Uniform decay of weak solutions to a von Kármán plate with nonlinear boundary dissipation (Q1328935)

Asymptotic properties of solutions to a von Kármán plate model without accounting rotational forces is treated. Decay behaviors are analysed for nonlinear boundary feedback. The nonlinear boundary feedback is of damping type and its form is given by the equations \[ \Delta w + (1 - \mu) B_ 1w = - f \left( {\partial \over \partial \nu} w_ t \right) \quad \text{on the boundary curve}, \tag{1} \] \[ {\partial \over \partial \nu} \Delta w + (1 - \mu) B_ 2w = g(w_ t) \quad \text{on the boundary curve}. \tag{2} \] In equations (1), (2) \(B_ 1\) and \(B_ 2\) are known linear operators, \(0<\mu<0.5\) is Poisson's ratio, \(\nu\) is the outer normal direction to the boundary curve, \(t\) is the time variable. The controls \(f\) and \(g\) are continuous monotone functions and are subjected to the following constraints: \[ f(s)s>0, \quad g(s)s>0 \quad \text{for} \quad s \neq 0, \tag{3} \] \[ m | s | \leq | f(s) | \leq M | s |, \quad m | s | \leq | g(s) | \leq M | s | \quad \text{for} \quad | s | > 1. \tag{4} \] The main goal of this paper is to prove that the energy of the von Kármán plate model, with nonlinear boundary feedback decays uniformly for all weak solutions of finite energy.