equations with localized internal damping and a nonlinear acoustic boundary conditions of Kirchhoff type is considered. Existence, uniqueness, and stability of solutions are established. To prove the existence of solution, we use Faedo–Galerkin method, and due to the presence of nonconstant coefficients in the boundary equation, new estimates are necessary. To prove the stability, we use multiplier method.