The Stabilization of a Linearized Self-Excited Wave Equation by an Energy Absorbing Boundary

We study the linearised self-excited wave equation xtt - ¿ x-P(x) xt=0, where P(x)¿0, P(x) L^¿(¿), in a bounded domain ¿ ¿ Rⁿ with smooth boundary ¿ where boundary damping is present. Considering the partition {¿+, ¿-} of the boundary ¿ on which x=0 on ¿, and un+ Kut + Lu= 0, on ¿, we find two different bounds for P such that the energy decays exponentially in the energy space as t tends to infinity (Here we assume ¿+ ¿ ¿- = ¿ for n ≫ 3). Both bounds depend on ¿ (The domain of wave equation in Rⁿ, n ¿ 1). The second bound also depends on the feed-back functions K,L L^¿(¿+) or more precisely depends on a positive function k(x) L^¿(¿+) which determines K and L on the partition ¿+.