id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-148	Ye, Xiaobing; Wang, Liangchen	Boundedness and asymptotic stability in a chemotaxis model with indirect signal production and logistic source	2022	17	.pdf	application/pdf	7586	412	82	Testing the first equation in (1.1) by lnu+ 1 and integrating we have d dt ∫ Ω u lnu = ∫ Ω (lnu+ 1)∆u− ∫ Ω (lnu+ 1)∇ · (u∇v) + µ ∫ Ω (lnu+ 1)u(1− u) = − ∫ Ω |∇u|2 u + ∫ Ω ∇u · ∇v + µ ∫ Ω (lnu+ 1)u(1− u) ≤ − ∫ Ω u∆v + µ ∫ Ω (lnu+ 1)u(1− u) ≤ ∫ Ω uw + µ ∫ Ω u+ µ ∫ Ω u lnu− µ ∫ Ω u2 − µ ∫ Ω u2 lnu (3.2) for all t ∈ (0, Tmax). [1, 8], testing the first equation of (1.1) by up−1(p ≥ 2) and integrating by parts over Ω, using (4.22) and Young’s inequality we have 1 p d dt ∫ Ω up + (p− 1) ∫ Ω up−2|∇u|2 + µ ∫ Ω up+1 = (p− 1) ∫ Ω up−1∇u · ∇v + µ ∫ Ω up ≤ c2(p− 1) ∫ Ω up−1|∇u|+ µ(p− 1) ∫ Ω up ≤ p− 1 2 ∫ Ω up−2|∇u|2 + (c22 2 + µ ) (p− 1) ∫ Ω up (4.23) for all t ∈ (0, Tmax).	cache/ejde-148.pdf	txt/ejde-148.txt