id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-236	Engu, Satyanarayana; Sahoo, Manas R.; Berke, Venkatramana P.	Solutions to viscous Burgers equations with time dependent source term	2021	16	.pdf	application/pdf	7006	351	80	∫ 0 −∞ −wwt dx dt+ ∫ 0 −∞ w2(x, t) dx − 1 2 [ ∫ T 0 ∫ 0 −∞ (u+ v)wwx dx dt− ∫ T 0 (w (u+ v)w)(0, t) dt ] + ∫ T 0 ∫ 0 −∞ w2 x dx dt− ∫ T 0 (wxw)(0, t) dt = 0. (3.8) Similarly for φ = w(x, t)H(T − t)H(x), integral equation (3.7) yields∫ T 0 ∫ ∞ 0 −wwt dx dt+ ∫ ∞ 0 w2(x, t) dx + ∫ T 0 ∫ R w2 x dx dt+ ∫ T 0 [ (wxw)(0+, t)− (wxw)(0−, t) ] dt = 1 2 [ ∫ T 0 ∫ R (u+ v)wwx dx dt+ ∫ T 0 [ ((u+ v)w2)(0+, t)− ((u+ v)w2)(0−, t) ] dt ] , which implies ‖w(·, T )‖22 + 2 ∫ T 0 ‖wx(· , t)‖22 dt ≤ 1 2 ∫ T 0 ∫ R ‖(u+ v)(t)‖∞|w(x, t)‖wx(t)| dx dt ≤ 1 2 ∫ T 0 ‖(u+	cache/ejde-236.pdf	txt/ejde-236.txt