id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-179	Camasta, Alessandro; Fragnelli, Genni	Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions	2022	22	.pdf	application/pdf	9200	521	78	Thus, we introduce Y := { u ∈ H2 1/a(0, 1) : u(x0) = (au′)(x0) = 0 } and, proceeding as in [9] and [21] (if x0 ∈ {0, 1}) or as in [7] (if x0 ∈ (0, 1)), one can prove the following result. If u0 ∈ Xµ and h ∈ L2(0, T ;Xµ), a function u is said to be a weak solution of (3.7) if u ∈ C ( [0, T ];Xµ ) ∩ L2 ( 0, T ;H2 a(0, 1) ) and∫ 1 0 u(T, x)ϕ(T, x) dx− ∫ 1 0 u0(x)ϕ(0, x) dx− ∫ (0,T )×(0,1) u(t, x)ϕt(t, x) dx dt + a(1)u(T, 1)ϕ(T, 1) β1 − a(1)u0(1)ϕ(0, 1) β1 − a(1) β1 ∫ T 0 u(t, 1)ϕt(t, 1)dt + a(0)u(T, 0)ϕ(T, 0) β0 − a(0)u0(0)ϕ(0, 0) β0 − a(0) β0 ∫ T 0 u(t, 0)ϕt(t, 0)dt = − ∫ (0,T )×(0,1) a(x)uxx(t, x)ϕxx(t, x) dx dt− γ1 β1 ∫ T 0 a(1)u(t, 1)ϕ(t, 1)dt − γ0 β0 ∫ T 0 a(0)u(t, 0)ϕ(t, 0)dt+ ∫ (0,T )×(0,1) h(t, x)ϕ(t, x) dx dt + ∫ T 0 a(1)h(t, 1)ϕ(t, 1)	cache/ejde-179.pdf	txt/ejde-179.txt