Proving the Non-negativity Property of a Diffusion Equation Solution

[modeiry88]

Homework Statement

Let u(x,t) satisfy

Homework Equations

($$\partial$$u/$$\partial$$t) = ($$\partial$$$$^{2}$$u/$$\partial$$x$$^{2}$$)...(0<x<1,t>0)

u(0,t)=u(1,t)=0...(t$$\geq$$0)

u(x,0)=f(x)...(o$$\leq$$x$$\leq$$1),

where f$$\in$$C[0.1] show that for any T$$\geq$$0

$$\int$$ from 0..1 (u(x,T))$$^{2}$$dx $$\leq$$ $$\int$$ from 0..1 (f(x))$$^{2}$$dx

The Attempt at a Solution

not sure

 

[berkeman]

That's 3 posts and no work shown on anyone of them. You *must* show an attempt at a solution, or we cannot be of help to you. Please click on the "Rules" link at the top of the page.

 

[modeiry88]

im sorry, I am obviously knew to this forum...

For this problem, I am trying to use the identity as follows

2u(($$\partial$$u/$$\partial$$t)-($$\partial$$$$^{2}$$u/$$\partial$$x$$^{2}$$)) = ($$\partial$$u$$^{2}$$/$$\partial$$t)-($$\partial$$/$$\partial$$x)*(u*($$\partial$$u/$$\partial$$x))+2*($$\partial$$u/$$\partial$$x)$$^{2}$$

 

[Dick]

It's a diffusion equation, so you might expect this sort of behavior. Your 'identity' is a little messed up. Can you fix it? Once you've done that substitute the PDE in. You should be able to show that ((u^2),t)/2-(u*(u,x)),x=(-(u,x)^2)<=0. I'm using commas for partial derivatives, forgive my laziness. Now integrate dx between 0 and 1. Can you show the (u*(u,x)),x term vanishes? Once you have integal (u^2),t<=0 you are home free.