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

We study the linearised self-excited wave equation x
_tt
- Δ x-P(x) x
_t
=0, where P(x)≥0, P(x) L
^∞
(Ω), in a bounded domain Ω ⊂ R
ⁿ
with smooth boundary Γ where boundary damping is present. Considering the partition {Γ
₊
, Γ
_-
} of the boundary Γ on which x=0 on Γ, and u
_n
+ Ku
_t
+ Lu= 0, on Γ, 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 Γ
₊
∩ Γ
_-
= ø for n > 3). Both bounds depend on Ω (The domain of wave equation in R
ⁿ
, n ≥ 1). The second bound also depends on the feed-back functions K,L L
^∞
(Γ
₊
) or more precisely depends on a positive function k(x) L
^∞
(Γ
₊
) which determines K and L on the partition Γ
₊
.