American Institute of Mathematical Sciences

Advanced
October  2020, 25(10): 3983-3999. doi: 10.3934/dcdsb.2019228

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

Yoshikazu Giga 1,, , Hiroyoshi Mitake 1, and  Hung V. Tran 2,

1. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
2. 

Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706, USA

* Corresponding author: Yoshikazu Giga

Received  June 2019 Revised  June 2019 Published  October 2020 Early access  September 2019

We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

Keywords: Asymptotic speed, large time behavior, birth and spread type nonlinear PDEs, fully nonlinear parabolic equations, forced mean curvature flow, crystal growth.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35K93, 35K20.
Citation: Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3983-3999. doi: 10.3934/dcdsb.2019228
References:
[1]

G. Barles, O. Ley, T.-T. Nguyen and T.V. Phan, Large time Behavior of unbounded solutions of first-order Hamilton-Jacobi in $\mathbb{R}^N$, Asymptot. Anal., 112 (2019), 1-22.  doi: 10.3233/ASY-181488.

[2]

G. Barles and P.E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939.  doi: 10.1137/S0036141099350869.

[3]

F. Cagnetti, D. Gomes, H. Mitake and H.V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 183-200.  doi: 10.1016/j.anihpc.2013.10.005.

[4]

A. Cesaroni and M. Novaga, Long-time behavior of the mean curvature flow with periodic forcing, Comm. Partial Differential Equations, 38 (2013), 780-801.  doi: 10.1080/03605302.2013.771508.

[5]

Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.  doi: 10.4310/jdg/1214446564.

[6]

M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[7]

A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.  doi: 10.1137/050621955.

[8]

L.C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.

[9]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  doi: 10.1016/S0764-4442(98)80144-4.

[10]

Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99. Birkhäuser, Basel-Boston-Berlin, 2006. doi: 10.1007/3-7643-7391-1.

[11]

Y. Giga, On large time behavior of growth by birth and spread, Proc. Int. Cong. of Math. 2018 Rio de Janeiro, 3 (2018), 2287-2310. 

[12]

M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333.  doi: 10.1007/s002050100154.

[13]

Y. Giga and N. Hamamuki, Hamilton-Jacobi equations with discontinuous source terms, Comm. Partial Differential Equations, 38 (2013), 199-243.  doi: 10.1080/03605302.2012.739671.

[14]

Y. Giga, H. Mitake and H.V. Tran, On asymptotic speed of solutions to level-set mean curvature flow equations with driving and source terms, SIAM J. Math. Anal., 48 (2016), 3515-3546.  doi: 10.1137/15M1052755.

[15]

Y. Giga, H. Mitake, T. Ohtsuka and H. V. Tran, Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations, to appear in Indiana Univ. Math. J., https://www.iumj.indiana.edu/IUMJ/Preprints/8305.pdf.

[16]

Y. Giga, M. Ohnuma and M.-H. Sato, On the strong maximum principle and the large time behavior of generalized mean curvature flow with the Neumann boundary condition, J. Differential Equations, 154 (1999), 107-131.  doi: 10.1006/jdeq.1998.3569.

[17]

Y. Giga, H.V. Tran and L.J. Zhang, On obstacle problem for mean curvature flow with driving force, Geom. Flows, 4 (2019), 9-29. 

[18]

N. Hamamuki, On large time behavior of Hamilton-Jacobi equations with discontinuous source terms, Nonlinear Analysis in Interdisciplinary Sciences – Modellings, Theory and Simulations, 83–112, GAKUTO Internat. Ser. Math. Sci. Appl., 36, Gakkotosho, Tokyo, 2013.

[19]

N. Hamamuki and K. Misu, Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms, work in progress.

[20]

N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419.  doi: 10.1007/s00205-008-0170-0.

[21]

H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean $n$ space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 231-266.  doi: 10.1016/j.anihpc.2006.09.002.

[22]

N. Q. Le, H. Mitake and H. V. Tran, Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations, Lecture Notes in Mathematics, 2183, Springer, Cham, 2017. doi: 10.1007/978-3-319-54208-9.

[23]

H. Mitake and H.V. Tran, On uniqueness sets of additive eigenvalue problems and applications, Proc. Amer. Math. Soc., 146 (2018), 4813-4822.  doi: 10.1090/proc/14152.

[24]

G. Namah and J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.  doi: 10.1080/03605309908821451.

[25]

L. J. Zhang, On curvature flow with driving force starting as singular initial curve in the plane, to appear in J. Geom. Anal.

show all references

References:
[1]

G. Barles, O. Ley, T.-T. Nguyen and T.V. Phan, Large time Behavior of unbounded solutions of first-order Hamilton-Jacobi in $\mathbb{R}^N$, Asymptot. Anal., 112 (2019), 1-22.  doi: 10.3233/ASY-181488.

[2]

G. Barles and P.E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939.  doi: 10.1137/S0036141099350869.

[3]

F. Cagnetti, D. Gomes, H. Mitake and H.V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 183-200.  doi: 10.1016/j.anihpc.2013.10.005.

[4]

A. Cesaroni and M. Novaga, Long-time behavior of the mean curvature flow with periodic forcing, Comm. Partial Differential Equations, 38 (2013), 780-801.  doi: 10.1080/03605302.2013.771508.

[5]

Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.  doi: 10.4310/jdg/1214446564.

[6]

M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[7]

A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.  doi: 10.1137/050621955.

[8]

L.C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.

[9]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  doi: 10.1016/S0764-4442(98)80144-4.

[10]

Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99. Birkhäuser, Basel-Boston-Berlin, 2006. doi: 10.1007/3-7643-7391-1.

[11]

Y. Giga, On large time behavior of growth by birth and spread, Proc. Int. Cong. of Math. 2018 Rio de Janeiro, 3 (2018), 2287-2310. 

[12]

M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333.  doi: 10.1007/s002050100154.

[13]

Y. Giga and N. Hamamuki, Hamilton-Jacobi equations with discontinuous source terms, Comm. Partial Differential Equations, 38 (2013), 199-243.  doi: 10.1080/03605302.2012.739671.

[14]

Y. Giga, H. Mitake and H.V. Tran, On asymptotic speed of solutions to level-set mean curvature flow equations with driving and source terms, SIAM J. Math. Anal., 48 (2016), 3515-3546.  doi: 10.1137/15M1052755.

[15]

Y. Giga, H. Mitake, T. Ohtsuka and H. V. Tran, Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations, to appear in Indiana Univ. Math. J., https://www.iumj.indiana.edu/IUMJ/Preprints/8305.pdf.

[16]

Y. Giga, M. Ohnuma and M.-H. Sato, On the strong maximum principle and the large time behavior of generalized mean curvature flow with the Neumann boundary condition, J. Differential Equations, 154 (1999), 107-131.  doi: 10.1006/jdeq.1998.3569.

[17]

Y. Giga, H.V. Tran and L.J. Zhang, On obstacle problem for mean curvature flow with driving force, Geom. Flows, 4 (2019), 9-29. 

[18]

N. Hamamuki, On large time behavior of Hamilton-Jacobi equations with discontinuous source terms, Nonlinear Analysis in Interdisciplinary Sciences – Modellings, Theory and Simulations, 83–112, GAKUTO Internat. Ser. Math. Sci. Appl., 36, Gakkotosho, Tokyo, 2013.

[19]

N. Hamamuki and K. Misu, Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms, work in progress.

[20]

N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419.  doi: 10.1007/s00205-008-0170-0.

[21]

H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean $n$ space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 231-266.  doi: 10.1016/j.anihpc.2006.09.002.

[22]

N. Q. Le, H. Mitake and H. V. Tran, Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations, Lecture Notes in Mathematics, 2183, Springer, Cham, 2017. doi: 10.1007/978-3-319-54208-9.

[23]

H. Mitake and H.V. Tran, On uniqueness sets of additive eigenvalue problems and applications, Proc. Amer. Math. Soc., 146 (2018), 4813-4822.  doi: 10.1090/proc/14152.

[24]

G. Namah and J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.  doi: 10.1080/03605309908821451.

[25]

L. J. Zhang, On curvature flow with driving force starting as singular initial curve in the plane, to appear in J. Geom. Anal.

Figure 1.  $ U $ and semicircles of radii $ 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Feng Li, Erik Lindgren. Large time behavior for a nonlocal nonlinear gradient flow. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022079
[2]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[3]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[4]

Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256
[5]

Yannan Liu, Hongjie Ju. Non-collapsing for a fully nonlinear inverse curvature flow. Communications on Pure and Applied Analysis, 2017, 16 (3) : 945-952. doi: 10.3934/cpaa.2017045
[6]

P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
[7]

Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69
[8]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[9]

Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199
[10]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[11]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
[12]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[13]

Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 465-475. doi: 10.3934/dcdsb.2017021
[14]

Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 463-473. doi: 10.3934/cpaa.2005.4.463
[15]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[16]

Ziyi Cai, Haiyang He. Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4349-4362. doi: 10.3934/cpaa.2020196
[17]

Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks and Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749
[18]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure and Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[19]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[20]

Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (255)
  • HTML views (531)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy