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Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. (English) Zbl 07272910
Summary: 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.
##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K93 Quasilinear parabolic equations with mean curvature operator 35K20 Initial-boundary value problems for second-order parabolic equations 53E10 Flows related to mean curvature
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##### References:
 [1] G. Barles; O. Ley; T.-T. Nguyen; T. V. Phan, Large time Behavior of unbounded solutions of first-order Hamilton-Jacobi in $$\mathbb{R}^N$$, Asymptot. Anal., 112, 1-22 (2019) · Zbl 1431.35194 [2] G. Barles; P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31, 925-939 (2000) · Zbl 0960.70015 [3] F. Cagnetti; D. Gomes; H. Mitake; 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, 183-200 (2015) · Zbl 1312.35020 [4] A. Cesaroni; M. Novaga, Long-time behavior of the mean curvature flow with periodic forcing, Comm. Partial Differential Equations, 38, 780-801 (2013) · Zbl 1288.35071 [5] Y. G. Chen; Y. Giga; S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33, 749-786 (1991) · Zbl 0696.35087 [6] M. G. Crandall; H. Ishii; P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1-67 (1992) · Zbl 0755.35015 [7] A. Davini; A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38, 478-502 (2006) · Zbl 1109.49034 [8] L. C. Evans; J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33, 635-681 (1991) · Zbl 0726.53029 [9] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327, 267-270 (1998) · Zbl 1052.37514 [10] Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99. Birkhäuser, Basel-Boston-Berlin, 2006. [11] Y. Giga, On large time behavior of growth by birth and spread, Proc. Int. Cong. of Math. 2018 Rio de Janeiro, 3, 2287-2310 (2018) · Zbl 1448.35007 [12] M.-H. Giga; Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159, 295-333 (2001) · Zbl 1004.35075 [13] Y. Giga; N. Hamamuki, Hamilton-Jacobi equations with discontinuous source terms, Comm. Partial Differential Equations, 38, 199-243 (2013) · Zbl 1263.35066 [14] Y. Giga; H. Mitake; 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, 3515-3546 (2016) · Zbl 1355.35123 [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; 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, 107-131 (1999) · Zbl 0930.35029 [17] Y. Giga; H. V. Tran; L. J. Zhang, On obstacle problem for mean curvature flow with driving force, Geom. Flows, 4, 9-29 (2019) · Zbl 1442.35197 [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; H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194, 383-419 (2009) · Zbl 1243.70017 [21] H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean $$n$$ space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 231-266 (2008) · Zbl 1145.35035 [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. · Zbl 1373.35006 [23] H. Mitake; H. V. Tran, On uniqueness sets of additive eigenvalue problems and applications, Proc. Amer. Math. Soc., 146, 4813-4822 (2018) · Zbl 1401.37069 [24] G. Namah; J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24, 883-893 (1999) · Zbl 0924.35028 [25] L. J. Zhang, On curvature flow with driving force starting as singular initial curve in the plane, to appear in J. Geom. Anal. · Zbl 1439.35309
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