id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-303	Pereira, Jardel Morais	Attractors for dissipative lattice differential equations with local and nonlocal nonlinearities	2021	36	.pdf	application/pdf	14254	687	81	Since u = (un) and v = (vn) belong to `2, we have∑ n∈Zd (∆dun)vn = d∑ i=1 ∑ n∈Zd (∂+i un)vn − d∑ i=1 ∑ n∈Zd (∂−i un)vn = d∑ i=1 ∑ n∈Zd (∂+i un)vn − d∑ i=1 ∑ n∈Zd (∂+i un)vn+ei = − ∑ n∈Zd d∑ i=1 ∂+i un∂ + i vn = − ∑ n∈Zd ∇+un · ∇+vn. This proves Lemma 2.2 if p = 1. ∈ C1(R+; `2), from (3.34), we obtain (−1)2k+1 ∑ n∈Zd ∆2k+1 d un(θnu̇n) = 1 2 d dt ∑ n∈Zd θn|D2k−1vn|2 + ∑ n∈Zd d∑ i=1 (∂+i θn)z (i) 2k−1,n + ∑ n∈Zd d∑ i=1 ∂+i θn [ (∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n) ] , (3.35) where, in view of Lemma 2.1, ∑ n∈Zd d∑ i=1 |z(i)2k−1,n| ≤ C(2k − 1, d)‖(v, v̇)‖2H ≤ 16d2C(2k − 1, d)‖(u, u̇)‖2H , ∑ n∈Zd d∑ i=1 |(∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n)| ≤ (4d)4k+1‖(u, u̇)‖2H .	cache/ejde-303.pdf	txt/ejde-303.txt