Nonparametric Mean Curvature Type Flows of Graphs with Contact Angle Conditions

In this article, we study nonparametric mean curvature type flows in $M{\mkern-1mu\times\mkern-1mu}\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $\psi(x,u,Du)$. Long time existence and uniformly convergence are established if $\psi(x,u, Du)\equiv 0$ with vertical contact angle and $\psi(x,u,Du)=h(x,u)\sqrt{1+|Du|^2}$ with $h_u(x,u)\geq h_0>0$. Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymptotic behavior of nonparametric mean curvature flows of graphs over a convex domain in $M^2$ which is a surface with nonnegative Ricci curvature.