Singular Internal Stabilization of the Wave Equation

We consider an initial and boundary value problem for the one and two dimensional wave equation with nonlinear damping concentrated on an interior point and respectively on an interior curve. In the two dimensional case our main result asserts that generically (i.e., for almost all interior curves) the solutions decay to zero in the energy space. When the domain is strictly convex we show that, whatever the interior curve is, the decay is not uniform. We generalize in this way results known in one space dimension. Our main improvement of existing one-dimensional results consists in giving sharp decay rates, provided that the initial data are regular and the damping term is linear. A crucial intermediate step is the proof of a generalization of Inghamʹs inequality on nonharmonic Fourier series.