We investigate the large-time asymptotics of the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with step-like initial data \begin{align*} q(x,0)=q_{0}(x)\rightarrow\left\{ \begin{aligned} &c_{l}, \quad x\rightarrow -\infty, &c_{r}, \quad x\rightarrow +\infty, \end{aligned} \right. \end{align*} where $c_l>c_r>0$. It follows from the standard direct and inverse scattering theory that an RH characterization for the step-like problem is constructed. By performing the nonlinear steepest descent analysis, we mainly derive the long-time asymptotics in the each of four asymptotic zones in the $(x,t)$-half plane.
Comment: 38 pages, 15 figures