= Solution
For the <scalar conservation law>, put $v_0(\xi)=f'(u_0(\xi))$ and $m=\min_\xi v_0'(\xi)=\min_\xi f''(u_0(\xi))u_0'(\xi)$. The <concave-flux characteristic lifespan> and <characteristic flow map> are
$$
\boxed{t^*=\begin{cases}-1/m,&m<0,\\ \infty,&m\ge0,\end{cases}\qquad X_t(\xi)=\xi+t f'(u_0(\xi)),\qquad u(t,X_t(\xi))=u_0(\xi).}
$$
For $t<t^*$, $X_t'=1+t v_0'>0$, and $X_t(\xi)=\xi+ct$ outside the support of $u_0$. Thus $X_t$ is a global smooth <diffeomorphism> and the formula is $u(t,x)=u_0(X_t^{-1}(x))$. The <method of characteristics> proves existence and uniqueness among smooth solutions.
If $m<0$, at a minimizer $\xi_*$ the numerator $u_0'(\xi_*)$ is nonzero, since its product with $f''(u_0(\xi_*))$ is negative. Therefore
$$
u_x(t,X_t(\xi_*))=\frac{u_0'(\xi_*)}{1+t v_0'(\xi_*)}
$$
blows up as $t\uparrow t^*$, showing that this is the maximal smooth lifespan. Strict <concavity> alone does not require $f''$ to be negative at every point; the argument uses no such extra hypothesis.
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