Existence and uniqueness of asymptotic traveling waves in a time-fractional bistable reaction-diffusion equation

We study asymptotic traveling waves for a reaction-diffusion equation with a Caputo time derivative. For general C¹ bistable imbalanced nonlinearities, we prove the existence of an asymptotic traveling wave connecting the two stable states. The asymptotic wave speed is unique, and the asymptotic profile is strictly increasing and unique up to translation. The proof uses continuation from a stationary front and the sliding method. For a cubic nonlinearity, we also derive an asymptotic formula for the speed near the balanced case, with an explicit leading coefficient. Finally, we perform numerical simulations to compare the speed predicted by the asymptotic formula with the front propagation speed in the original equation.