# American Institute of Mathematical Sciences

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

 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.

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [19] N. Hamamuki and K. Misu, Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms, work in progress. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] L. J. Zhang, On curvature flow with driving force starting as singular initial curve in the plane, to appear in J. Geom. Anal. Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [19] N. Hamamuki and K. Misu, Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms, work in progress. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] L. J. Zhang, On curvature flow with driving force starting as singular initial curve in the plane, to appear in J. Geom. Anal. Google Scholar
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