Question: Please provide full solutions 2. Let ( c R be a bounded open set with piecewise C' boundary. Let u : [0, co) x ?

Please provide full solutions

2. Let ( c R" be a bounded open set with piecewise C' boundary. Let u : [0, co) x ? - R be a solution of the heat equation ut - Au =0 in (0, co) x 1; u =0 on (0, co) x 80; u(0, x) = f(x) on !. Assume that f(x) 20 on !. Prove that the quantity E(t)= u(t, r) dr is non-increasing in t

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