Characteristic crossing
= Characteristic crossing
Characteristic crossing occurs when the map from an initial point to its position at time $t$ loses injectivity. For $u_t+f(u)_x=0$, its Jacobian is $1+t f''(u_0)u_0'$ in the autonomous case, so a negative value of $f''(u_0)u_0'$ causes finite-time crossing.