Exponential stability of wave equations with potential and indefinite damping

First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If a¯:=1L∫0La(x)dx is positive and the spectrum of the operator (∂xx−b) is negative, exponential stability is proved for small ‖a¯−a‖L2. Explicit estimates of the decay rate ω are given in terms of a¯ and the largest eigenvalue of (∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σ(ux)x+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω.