id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1792	Munoz Rivera, Jaime; Ochoa Ochoa, Elena; Quintanilla, Ramon 	Thermoelastic plates with type I heat conduction with second gradient	2025	11	.pdf	application/pdf	5138	318	73	d iω ∫ Ω ∆θ(iρωv − η∆θ − g2)dΩ = dρ ∫ Ω ∆θ v dΩ+ d iω ∫ Ω η|∆θ|2dΩ+ d iω ∫ Ω ∆θg2dΩ. Finally, multiplying equation (2.6) by u we find that iρ ∫ Ω ωvu dΩ+ c ∫ Ω |∆u|2 dΩ− η ∫ Ω ∆θu dΩ = ∫ Ω g2u dΩ. Using equation (2.5) we obtain ρ ∫ Ω |v|2 dΩ = −ρ ∫ Ω vg1 dΩ+ c ∫ Ω |∆u|2 dΩ− η ∫ Ω ∆θu dΩ− ∫ Ω g2u dΩ. The above inequality implies∫ Ω |v|2 dΩ ≤ c ∫ Ω |∆u|2 dΩ+ c ∫ Ω |∆θ|2 dΩ+ c̃ϵ∥U∥H∥G∥H. Using (3.3) and (2.8) we obtain∫ Ω |v|2 dΩ ≤ c̃ϵ∥U∥H∥G∥H + c̃∥G∥2H, for ϵ small.	cache/ejde-1792.pdf	txt/ejde-1792.txt