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Math Help - heat equation problem

  1. #1
    Newbie
    Joined
    Jan 2011
    Posts
    24

    heat equation problem

    Hi!

    I've got the following problem:

    f:\mathbb{R}\rightarrow\mathbb{R} a C^2-function such that f is convex and f(0)=f'(0)=0. g\in C^{\infty}([0,\infty),\mathscr{S}(\mathbb{R}^d)) a real-valued solution to the heat equation g_t=\Delta g. Using this show that F\in C^1 and F decrasing, where:

    F(t)=\int_{\mathbb{R}^d}f(g(x,t))dx.

    ----------------------------------------------------------------
    'I started by just diff. F and I got

    F'(t) = \frac{d}{dt}\int_{\mathbb{R}^d}f(g(x,t))dx = \int_{\mathbb{R}^d} \frac{d}{dt}f(g(x,t))dx = \int_{\mathbb{R}^d}\frac{df(g)}{dg}g_tdx = \int_{\mathbb{R}^d} \frac{df}{dg}\Delta gdx.

    What next?
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  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721

    Re: heat equation problem

    Notice

    F'(t)= \int_{ \mathbb{R}^d } f'(g)\Delta g dx = - \int_{ \mathbb{R}^d } \nabla \left( f'(g) \right) \cdot \nabla (g)dx = -\int_{ \mathbb{R}^d } f''(g)|\nabla(g)|^2 dx\leq0

    where we first integrated by parts and then used that f''\geq 0.
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