The modified Camassa-Holm equation in Lagrangian coordinates

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: $T_{max}\geq \frac{1}{||m_0||{L^\infty}||m_0||{L^1}}.$Andthereisauniquesolution$X(\xi,t)$totheLagrangedynamicswhichisastrictlymonotonicfunctionof$\xi$forany$t\in[0,T_{max})$:$X_\xi(\cdot,t)>0$\mathrm{.}As$t$approaching$T_{max}$,weprovethatclassicalsolution$m(\cdot ,t)$ 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.