= Concave-flux characteristic lifespan
{title2=$t_*$}
For smooth compactly supported data in the <scalar conservation law> $u_t+f(u)_x=0$, put $m=\min_\xi f''(u_0(\xi))u_0'(\xi)$. Its smooth <characteristic solution of a scalar conservation law> exists until $t_*=-1/m$ if $m<0$, and for all times if $m\ge0$. The <characteristic flow map> $X_t(\xi)=\xi+t f'(u_0(\xi))$ has derivative $1+t f''(u_0)u_0'$; its first zero produces gradient blowup. The formula applies to smooth fluxes of either concavity, although it is particularly useful for a <concave function> flux.
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