A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

In this paper we consider a mean curvature flow \(V=H+A\) in a high dimensional cylinder \(\Omega\times \R\), where, \(A\) is a constant, \(\Omega\) is a bounded domain in \(\R^n\), and, for a hypersurface \(y=u(x,t)\) over \(\Omega\), \(V\) and \(H\) denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary \(\partial \Omega \times \R\) with prescribed angle \(\theta(x)\). Under certain assumptions such as \(\Omega\) is strictly convex and \(\|\cos\theta\|_{C^2}\) is small, or \(\Omega\) is not necessarily convex but \(|A|\) is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when \(I:= A|\Omega|+\int_{\partial \Omega} \cos\theta(x) d\sigma>0\) (resp. \(=0\), \(