 Domain determined by a finite segment under a radial characteristic flow

For a first-order partial differential equation with the displayed projected characteristic field, label the initial segment by $r=y(0)\in [0,c]$. The characteristic curves are $x=a-(a-b)e^s$, $y=r{e}^s$. They are the branches of straight lines through $(a,0)$ on the side $x<a$. The inverse coordinates areThey are unique exactly in the displayed sector. The Jacobian determinant is $-(a-b)e^{2s}\ne 0$, so the initial segment is noncharacteristic and its transported data determine a unique classical solution on the sector's interior whenever the characteristic equation for the dependent variable has a unique solution. The bounding rays inherit data from the two endpoints. The point $(a,0)$ is approached only at infinite parameter time, and characteristics on the opposite side cannot reach the initial segment.