Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 30A (Partial Differential Equations)

For the Cauchy problem for a partial differential equation, suppose $a,b,c$ are $C^1$ near the lifted initial data, $\Gamma$ has a regular $C^1$ parametrization $(\xi (s),\eta (s))$, and the initial function $u_0(s)$ is $C^1$. The local existence and uniqueness theorem requires the Noncharacteristic Cauchy data conditionat the starting point. Then there is a neighbourhood of that point carrying a unique $C^1$ classical solution attaining the initial values.

The method of characteristics proves this by solvingwith initial values $(\xi (s),\eta (s),u_0(s))$ at $\tau =0$. Local existence for these ordinary differential equations gives a $C^1$ map. The noncharacteristic determinant is, up to sign, the Jacobian determinant of $(s,\tau )\mapsto (x_1,x_2)$ initially. The inverse function theorem makes this map locally invertible, defining $u(x_1,x_2)=U(s,\tau )$ and proving uniqueness along characteristics.

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.