Prove Solution to PDE Vanishes

I am not getting anywhere on this problem

Let be a solution of

of class in the closed unit disk in the -plane. Let on the boundary of Prove that vanishes

identically. (Hint: Show that 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 Then the characteristic ODE is

Therefore, . So if then is a strictly decreasing function with respect to . If then is a strictly increasing function with respect to . 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.