How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

Question

I am having a test in few days and I saw an interesting question while I was skimming through the book problems.

The problem is concerned about initial-boundary value problem of 2nd order PDEs. To be specific, a IBVP problem of the diffusion (heat) equation.

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Let us say I have the diffusion equation in 1-D, with the diffusivity constant to be unity (for simplicity):

with the following conditions imposed on the PDE:

Now, I want to prove that the norm of is a non-increasing function of time (assuming the norm of to be finite.)

• Well I know that this norm is defined as
• How can I start the proof? Shall I use the Poincare inequality?

Answer 1

Write:

$$\frac{d}{dt} \int_0^L u^2(t,x) dx = \int_0^L 2 \frac{\partial u}{\partial t} u dx = \int_0^L 2 \frac{\partial^2 u}{\partial x^2} u dx$$

Now integrate by parts. Your boundary term will vanish, and the remaining integral term will be the integral of a nonpositive function, so it will be nonpositive. So all that's left is to justify the differentiation under the integral sign.