id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-759	Rassias, John Michael; Karimov, Erkinjon Tulkinovich	Boundary-value Problems with Non-local Initial Condition for Parabolic Equations with Parameter	2010	10	.pdf	application/pdf	3669	239	79	Then we get ∫ ∂Ω ∫ Re � ymuux cos (ν , x) + xnuuy cos � ν , y �− 1 2 xn ym |u|2 cos (ν , t) � dτ = ∫ ∫ Ω ∫ � ym ��ux ��2 + xn ��uy ��2 +λ1 xn ym |u| � dσ where ν is outer normal. Considering Re � uux � = Re � uux � , Re � uuy � = Re � uuy � we obtain Re ∫ ∫ S1 1 2 xn ym |u|2 dτ1 + ∫ ∫ S2 ymRe � uux � dτ2− ∫ ∫ S3 xnRe � uuy � dτ3− ∫ ∫ S4 ymRe � uux � dτ4 + ∫ ∫ S5 xnRe � uuy � dτ5−Re ∫ ∫ S6 1 2 xn ym |u|2 dτ6 (8) = ∫ ∫ Ω ∫ � ym ��ux ��2 + xn ��uy ��2 +λ1 xn ym |u| � dσ. From (8) and by using conditions (6), (8), we find 1 2 � 1− � α2 1 +α 2 2 �� 1∫ 0 1∫ 0 xn ym ��u�x , y, 1 ���d xd y + ∫ ∫ Ω ∫ � ym ��ux ��2 + xn ��uy ��2 +λ1 xn ym |u| � dσ= 0. (9) J. Rassias, E. Karimov / Eur.	cache/ejpam-759.pdf	txt/ejpam-759.txt