= Parabolic characteristic strip from a finite initial segment
{title2=$yu_x+u_y=u,\quad u(x,0)=x\ (1\leq x\leq2)$}
The equations of the <method of characteristics> give $(x,y,u)=(s+t^2/2,t,se^t)$. 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 $1\leq x-y^2/2\leq2$. Both positive and negative $y$ occur. Outside this closed parabolic strip, $u=e^yF(x-y^2/2)$ allows arbitrary smooth changes of $F$ outside the initial parameter interval. The boundary <characteristic curves> carry the data from its endpoints.
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