Influence of the Domain Boundary on the Speeds of Traveling Waves

Abstract

Let H > 0 be a constant, g ≥ 0 be a periodic function and Ω ={(x, y) | ｜ x ｜ < H + g ( y ), y ∈ R} . We consider a curvature flow equation V = κ + A in Ω, where for a simple curve γ t Ω , V denotes its normal velocity, κ denotes its curvature and A > 0 is a constant. [1] proved that this equation has a periodic traveling wave U , and that the average speed c of U is increasing in A and H , decreasing in max g' when the scale of g is sufficiently small. In this paper we study the dependence of c on A , H , max g' and on the period of g when the scale of g is large. We show that similar results as [1] hold in certain weak sense.