  A modification of the parabolic Allen-Cahn equation, determined by the
substitution of Fick's diffusion law with a relaxation relation of
Cattaneo-Maxwell type, is considered. The analysis concentrates on traveling
fronts connecting the two stable states of the model, investigating both the
aspects of existence and stability. The main contribution is the proof of the
nonlinear stability of the wave, as a consequence of detailed spectral and
linearized analyses. In addition, numerical studies are performed in order to
determine the propagation speed, to compare it to the speed for the parabolic
case, and to explore the dynamics of large perturbations of the front.
