id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1531	Ha, Tae Gab	Global solutions and blow-up for wave equations with variable coefficients and boundary supercritical source	2025	31	.pdf	application/pdf	13930	760	84	Indeed, considering w = uηm in (3.1) and then integrating over (0, T ), we have∫ T 0 ⟨uηmtt , uηm⟩ dt+ ∫ T 0 µ(t)∥ |∇gu ηm|g∥22 dt+ η ∫ T 0 ⟨uηmt , uηm⟩Γ1 dt 10 T. G. HA EJDE-2025/104 + ∫ T 0 ⟨q(uηmt ), uηm⟩Γ1 dt− ∫ T 0 ⟨h(uηm), uηm⟩Γ1 dt = ∫ T 0 ⟨f, uηm⟩ dt. Next, considering w = uηmt in (3.1) and then integrating over (0, T ), we have∫ T 0 ⟨uηmtt , u ηm t ⟩ dt+ ∫ T 0 µ(t) ∫ Ω ⟨∇gu ηm,∇gu ηm t ⟩g dx dt+ η ∫ T 0 ∥uηmt ∥22,Γ1 dt + ∫ T 0 ⟨q(uηmt ), uηmt ⟩Γ1 dt− ∫ T 0 ⟨h(uηm), uηmt ⟩Γ1 dt = ∫ T 0 ⟨f, uηmt ⟩ dt. From (3.14)-(3.18) and (3.22), we arrive at lim m→∞,η→0 ∫ T 0 ⟨q(uηmt ), uηmt ⟩Γ1 dt = ∫ T 0 ⟨ψ, ut⟩Γ1 dt.	cache/ejde-1531.pdf	txt/ejde-1531.txt