id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-71	Kurima , Shunsuke	Existence for a nonlocal Penrose-Fife type phase field system with inertial term	2023	18	.pdf	application/pdf	6422	393	92	Therefore, since vh − vτ = vh − v̂h + v̂τ − vτ + v̂h − v̂τ , we can prove Lemma 5.1 by (5.3)-(5.7), the Schwarz inequality, the Young inequality, (2.11), (2.13), Lemmas 4.1, 4.3, 4.8. − 1 2 ‖θn‖2L2(Ω) + 1 2 ‖θn+1 − θn‖2L2(Ω) + h(−∆un+1, θn+1)L2(Ω) = h(fn+1, θn+1)L2(Ω) − h(vn+1, θn+1)L2(Ω). (4.11) Here, since un+1 = − 1 θn+1 , θn+1 > 0, and gn+1 ≤ 0, we have that h(−∆un+1, θn+1)L2(Ω) = h ∫ Ω ∇un+1 · ∇θn+1 + h ∫ ∂Ω un+1θn+1 − h ∫ ∂Ω gn+1θn+1 ≥ h ∫ Ω |∇ ln θn+1|2 − h|∂Ω|.	cache/ejde-71.pdf	txt/ejde-71.txt