id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1051	Wu, Chun	Global boundedness in an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion	2025	17	.pdf	application/pdf	7365	432	82	Using wq−1 to test the third equation of (1.6) and integrating gives 1 q d dt ∫ Ω wq = −δ ∫ Ω wq + ∫ Ω uwq−1 ≤ −δ ∫ Ω wq + δ 2 ∫ Ω wq + C ∫ Ω uq (2.6) for all t ∈ (0, Tmax) which implies (2.5) with (2.3) and Lemma 2.4. □ Lemma 2.7. Applying integration by parts to the second equation in (1.6) and using the well-known equation 2∇v · ∇∆v = ∆|∇v|2 − 2|D2v|2, we find that d dt ∫ Ω v1−p|∇v|p = p ∫ Ω v1−p|∇v|p−2∇v · ∇(∆v − uvw)− (p− 1) ∫ Ω v−p|∇v|p(∆v − uvw) = p 2 ∫ Ω v1−p|∇v|p−2(∆|∇v|2 − 2|D2v|2)− p ∫ Ω v1−p|∇v|p−2∇v · ∇(uvw) − (p− 1) ∫ Ω v−p|∇v|p∆v + (p− 1) ∫ Ω wv1−p|∇v|p = p(p− 1) ∫ Ω v−p|∇v|p−2∇v · ∇|∇v|2 − p ∫ Ω v1−p|∇v|p−2|D2v|2 − p(p− 2) 4 ∫ Ω v1−p|∇v|p−4|∇|∇v|2|2 − p(p− 1) ∫ Ω v−p−1|∇v|p+2 + p 2 ∫ ∂Ω v1−p|∇v|p−2 · ∂|∇v|2 ∂ν + p(p− 2) 2 ∫ Ω wv−p+2|∇v|p−4∇v · ∇|∇v|2 + p ∫ Ω wv−p+2|∇v|p−2∆v − (p− 1)2 ∫ Ω wv1−p|∇v|p. (3.5) EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 9 The pointwise identity [49, Lemma 3.2] and ∇|∇v|2 = 2D2v · ∇v imply that p(p− 1) ∫ Ω v−p|∇v|p−2∇v ·	cache/ejde-1051.pdf	txt/ejde-1051.txt