id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-333	Liu, Zhiqing; Gao, Cunchen; Fang, Zhong Bo	Well-posedness and energy decay of a transmission problem of Kirchhoff type wave equations with damping and delay terms	2021	23	.pdf	application/pdf	8566	431	82	Substituting (3.19)-(3.22) into (3.17), we can derive d dt E (n) 1 (t) + r(1− 3η)‖u(n) tt ‖2Γ2 ≤ 3‖∇u(n)‖Ω1‖∇u (n) t ‖3Ω1 + 3‖∇v(n)‖Ω2‖∇v (n) t ‖3Ω2 − (µ1 − µ2 2 − ζ 2τ )‖u(n) tt ‖2Ω1 − ( ζ 2τ − µ2 2 )‖z(n) t (x, 1, t)‖2Ω1 + r 4η (k′(t))2‖k′′(t− s)‖L1(0,+∞) ∫ t 0 k′′(t− A direct calculation shows that (h ∗ u, ut)Γ2 =− 1 2 d dt [ ∫ Γ2 (h � u)(t)dΓ− (∫ t 0 h(s)ds ) ‖u‖2Γ2 ] − 1 2 h(t)‖u‖2Γ2 + 1 2 ∫ Γ2 (h′ � u)(t)dΓ, (2.1) and ‖(h ◦ u)(t)‖2Γ2 ≤ (∫ t 0 |h(s)|ds )∫ Γ2 (|h| � u)(t)dΓ. (2.2) Differentiating (1.3), we arrive at the following Volterra equation (1 + ‖∇u‖2Ω1 )	cache/ejde-333.pdf	txt/ejde-333.txt