WELL-POSED AND LIMIT PROBLEM FOR NONLINEAR DAMPED WAVE EQUATION

Abstract

This paper studies a nonlinear damped plate equation that models the vibration of thin elastic structures. We give a clear and rigorous analysis of the initial–boundary value problem. Under natural energy conditions and standard growth on the nonlinearity, we prove local and global existence and uniqueness of weak solutions with continuous dependence on the initial data.  We then study two small-parameter limits with explicit rates. First, we show that the solution depends continuously on the stiffness parameter \(a\) and that, as \(a\to 0\), the solution \(v_a\) converges to the solution \(v^\ast\) of the limiting biharmonic problem  in the energy norm. Second, we show that, as the inertia parameter \(\gamma\to 0\), the hyperbolic solution \(u_\gamma\) converges to the parabolic solution \(u\) in a suitable norm. These results give simple, quantitative error bounds for reduced models and link the hyperbolic and parabolic pictures of plate dynamics.        Our method is of using  the Faedo--Galerkin scheme, energy estimates that track the parameters, and Steklov averaging to handle time derivatives for weak solutions. Compactness comes from the Aubin--Lions--Simon lemma, and Gronwall-type inequalities yield clean bounds in time. The approach is easy to apply and extends to related plate models with damping and stiffness terms. \\