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

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.