id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-257	Pereira, Ducival; Cordeiro, Sebastiao; Raposo, Carlos; Maranhao, Celsa	Solutions of Kirchhoff plate equations with internal damping and logarithmic nonlinearity	2021	14	.pdf	application/pdf	5430	382	83	Integrating (3.7) from 0 to t, 0 ≤ t ≤ tm, we obtain 1 2 ‖umt (t)‖2 + 1 2 ‖∆um(t)‖2 + 1 2 M̂(‖∇um(t)‖2) + 1 2 ‖um(t)‖2 + ∫ t 0 ‖umt (s)‖2ds = 1 2 ‖u1m‖2 + 1 2 ‖∆u0m‖2 + 1 2 M̂(‖∇u0m‖2)− 1 2 ∫ Ω (u0m)2 ln |u0m|2 dx (3.8) + 1 2 ∫ Ω (um(t))2 ln |um(t)|2 dx. Introduction In this article we study the existence and decay properties of global solutions for the nonlinear initial boundary value problem utt + ∆2u+M(‖∇u‖2)(−∆u)	cache/ejde-257.pdf	txt/ejde-257.txt