doi: 10.3934/dcds.2021115
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On a curvature flow in a band domain with unbounded boundary slopes

1. 

Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China

2. 

School of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

* Corresponding author: Wei Zhao

Received  March 2021 Revised  June 2021 Early access August 2021

Fund Project: The first author is supported by by NNSFC (No. 12001375). The second author is supported NNSFC (No. 11761058) and NSFS (No. 19ZR1411700, No. 21ZR1418300)

This paper is devoted to an anisotropic curvature flow of the form $ V = A(\mathbf{n})H + B(\mathbf{n}) $ in a band domain $ \Omega : = [-1,1]\times {\mathbb{R}} $, where $ \mathbf{n} $, $ V $ and $ H $ denote respectively the unit normal vector, normal velocity and curvature of a graphic curve $ \Gamma_t $. We require that the curve $ \Gamma_t $ contacts $ \partial \Omega $ with slopes equaling to the heights of the contact points (which corresponds to a kind of Robin boundary conditions). In spite of the unboundedness of the boundary slopes, we are able to obtain the uniform interior gradient estimates for the solutions by using the zero number argument. Furthermore, when $ t\to \infty $, we show that $ \Gamma_t $ converges to a traveling wave with cup-shaped profile and infinite boundary slopes in the $ C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}}) $-topology.

Citation: Lixia Yuan, Wei Zhao. On a curvature flow in a band domain with unbounded boundary slopes. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021115
References:
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[22]

H. MatanoK. I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568.  doi: 10.3934/nhm.2006.1.537.  Google Scholar

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J. RubinsteinP. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.  doi: 10.1137/0149007.  Google Scholar

show all references

References:
[1]

M. AlfaroD. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565.  doi: 10.1016/j.jde.2008.01.014.  Google Scholar

[2]

S. J. Altschuler and L. F. Wu, Convergence to translating solutions for a class of quasilinear parabolic boundary problems, Math. Ann., 295 (1993), 761-765.  doi: 10.1007/BF01444916.  Google Scholar

[3]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.  Google Scholar

[4]

S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[5]

J. Cai and B. Lou, Convergence in a quasilinear parabolic equation with Neumann boundary conditions, Nonl. Anal., 74 (2011), 1426-1435.  doi: 10.1016/j.na.2010.10.016.  Google Scholar

[6]

Y. L. ChangJ. S. Guo and Y. Kohsaka, On a two-point free boundary problem for a quasilinear parabolic equation, Asymptotic Anal., 34 (2003), 333-358.   Google Scholar

[7]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[8]

X. Chen and J.-S. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311.  doi: 10.1007/s00208-010-0558-7.  Google Scholar

[9]

X. Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630.  doi: 10.1007/s002080050202.  Google Scholar

[10]

K. -S. Chou and X. L. Wang, The curve shortening problem under Robin boundary condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 177-194.  doi: 10.1007/s00030-011-0123-4.  Google Scholar

[11]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312.  doi: 10.4171/JEMS/198.  Google Scholar

[12]

M.-H. GigaY. Giga and H. Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226.  doi: 10.1137/040614372.  Google Scholar

[13]

J.-S. Guo and B. Hu, On a two-point free boundary problem, Quart. Appl. Math., 64 (2006), 413-431.  doi: 10.1090/S0033-569X-06-01021-1.  Google Scholar

[14]

J.-S. GuoH. MatanoM. Shimojo and C. H. Wu, On a free boundary problem for the curvature flow with driving force, Arch. Ration. Mech. Anal., 219 (2016), 1207-1272.  doi: 10.1007/s00205-015-0920-8.  Google Scholar

[15] M. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 1993.   Google Scholar
[16]

G. Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differential Equations, 77 (1989), 369-378.  doi: 10.1016/0022-0396(89)90149-6.  Google Scholar

[17]

Y. Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonl. Anal., 45 (2001), 865-894.  doi: 10.1016/S0362-546X(99)00422-8.  Google Scholar

[18]

B. Lou, Periodic traveling waves of a mean curvature flow in heterigeneous media, Discrete Contin. Dynam. Syst., 15 (2009), 231-249.  doi: 10.3934/dcds.2009.25.231.  Google Scholar

[19]

B. Lou, The zero number diminishing property under general boundary conditions, Appl. Math. Lett., 95 (2019), 41-47.  doi: 10.1016/j.aml.2019.03.016.  Google Scholar

[20]

B. LouH. Matano and K. Nakamura, Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed, J. Differential Equations, 255 (2013), 3357-3411.  doi: 10.1016/j.jde.2013.07.038.  Google Scholar

[21]

B. Lou, X. Wang and L. Yuan, Convergence to a grim reaper for a curvature flow with variable boundary slopes, Calc. Var. Partial Differential Equations, 60 (2021), https: //doi.org/10.1007/s00526-021-01991-x Google Scholar

[22]

H. MatanoK. I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568.  doi: 10.3934/nhm.2006.1.537.  Google Scholar

[23]

W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), 900-904.  doi: 10.1063/1.1722511.  Google Scholar

[24]

N. C. OwenJ. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition, Proc. Roy. Soc. London Ser. A, 429 (1990), 505-532.   Google Scholar

[25]

J. RubinsteinP. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.  doi: 10.1137/0149007.  Google Scholar

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