Head and Tail Speeds of Mean Curvature Flow with Forcing

In this paper, we investigate the large time behavior of interfaces moving with motion law V = - κ + g ( x ) , where g is positive, Lipschitz and Z n -periodic. We show that the behavior of the interface can be characterized by its head and tail speeds s ¯ and s ̲ , which only depend on its overall direction of propagation ν . We discuss the large time behavior of the moving interface in terms of s ¯ and s ̲ , which is shown to vary continuously in ν . In the laminar setting we show that when s ¯ > s ̲ there exists an unbounded stationary solution as well as localized traveling waves with different speeds.