Energy Loss Over Time: Showing F(t) - F(0) <= 0

[eckiller]

I need to show Integral( u(x, t)^2 dx from a to b) <= Integral( u(x, 0)^2 dx from a to b). In other words, energy is lost with time.

This is what I have so far, but I get stuck near the end.

Let F(t) = Integral( u(x, t)^2 dx from a to b)

F(t) - F(0) = Integral( dF/dt dt from 0 to t)

= Integral( Integral( 2u(x, t) * u_t(x, t) dx from a to b ) dt from 0 to t)

By heat equation: u_t = u_xx

= Integral( Integral( 2u(x, t) * u_xx(x, t) dx from a to b ) dt from 0 to t)

Now I integrate by parts and get:

= 2 Integral( u(b,t) u_x(b, t) - u(a, t) u_x(a, t) - Integral( u_x(x, t)^2 dx from a to b ) dt from 0 to t)

At this point I know Integral( u_x(x, t)^2 dx from a to b ) >= 0.

I would like to conclude that the entire RHS is <= 0 so that F(t) - F(0) <= 0, but for arbitrary boundary conditions I am stuck.

 

[mighty2000]

To prove that the RHS is less than or equal to 0, you can use the Cauchy-Schwarz inequality:

|u(b,t) u_x(b,t) - u(a,t) u_x(a,t)| <= sqrt(Integral( u(b,t)^2 dx from a to b)) * sqrt(Integral( u_x(b,t)^2 dx from a to b))

= sqrt(F(t)) * sqrt(Integral( u_x(b,t)^2 dx from a to b))

= sqrt(F(t)) * sqrt(Integral( u_x(b,t)^2 dx from a to b))

= sqrt(F(t)) * sqrt(F(t)) = F(t)

Therefore, the RHS is less than or equal to F(t), which is less than or equal to 0. This means that F(t) - F(0) <= 0, or in other words, energy is lost with time.