Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 29C / ii / Solution

Here the characteristic curves are straight lines $x=y+sa$, $y\cdot n=0$, along which $\dot{u}=c\cdot (y+sa)+d$. If $a\cdot n\ne 0$, every nearby point determines a unique starting point and travel parameter:Integrating along the Characteristic curve givesThe non-characteristic condition is exactly what permits division by $a\cdot n$ and ensures transversality to the data hyperplane. If $a\cdot n=0$, the Characteristic curve lines starting there never leave it. Moreover the prescribed data must satisfy $a\cdot {\nabla }_S{u}_0=c\cdot x+d$ on the hyperplane even to be compatible. If they do, the Characteristic curve construction still leaves off-hyperplane data undetermined; if they do not, no $C^1$ solution exists.