id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1668	Zheng, Jiashan; Wang, Yuying	Blow-up prevention and rate of convergence of solutions for N-dimensional parabolic-parabolic systems with consumption of chemoattractant	2025	21	.pdf	application/pdf	10636	464	62	+ ∫ t 0 ∥∇e(t−s)∆u(·, s)v(·, s)∥Lq(Ω)ds ≤ C1 + C2 ∫ t 0 ( 1 + (t− s)− 1 2− N 2 ( 1 k− 1 q ) ) e−λ1(t−s)∥u(·, s)v(·, s)∥Lk(Ω) (4.22) for all t ∈ (0, Tmax), where λ1 is the first positive eigenvalue of −∆ under homogeneous Neumann boundary conditions. From (4.21)and the identity −1 2 − N 2 (1 k − 1 q ) > −1 (4.23) we obtain ∫ t 0 ( 1 + (t− s)− 1 2− N 2 ( 1 k− 1 q ) ) e−λ1(t−s)ds ≤ ∫ ∞ 0 ( 1 + σ− 1 2− N 2 ( 1 k− 1 q ) ) e−λ1σdσ < +∞ for all t ∈ (0, Tmax), (4.24) and that for some C3 > 0, the latter on the right-hand side of (4.22) bringing together (4.23)-(4.24) also trivially provides C2 ∫ t 0 ( 1 + (t− s)− 1 2− N 2 ( 1 k− 1 q ) )	cache/ejde-1668.pdf	txt/ejde-1668.txt