If \phi:R\to R is a non-convex energy density, the initial-boundary value problem for the equation (1) \begin{equation} \left\{ \begin{array}{ll} u_t+\frac{1}{2}(\phi(u_x))_x,x\in(0,1) \\ u(\cdot,0)=u_0, \end{array} \right. \end{equation} is not well posed. In the attempt to define a notion of solution for equation (1) we consider, for 0<\epsilon<<1, the regularization: u_t=-\epsilon^2u_{xxxx}+\frac{1}{2}\phi''(u_x)u_{xx} of problem (1) and, motivated by numerical experiments, we reinterpret equation (1) as a suitable free-boundary problem which is well defined for any given initial datum.