id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-349	Mustafa, Muhammad I.	Optimal energy decay rates for viscoelastic wave equations with nonlinearity of variable exponent	2023	17	.pdf	application/pdf	6457	365	83	≤ − (∫ t 0 g(s)ds− δ )∫ Ω u2 t dx+ δ ∫ Ω |∇u|2 dx+ c[Cα + 1] δ (h ◦ ∇u)(t) + cδ(g ◦ ∇u)(t) + a2 4 ∫ Ω∗ |ut|2m(x)−2 dx+ a ∫ Ω Cδ(x)|ut|m(x) dx. (3.7) Proof. [see (2.4)], then the use of hypothesis (2.2), (3.1), and Jensen’s inequality leads to∫ t 0 g(s) ∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ≤ I(t) I(t) ∫ t 0 H −1 (−g′(s) ξ(s) )∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ≤ I(t)H −1 ( 1 I(t) ∫ t 0 (−g′(s) ξ(s) )∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ) ≤ I(t)H −1 ( 1 I(t)ξ(t) ∫ t 0 ( − g′(s) )∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ) ≤ I(t)H −1 (−2E′(t) I(t)ξ(t) ) .	cache/ejde-349.pdf	txt/ejde-349.txt