Parabolic characteristic strip from a finite initial segment ($y{u}_x+u_y=u,u(x,0)=x(1\le x\le 2)$)

The equations of the method of characteristics give $(x,y,u)=(s+t^2/2,t,s{e}^t)$. Their parameter map to the plane has Jacobian determinant one, so it is globally invertible and $u=(x-y^2/2)e^y$ is uniquely determined exactly where $1\le x-y^2/2\le 2$. Both positive and negative $y$ occur. Outside this closed parabolic strip, $u=e^{y}F(x-y^2/2)$ allows arbitrary smooth changes of $F$ outside the initial parameter interval. The boundary characteristic curves carry the data from its endpoints.