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Math Help - Heat Equation PDE

  1. #1
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    Heat Equation PDE

    Let u(x,t) satisfy u_{t}=u_{xx} for x \in (0,1) and t>0, the boundary conditions u(0,t)=u(1,t)=0 for t\geq 0, and the initial condition u(x,0)=f(x) for x\in [0,1] with f being a continuously differentiable function. Prove that
    \int_{0}^{1} |u(x,t)|^2 x\leq \int_{0}^{1} |f(x)|^2 x, for any t\geq 0

    Hint: use 2uu_{t}=(u^2)_t and 2uu_{xx}=(uu_{x})_{x}-2(u_x)^2
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  2. #2
    Super Member PaulRS's Avatar
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    Consider the function g(t) =  \displaystyle\int_0^1{\left(u(x,t)\right)^2dx} and show that it is not increasing! -differentiate.
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