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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Heat pde proof

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