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Thread: Prove Solution to PDE Vanishes

  1. #1
    Senior Member
    Joined
    Mar 2009
    Posts
    378

    Prove Solution to PDE Vanishes

    I am not getting anywhere on this problem
    Let $\displaystyle u$ be a solution of
    $\displaystyle
    a(x,y)u_x + b(x,y)u_y = -u
    $
    of class $\displaystyle C^1$ in the closed unit disk $\displaystyle \Omega$ in the $\displaystyle xy$-plane. Let $\displaystyle a(x,y)x + b(x,y)y>0$ on the boundary of $\displaystyle \Omega.$ Prove that $\displaystyle u$ vanishes
    identically. (Hint: Show that $\displaystyle \max_{\Omega} u \le 0, \, \min_{\Omega} u \ge 0,$ using conditions for a maximum at a boundary point.)

    Since we have been using method of characteristics, I assume we need to use that.
    So let $\displaystyle \Gamma: x = s,\, y = 0, \, z = w.$ Then the characteristic ODE is
    $\displaystyle
    \frac{dx}{dt} = a(x,y)
    $
    $\displaystyle
    \frac{dy}{dt} = b(x,y)
    $
    $\displaystyle
    \frac{dz}{dt} = -z
    $
    Therefore, $\displaystyle z = c e^{-t}$. So if $\displaystyle c > 0$ then $\displaystyle u$ is a strictly decreasing function with respect to $\displaystyle t$. If $\displaystyle c < 0$ then $\displaystyle u$ is a strictly increasing function with respect to $\displaystyle t$. However, I am not sure how this even helps me.

    EDIT: I will even take some brain storming from people. I could use any help offered.
    Last edited by lvleph; Jan 31st 2010 at 04:40 PM.
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  2. #2
    Senior Member
    Joined
    Mar 2009
    Posts
    378
    So, the fact that $\displaystyle u = ce^{-z}$ must imply that there is no min/max inside the region $\displaystyle \Omega$ or that $\displaystyle u$ is constant on $\displaystyle \Omega$. However, I can't seem to get to the vanishes part.
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