The authors consider the model equation \(v_{tt}+\Delta ^2v+a(t)v_t=0\), where \(a\) is a nonlinear function depending on the \(L^2\)-norm of \(\nabla v(t)\). In their main result the authors show that a globally existing smooth solution \(v=v(x,t)\) decays at a uniform rate \(t^{-\gamma}\) as \(t\to \infty \) for some \(\gamma =\gamma (n)>0\), where \(n\) is the space dimension. This rate of decay is obtained using Fourier transform techniques. Moreover, suitable functionals of the solutions are studied, leading to the conclusion that they approach \(0\) as \(t\to \infty \). The connection of the studied equation to the nonlinear Schrödinger-type equation \(i w_{t}+\Delta w+i a(t)u=0\), \(w(x,0) = h(x)+i\varphi (x)\), is discussed. In an Appendix, the global existence of a smooth solution is recalled, making the paper self-contained.