Sharp stability estimates for quasi-autonomous evolution equations of hyperbolic type

We study the energy decay of the difference of two solutions for dissipative evolution problems of the type: \[ u” + Lu + g(u’) = h(t), \qquad t \ge 0 ,\] including wave and plate equations and ordinary differential equations. In the general case, when the damping term g behaves like a power of the velocity ú, the energy decreases like a negative power of time, multiplied by a constant depending on the initial energies. We provide estimates on these constants and prove their optimality. In the special case of the ordinary differential equation with periodic forcing, we establish, relying on a controllability-like technique, that the decay is in fact exponential, even under very weak damping.