SINGULAR INTERNAL STABILIZATION OF THE WAVE-EQUATION

We consider an initial and boundary value problem for the one aid two dimensional wave equation with nonlinear damping concentrated on an in terior point and respectively on an interior curve. In the two dimensi onal case our main result asserts that generically (i.e., for almost a ll interior curves the solutions decay to zero in the energy space. Wh en the domain is strictly convex we show that, whatever the interior c urve is, the decay is not uniform. We generalize in this way results k now in one space dimension. Our main improvement of existing one-dimen sional results consists in giving sharp decay rates, provided that the initial data are regular and the damping term is linear. A crucial in termediate step is the proof of a generalization of Inghams inequality on nonharmonic Fourier series. (C) 1998 Academic Press.