id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1250	Xu, Ling	Asymptotic behavior of Kirchhoff type plate equations with nonlocal weak damping, anti-damping and subcritical nonlinearity	2025	25	.pdf	application/pdf	10466	520	81	+ ∫ T 0 Im(t)dt ≤ C(R) {∫ T 0 ∥ιt(t)∥2dt+ k ∫ T 0 (∥wt∥pwt − ∥vt∥pvt, ιt)dt + k ∫ T 0 |(∥wt∥pwt − ∥vt∥pvt, ι)|dt+ ∫ T 0 ∥∇ι∥2dt+ ∫ T 0 dt ∫ T t ∥∇ι(τ)∥2dτ + ∫ T 0 dt ∫ T t ∥∇ι(τ)∥∥ιt(τ)∥dτ + ∣∣∣ ∫ T 0 (N (ιt), ιt)dt ∣∣∣+ ∣∣∣ ∫ T 0 (N (ιt), ι)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t (N (ιt), ιt)dτ ∣∣∣+ ∣∣∣ ∫ T 0 (g(w)− g(v), ιt)dt ∣∣∣+ ∣∣∣ ∫ T 0 (g(w)− g(v), ι)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t (g(w)− g(v), ιt(τ))dτ ∣∣∣}, ∀T ≥ + ∫ T 0 dt ∫ T t (N (ιt), ιt)dτ ≤ C {∫ T 0 ∥ιt∥2dt+ k ∫ T 0 (∥wt∥pwt − ∥vt∥pvt, ιt)dt + k ∫ T 0 |(∥wt∥pwt − ∥vt∥pvt, ι)|dt+ ∣∣∣ ∫ T 0 ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dt ∣∣∣ + ∣∣∣ ∫ T 0 ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ι)dt ∣∣∣+ ∣∣∣ ∫ T 0 m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ ∣∣∣ + ∣∣∣ ∫ T 0 (N (ιt), ιt)dt ∣∣∣+ ∣∣∣ ∫ T 0 (N (ιt), ι)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t (N (ιt), ιt)dτ ∣∣∣+ ∣∣∣ ∫ T 0 (g(w)− g(v), ιt)dt ∣∣∣ + ∣∣∣ ∫ T 0 (g(w)− g(v),	cache/ejde-1250.pdf	txt/ejde-1250.txt