Exponential stability for wave equations with non-dissipative damping

We consider the nonlinear wave equation utt−σ(ux)x+a(x)ut=0utt−σ(ux)x+a(x)ut=0 in a bounded interval (0,L)⊂R1(0,L)⊂R1. The function aa is allowed to change sign, but has to satisfy a¯=1L∫0La(x)dx>0. For this non-dissipative situation we prove the exponential stability of the corresponding linearized system for: (I) possibly large ‖a‖L∞‖a‖L∞ with small ‖a(⋅)−a¯‖L2, and (II) a class of pairs (a,L)(a,L) with possibly negative moment ∫0La(x)sin2(πx/L)dx. Estimates for the decay rate are also given in terms of a¯. Moreover, we show the global existence of smooth, small solutions to the corresponding nonlinear system if, additionally, the negative part of aa is small enough.