Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems

In this paper, we consider two damped wave problems for which the damping terms are allowed to change their sign. Using a careful spectral analysis, we nd critical values of the damping coecients for which the prob- lem becomes exponentially or polynomially stable up to these critical values. 1. Introduction. We consider a one-dimensional wave equation with an indenite sign damping and a zero order potential term which is either internally damped of the form utt(x;t) uxx(x;t) + 2 (0;1)(x)ut(x;t)+ 2 ( 1;0)(x)ut(x;t) = 0; x2 ( 1; 1); t > 0; u(1;t) = u( 1;t) = 0; t > 0; u(x; 0) = u0(x); ut(x; 0) = u1(x)