id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-137	Lagha, Aesha; Hattori, Hattori	Cauchy problems for chemotaxis systems with chemo attractant and repellent	2022	23	.pdf	application/pdf	7766	399	81	There exists a positive number ε0 such that if ‖(n0 − n∞, u0, c1,0, c2,0)‖N ≤ ε0, the Cauchy problem (1.3)-(1.4) has a unique solution (n, u, c1, c2)(t) globally in time which satisfies (n− n∞, u)(t) ∈ C([0,∞);HN (R3)) ∩ C1([0,∞);HN−1(R3)), (c1, c2)(t) ∈ C([0,∞);HN (R3)) ∩ C1([0,∞);HN−2(R3)) and there are constants λ1 > 0, λ2 > 0, λ3 > 0 and C0 > 0 such that ‖(n− n∞, u, c1, c2)‖2N + λ1 ∫ t 0 ‖∇(n− n∞)‖2N−1 + λ2 ∫ t 0 ‖∇(c1, c2)‖2N + λ3 ∫ t 0 ‖(u, c1, c2)‖2N ≤ C0‖(n0 − n∞, u0, c1,0, c2,0)‖2HN . C‖∂αρ0‖+ C‖ρ‖N ∫ t 0 (‖∂αu‖2 + ‖∂αρ‖2)ds + C‖u‖N ∫ t 0 ‖∂αρ‖2ds+ C‖u‖N ∫ t 0 ‖∂αu‖2ds + C‖c1‖N ∫ t 0 (‖∂αu‖2 + ‖∂α∇c1‖2)ds + C‖c2‖N ∫ t 0 (‖∂αu‖2 + ‖∂α∇c2‖2)ds.	cache/ejde-137.pdf	txt/ejde-137.txt