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

Analytical and numerical investigation of traveling waves for the Allen-Cahn model with relaxation