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Math Help - Prove Solution to PDE Vanishes

  1. #1
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    Mar 2009
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    Prove Solution to PDE Vanishes

    I am not getting anywhere on this problem
    Let u be a solution of
    <br />
a(x,y)u_x + b(x,y)u_y = -u<br />
    of class C^1 in the closed unit disk \Omega in the xy-plane. Let a(x,y)x + b(x,y)y>0 on the boundary of \Omega. Prove that u vanishes
    identically. (Hint: Show that \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 \Gamma: x = s,\, y = 0, \, z = w. Then the characteristic ODE is
    <br />
\frac{dx}{dt} = a(x,y) <br />
    <br />
\frac{dy}{dt} = b(x,y)<br />
    <br />
\frac{dz}{dt} = -z<br />
    Therefore, z = c e^{-t}. So if c > 0 then u is a strictly decreasing function with respect to t. If c < 0 then u is a strictly increasing function with respect to 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; January 31st 2010 at 05:40 PM.
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  2. #2
    Senior Member
    Joined
    Mar 2009
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    So, the fact that u = ce^{-z} must imply that there is no min/max inside the region \Omega or that u is constant on \Omega. However, I can't seem to get to the vanishes part.
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