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## Homework Statement

Let u(x,t) satisfy

## Homework 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

## The Attempt at a Solution

not sure