January  2022, 42(1): 261-283. doi: 10.3934/dcds.2021115

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 Published  January 2022 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 and Continuous Dynamical Systems, 2022, 42 (1) : 261-283. doi: 10.3934/dcds.2021115
References:
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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.

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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.

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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.

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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.

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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.

[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. 

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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.

[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.

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

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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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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

[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.

[23]

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

[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. 

[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.

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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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

[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.

[23]

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

[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. 

[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.

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