id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ma-55	 Pereira, Ducival C.; Araújo, Geraldo M. de; Raposo, Carlos A.	Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping and Logarithmic Source	2022-01-06	20	.pdf	application/pdf	6852	456	84	in Q, (5.38) u� ⇀ u in L∞(0,T ; H3Γ(Ω)), (5.39) u�t ⇀ ut in L2(0,T ; H10 (Ω) ∩H2(Ω)), (5.40) u�tt ⇀ utt in L2(0,T ; H−1(Ω)), (5.41) |u�|r−2u� ln |u�| ⇀ − (−∆u � p(t)),v〉pθ(t)dt ∣∣∣∣ = ∣∣∣∣∫ T 0 ∫ Ω ( |∇u�m(t)|p−2∇u�m(t) −|∇u�(t)|p−2∇u�(t) ) ∇v dx θ(t) dt ∣∣∣∣ ≤ C|θ|∞ ∫ T 0 ∫ Ω ( |∇u�m(t)|p−2 + |∇u�(t)|p−2 ) |∇u�m(t) −∇u�(t)||∇v|dx dt ≤ C1 ∫	cache/ma-55.pdf	txt/ma-55.txt
