# Prove Solution to PDE Vanishes

• Jan 31st 2010, 05:29 PM
lvleph
Prove Solution to PDE Vanishes
I am not getting anywhere on this problem
Let $u$ be a solution of
$
a(x,y)u_x + b(x,y)u_y = -u
$

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
$
\frac{dx}{dt} = a(x,y)
$

$
\frac{dy}{dt} = b(x,y)
$

$
\frac{dz}{dt} = -z
$

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.
• Feb 1st 2010, 07:34 AM
lvleph
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.