Generalized motion of hypersurfaces whose growth speed depends superlinearly on the curvature tensor

We prove a comparison principle for viscosity solution with finite speed for its level set, which solves degenerate parabolic equations with discontinuity. We also prove the (global) existence of solution in the class of viscosity solution with finite speed for the initial value problem. Our comparison and existence results yield a unique global-in-time generalized solution to interface evolution equations whose speed grows superlinearly in curvature tensors.