Problem 3. Consider the classical solution to the initial boundary-value problem for the heat equation: (t, x) E (0, 0) x (0,1), te (0, 0), л€ [0, 1]. dru – Kôrau = 0, - u(t,0) = u(t, 1) = 0, (1.6)...

Extracted text: Problem 3. Consider the classical solution to the initial boundary-value problem for the heat equation: (t, x) E (0, 0) x (0,1), te (0, 0), л€ [0, 1]. dru – Kôrau = 0, - u(t,0) = u(t, 1) = 0, (1.6) u(0, x) = 4x(1 – x), (1) Show that 0 < u(t,="" x)="">< 1="" for="" all="" (t,="" x)="" e="" (0,="" ∞)="" ×="" [0,="" 1].="" (2)="" show="" that="" u(t,="" x)="u(t," 1="" –="" x)="" for="" all="" (t,="" x)="" e="" [0,="" c)="" ×="" [0,="" 1].="" (3)="" use="" the="" energy="" method="" to="" show="" that="" e(t)="|" lu(t,="" x)|²dæ="" (1.7)="" is="" a="" decreasing="" function="" of="" time.="">