  In a previous work, we prove the existence of weak solutions to an
initial-boundary value problem, with $H^1(\Omega)$ initial data, for a system
of partial differential equations, which consists of the equations of linear
elasticity and a nonlinear, degenerate parabolic equation of second order.
Assuming in this article the initial data is in $H^2(\Omega)$, we investigate
the regularity of weak solutions that is difficult due to the gradient term
which plays a role of a weight. The problem models the behavior in time of
materials with martensitic phase transitions. This model with diffusive phase
interfaces was derived from a model with sharp interfaces, whose evolution is
driven by configurational forces, and can be thought to be a regularization of
that model. Our proof, in which the difficulties are caused by the weight in
the principle term, is only valid in one space dimension.
