δh∂tw(t), w(t)iα
β + δk∂tw(t)k2
α + δhα∂ttw(t), w(t)iW 1 .
Using (25) and (24), we can see that
hα∂ttw(t), w(t)iW 1 = −hβ∂tw(t), w(t)iW 1 + h∂tw∗(t), d w(t)iW 2
= −hβ∂tw(t), w(t)iW 1 − k∂tw
∗
(t)k2
γ .
With (30) and Young’s inequality, the first term can be further bounded by
−hβ∂tw(t), w(t)iW 1 ≤ Cβk∂tw(t)kαkw(t)kα
≤
Cβ
cβ
(cid:16)
- (CPCβ )2
2
(cid:17)
k∂tw(t)k2
α + δhα∂ttw(t), w(t)iW 1 .
Now let C1 denote the coefficcient in front of the first term. Then together with the previous estimates, we immediately obtain
Eδ(t) ≤ − min(cβ, δ)E0(t) ≤ − min(cβ, δ)
2+2Cβ /cβ + (CPCβ )2 , we further see that δ(1 + C1) ≤ cβ
cβ − δ(1 + C1) ≤ k∂tw(t)k2 α + δhα∂ttw(t), w(t)iW 1 .
Consequently, for any 0 ≤ δ ≤ ddt , we have Eδ(t) ≤ − min(cβ, δ)E0(t) ≤ − min(cβ, δ)k∂tw(t)k2 α + δhα∂ttw(t), w(t)iW 1 .
In summary, the article discusses the exponential stability of damped wave equations under certain conditions. The authors show that by using Young’s inequality and the coefficcient C1 in front of the first term, they can bound the error term Eδ(t) and prove that it is bounded by a constant times the norm of the initial data. This result provides a framework for analyzing the stability of damped wave equations in various applications, including elasticity, fluid dynamics, and electromagnetism.
