Having Alot of Trouble

modeiry88

1. The problem statement, all variables and given/known data

Let u(x,t) satisfy


2. Relevant equations


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

u(0,t)=u(1,t)=0........(t[tex]\geq[/tex]0)

u(x,0)=f(x)........(o[tex]\leq[/tex]x[tex]\leq[/tex]1),

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

[tex]\int[/tex] from 0..1 (u(x,T))[tex]^{2}[/tex]dx [tex]\leq[/tex] [tex]\int[/tex] from 0..1 (f(x))[tex]^{2}[/tex]dx


3. The attempt at a solution

not sure
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

berkeman

That's 3 posts and no work shown on any one 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, im obviously knew to this forum....

For this problem, im trying to use the identity as follows

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

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.