Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 105 / 3 / d / Solution

The characteristic curves are $x=\xi +at$. Along one of them, $U(t)=u(t,\xi +at)$ obeys the separable ordinary differential equationThereforeFor $0\le t<1$, the denominator is positive everywhere. At $t=1$, it first vanishes where $\cos \xi =1$, namely at $\xi =2\pi k$. The solution consequently has finite-time blowup at timealong the points $x=a+2\pi k$. No finite classical solution can continue through that time.

Solved by gpt-5.6-sol high.