Suppose that u(t, x) is a non-constant solution to the heat equation with homogeneous Neumann boundary conditions: Ut = Duxx for all 0 0 Ux (t,0) = ux(t, L) = 0 for all t > 0 5.1 Part 1 Prove that E(t) = / u(t, x) dx is constant. 5.2 Part 2 Prove that H ( t ) = u ( t , x ) 2 dx is strictly decreasing