id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-3003	Di, Huafei; Shang, Yadong	Existence, Nonexistence and Decay Estimate of Global Solutions for a Viscoelastic Wave Equation with Nonlinear Boundary Damping and Internal Source Terms	2017	34	.pdf	application/pdf	11397	596	85	If u ∈ L∞(0, T ;H1 Γ0 (Ω)) with ut ∈ L∞(0, T ;Lρ+1(Ω))∩Lq+1(0, T ;Lq+1(Γ1)) satisfy the following conditions (i) For any v ∈ H1 Γ0 (Ω) ∩ Lq+1(Γ1) ∩ Lρ+1(Ω) and a.e 0 ≤ t ≤ T , such that 1 ρ (|ut|ρ−1ut, v) + ∫ t 0 b1(u, v)ds+ ∫ t 0 b2(u, v)ds − ∫ t 0 (|u|p−1u, v)ds+ ∫ t 0 (|ut|q−1ut, v)Γ1ds = 1 ρ (|u1|ρ−1u1, v), (2.12) where b1(u, v) = (∇u,∇v), b2(u, v) = −( ∫ s 0 g(s− τ)∇u(τ)dτ,∇v); (ii) u(x, 0) = u0(x) in H1 Γ0 (Ω), ut(x, 0) = u1(x) in Lρ+1(Ω) ∩ Lq+1(Γ1). Math, 10 (4) (2017), 668-701 683 = ∣∣ ∫ t 0 g(t− s) ∫ Ω |∇u(s)−∇u(t)|2dxds− ∫ t 0 g(t− s) ∫ Ω |∇um(s)−∇um(t)|2dxds ∣∣ ≤ ∫ t 0 g(t− s)‖∇u(s)−∇um(s)‖2‖∇u(s) +∇um(s)‖2ds + ∫ t 0 g(t− s)‖∇u(s)−∇um(s)‖2ds‖∇u(t) +∇um(t)‖2 + ∫ t 0 g(t− s)‖∇u(s) +∇um(s)‖2ds‖∇u(t)−∇um(t)‖2 + ∫ t 0 g(s)ds‖∇u(t) +∇um(t)‖2‖∇u(t)−∇um(t)‖2 ≤ C ∫ t 0 g(t− s)‖∇u(s)−∇um(s)‖2ds+ C ∫ t 0 g(s)ds‖∇u(t)−∇um(t)‖2 → 0, (3.46) as m→∞. Taking into account the nonlinear term of the functional J(u), we deduce ‖um‖p+1 p+1 − ‖u‖ p+1 p+1 ≤ (p+ 1)| ∫ Ω |u+ θmum|p−1(u+ θmum)(um − u)dx| ≤ (p+ 1)‖u+ θmum‖pp+1‖um − u‖p+1 ≤ C‖um − u‖p+1 → 0, (3.47) as m→∞, where 0 < θm < 1.	cache/ejpam-3003.pdf	txt/ejpam-3003.txt