Global solution to a phase transition model with microscopic movements and accelerations in one space dimension

This note deals with a nonlinear system of PDEs accounting for
phase transition phenomena. The existence of solutions of a related Cauchy-
Neumann problem is established in the one-dimensional setting. A fixed point
procedure guarantees the existence of solutions locally in time. Next, an ar-
gument based on a priori estimates allows to extend such solutions in the
whole time interval. Hence, the uniqueness of the solution is proved by proper
contracting estimates.