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

Let Ω be the unit ball in ${‐{R}}^N$ . Consider the mean curvature equation $u_t=(1+|\mathit{Du}{|}^2{)}^{\sigma /2}\left[\mathrm{div}\frac{\mathit{Du}}{\sqrt{1+|\mathit{Du}{|}^2}}+A\right]\text{for}x\in \Omega \text{,}t>0\text{,}$ with capillarity boundary condition $\mathit{Du}·\gamma =k(t\text{,}u)\sqrt{1+|\mathit{Du}{|}^2}\text{for}x\in \vartheta \Omega \text{,}t>0\text{,}$ where δ and A (∂0) are real numbers, γ is the unit inner normal to ∂O 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 $k=\tilde{k}(u)$ is periodic.