# Where to specify boundary conditions

The problem asks me where I need to specify boundary values for the linear PDE problem:

$u_t + xu_x + yu_y = 0$ on the domain $\Omega = x^2 + y^2 \le 1$. Using characteristics I get that $u(x,y,t) = u_0(x_0, y_0)$ and the characteristic lines are $x(t) = x_0e^t$ and $y(t) = y_0e^t$. I'm not really sure how to proceed from there because this requires that the boundary value problem be an initial value problem...help?

• Nobody?? I would really appreciate any direction on this – user137302 Mar 23 '14 at 19:12

So, the characteristic curves beginning at $(x_0,y_0,0)$ eventually exit the space-time cylinder through its lateral surface, except if $x_0=y_0=0$. Two possibilities for boundary conditions:
• prescribe them on the cylinder $x^2+y^2=1$ and also at the point $(0,0,0)$.
• alternatively, prescribe them on the disk $x^2+y^2\le 1$, $t=0$.