Phase field equations with memory: The hyperbolic case

We present a phenomenological theory for phase transition dynamics with mem ory which yields a hyperbolic generalization of the classical phase field m odel when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in two dimensions for the inter face between two different phases. This equation can be considered as a hyp erbolic generalization of the classical ow by mean curvature equation, as w ell as a generalization of the Born-Infeld equation. We use a crystalline a lgorithm to study the motion of closed curves for our hyperbolic generaliza tion of ow by mean curvature and present some numerical results which indic ate that a certain type of two-dimensional damped oscillation may occur.