 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 $1+t{f}^{\prime \prime }(u_0)u_0^{\prime }$ in the autonomous case, so a negative value of $f^{\prime \prime }(u_0)u_0^{\prime }$ causes finite-time crossing.