  In this paper, we study the modified Camassa-Holm (mCH) equation in
Lagrangian coordinates. For some initial data $m_0$, we show that classical
solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$,
existence and uniqueness of classical solutions are established. Lifespan for
classical solutions is obtained: $T_{max}\geq
\frac{1}{||m_0||_{L^\infty}||m_0||_{L^1}}.$ And there is a unique solution
$X(\xi,t)$ to the Lagrange dynamics which is a strictly monotonic function of
$\xi$ for any $t\in[0,T_{max})$: $X_\xi(\cdot,t)>0$. As $t$ approaching
$T_{max}$, we prove that classical solution $m(\cdot ,t)$ in Eulerian
coordinate has a unique limit $m(\cdot,T_{max})$ in Radon measure space and
there is a point $\xi_0$ such that $X_\xi(\xi_0,T_{max})=0$ which means
$T_{max}$ is an onset time of collision of characteristics. We also show that
in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize
the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon
measure space.
