Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 30A / b

Parameterize the initial line by $(s,2)$ and use characteristic time $\tau$. The characteristic equations for a transport equation integrate toHere $dU/d\tau =U^2$ and $U(0)=h$. Since $\tau =\frac{1}{4}\log (x_2/2)$, the solution isDirect differentiation gives $u_{x_1}=0$, $u_{x_2}=u^2/(4x_2)$, verifying the equation and the initial values. The map $(s,\tau )\mapsto (x_1,x_2)$ has determinant $8e^{6\tau }\ne 0$, so every point of this strip is reached exactly once.

At the upper boundary $u\to +\infty$, a finite-time blow-up of an ordinary differential equation. At the lower boundary $u\to 0$, but $u_{x_2}=u^2/(4x_2)$ diverges as $x_2\downarrow 0$, so no $C^1$ continuation through that boundary is possible. Thus the displayed strip is the maximal connected domain containing the initial line on which this classical Cauchy solution exists.