Periodic traveling waves of a mean curvature equation in high dimensional cylinders

摘要

Let Ω be the unit ball in RN. Consider the mean curvature equation (E0)ut=(1+|Du|2)σ/2divDu1+|Du|2+Aforx∈Ω,t>0,with capillarity boundary condition (BC0)Du·γ=k(t,u)1+|Du|2forx∈∂Ω,t>0,where σ and A (⩾0) are real numbers, γ is the unit inner normal to ∂Ω and k is a smooth function with ∣k∣ < 1. We first study the time-global existence of radial solutions of (E0)-(BC0) with some initial datum, and then study the existence, uniqueness and stability of the radial periodic traveling wave of (E0)-(BC0) when k = k(t) or k=k˜(u) is periodic.