Uniform gradient bounds and convergence of mean curvature flows in a cylinder

We consider a mean curvature flow V = H + A in a cylinder Omega x R, where, Omega is a bounded domain in R-n, A is a constant driving force, Vand Hare the normal velocity and the mean curvature respectively of a moving hypersurface, which contacts the cylinder boundary with prescribed angle theta(x). Under certain conditions such as Omega is convex and parallel to cos theta parallel to(C2) is small, or Omega is non-convexand vertical bar A vertical bar is large, we derive the uniform gradient boundsfor bounded and unbounded solutions (which is crucial in the study of the asymptotic behavior of the solutions). Then we present a trichotomy result on the convergence of the solutions as well as its criterion: when A vertical bar Omega vertical bar + integral(partial derivative Omega) cos theta(x) d sigma > 0 (resp. = 0, < 0), the solution converges as t -> infinity to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed). (c) 2023 Elsevier Inc. All rights reserved.