Entire solutions of a mean curvature flow connecting two periodic traveling waves

Abstract

We consider the mean curvature flow V=H in a cylinder Ω≔B×R, where B is the unit ball in RN, V is the normal velocity of a moving surface Γt:xN+1=u(x,t)(x∈B) and H is the mean curvature of Γt. Assume that the surface is radially symmetric and it contacts the boundary of Ω with a prescribed angle θ=θ(t,u). In case θ tends to two (spatially or temporally) periodic functions as u→±∞, we show that the problem has a unique entire solution U, which propagates from −∞ to ∞ and connects two (spatially or temporally) periodic traveling waves at t=−∞ and at t=∞.