Where to specify boundary conditions

Question

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?

Answer 1

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 $x^2+y^2\le 1$, $t=0$.

Either way you get unique solution. The first condition is more of boundary value flavor, the second is of initial condition flavor. But in 1st order linear PDE the time and space are not strongly differentiated from each other.