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We construct strong solutions for a nonlinear wave equation for a thinvibrating plate described by nonlinear elastodynamics. For sufficiently smallthickness we obtain existence of strong solutions for large times underappropriate scaling of the initial values such that the limit system as $h\to0$ is either the nonlinear von K\'arm\'an plate equation or the linear fourthorder Germain-Lagrange equation. In the case of the linear Germain-Lagrangeequation we even obtain a convergence rate of the three-dimensional solution tothe solution of the two-dimensional linear plate equation.

Large Time Existence for Thin Vibrating Plates