https://neckpinch.com/authors/bendong-lou/New arXiv papers by Bendong Lou, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2302.07831arxiv:2302.07831Fri, 03 Feb 2023 00:00:00 GMTXinfu Chen, Bendong Lou, Xiaoliu Wang, Lixia Yuan — We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen–Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a fixed profile and finite speed. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient |Du| (away from the center) and the instantaneous speed uₜ diverge exponentially as t→∞. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time C⁰, C¹, or C² estimates are available, our analysis relies on a new approach based on the zero number argument.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactLong-time existence & convergenceRotational symmetry & warped productsPhase fields & multiphase flows https://arxiv.org/abs/2210.16475arxiv:2210.16475Sat, 29 Oct 2022 00:00:00 GMTZhenghuan Gao, Bendong Lou, Jinju Xu — In this paper we consider a mean curvature flow V=H+A in a high dimensional cylinder Ω× R, where, A is a constant, Ω is a bounded domain in Rⁿ, and, for a hypersurface y=u(x,t) over Ω, V and H denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary ∂Ω× R with prescribed angle θ(x). Under certain assumptions such as Ω is strictly convex and \|cosθ\|_(C²) is small, or Ω 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|Ω|+∫_∂Ω cosθ(x) dσ>0 (resp. =0, <0), the solution u converges as t→ ∞ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).Mean curvature flowSelf-shrinkers & translatorsFree boundary & capillary https://arxiv.org/abs/2205.14431arxiv:2205.14431Sat, 28 May 2022 00:00:00 GMTBendong Lou, Lixia Yuan — We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy a prescribed boundary angle in a cylinder. We then consider the flow starting at an initial hypersurface: showing the a priori estimates (especially the uniform-in-time bounds for the mean curvature which guarantee the uniform parabolicity of the corresponding fully nonlinear equation), giving the global existence for the solution of the initial boundary value problem, and proving its convergence to the corresponding translating solution. Our study provides a complete exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angles on the existence and stability of radially symmetric translating solutions.Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessLong-time existence & convergenceRotational symmetry & warped products