id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-371	Gopal, Lavanya	Identification of the Memory Kernels and Controllability for Parabolic Equations	2010	19	.pdf	application/pdf	6605	279	77	The integral D2 can be bounded by D2 ≤ ∫∫ Q e−2sαs3λ3φ3|q|2d xd t + ∫∫ Q e−2sαλ(sφ)−1 ���� ∫ T t mt(τ, t)∆q(τ)dτ ��� 2 + ��� ∫ T t nt(τ, t)qτ(τ)dτ ��� 2� d xd t = D21 + D22 (17) R. Lavanya / Eur. Moreover, in proving the main inequality, we need the following estimates for the functions φ and α: �� ∂ φ ∂ t �� = |T−2t| t2(T−t)2 eλψ ≤ C(Ω,ω)Tφ2 �� ∂ α ∂ t �� = |T−2t| t2(T−t)2 |e2λΨ − eλψ| ≤ C(Ω,ω) Teλψ t2(T−t)2) ≤ C(Ω,ω)Tφ2 �� ∂ 2α ∂ t2 �� = |2T2−6T t+6t2 | t3(T−t)3 |e2λΨ − eλψ| ≤ C(Ω,ω)T 2φ3    (6) where C(Ω,ω) is a generic constant.	cache/ejpam-371.pdf	txt/ejpam-371.txt