Nonparametric mean curvature type flows of graphs with contact angle conditions

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ 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 $\psi(x,u,Du)=h(x,u)\omega$ with $h_u(x,u)\geq h_0>0$ and $\omega=\sqrt{1+|Du|^2}$. 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.