id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-657	Zhao, Fengxiang; Tang, Haotian; Zheng, Jiashan; Li, Kaiqiang	Existence and boundedness of solutions for a parabolic-parabolic predator-prey model	2025	17	.pdf	application/pdf	7168	311	81	To deal with u, multiplying both sides of the second equation in (1.1) by ur̃−1 and integrating by parts, for any small ε ∈ (0, 1), we derive from Young’s inequality that 1 r̃ d dt ∫ Ω ur̃ + (r̃ − 1) ∫ Ω ur̃−2|∇u|2 = − r̃ − 1 r̃ χ ∫ Ω ur̃∆w + λ1 ∫ Ω ur̃ − µ1 ∫ Ω ur̃+r1−1 + a ∫ Ω ur̃v ≤ r̃ − 1 r̃ κ ∫ Ω ur̃|∆w|+ λ1 ∫ Ω ur̃ − µ1 ∫ Ω ur̃+r1−1 + ε ∫ Ω ur̃+r1−1 + C3 ∫ Ω v r̃+r1−1 r1−1 ∀t ∈ (0, Tmax), (3.29) where C3 = r̃ + r1 − 1 r1 − 1 (ε r̃ + r1 − 1 r̃ )− r̃ r1−1 a r̃+r1−1 r1−1 . − r̃ r1−1 × ( r̃ − 1 r̃ κ ) r̃+r1−1 r1−1 ∫ Ω |∆w| r̃+r1−1 r1−1 = λ0 ∫ Ω ur̃+r1−1 + Ã1λ − r̃ r1−1 0 κ r̃+r1−1 r1−1 ∫ Ω |∆w| r̃+r1−1 r1−1 ∀t ∈ (0, Tmax), (3.32) where Ã1 = r1 − 1 r̃ + r1 − 1 ( r̃ + r1 − 1 r̃ )− r̃ r1−1 ( r̃ − 1 r̃ ) r̃+r1−1 r1−1 .	cache/ejde-657.pdf	txt/ejde-657.txt