id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-744	Zhang, Guang; Chai, Shugen	Exact controllability for degenerate and singular wave equations	2025	17	.pdf	application/pdf	6665	372	75	= ∫ 1 0 uv + au′v′ dx, ∀u, v ∈ V 1 a (0, 1), V 2 a (0, 1) = {u ∈ V 1 a (0, 1)|au′ ∈ H1(0, 1)}; and V 1 a,b(0, 1) = {u ∈ L2(0, 1) ∩H1 loc(0, 1]|a(u′) 2 − λ b u2 ∈ L1(0, 1)}, ∥u∥2V 1 a,b(0,1) = ∫ 1 0 u2 + a(u′) 2 − λ b u2 dx, ∀u ∈ V 1 a,b(0, 1), ⟨u, v⟩V 1 a,b(0,1) = ∫ 1 0 uv + au′v′ − λ b uv dx, ∀u, v ∈ V 1 a,b(0, 1), V 2 a,b(0, 1) = {u ∈ V 1 a,b(0, 1)|(au′)′ − λ b u ∈ L2(0, 1)}. Then, for every T ≥ 0, a(1) ∫ T 0 u2 x(t, 1)dt ≥ − ( 4max{ 1 a(1)θ , 1}+ 2Ka 1√ θa(1) ) Eu(0) + T{(2−Ka) + λCa,bθ (2−Ka −Kb)}Eu(0), (5.3) for λ ∈ (0, 1 Ca,b ), and a(1) ∫ T 0 u2 x(t, 1)dt ≥ − ( 4max{ 1 a(1) , 1}+ 2Ka 1√ a(1) ) Eu(0) + T{(2−Ka)− |λ|Ca,b (2−Ka −Kb)}Eu(0), (5.4) for λ ∈ (−∞, 0], where the constants Ca,b and θ are given in (2.3) and (2.6), respectively.	cache/ejde-744.pdf	txt/ejde-744.txt