# American Institute of Mathematical Sciences

2011, 29(4): 1463-1470. doi: 10.3934/dcds.2011.29.1463

## On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow

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

Received  November 2009 Revised  October 2010 Published  December 2010

We study the motion of noncompact hypersurfaces moved by their mean curvature obtained by a rotation around $x$-axis of the graph a function $y=u(x,t)$ (defined for all $x\in \mathbb{R}$). We are interested to estimate its profile when the hypersurface closes open ends at the quenching (pinching) time $T$. We estimate its profile at the quenching time from above and below. We in particular prove that $u(x,T)$ ~ $|x|^{-a}$ as $|x|\to\infty$ if $u(x,0)$ tends to its infimum with algebraic rate $|x|^{-2a}$ (as $|x| \to \infty$ with $a>0$).
Citation: Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463
##### References:
 [1] M.-H. Giga, Y. Giga and J. Saal, "Nonliear Partial Differential Equations - Asymptotic Behaviour of Solutions and Self-Similar Solutions,", Progress in Nonlinear Differential Equations and Their Applications \textbf{79}, 79 (1999). [2] Y. Giga, "Surface Evolution Equations. A Level Set Approach,", Birkhäuser, (2006). [3] Y. Giga, Y. Seki and N. Umeda, Blow-up at space infinity for nonlinear heat equations,, in, (2007), 77. [4] Y. Giga, Y. Seki and N. Umeda, Mean curvature flow closes open sets of noncompact surface of rotation,, Comm. Partial Differential Equations, 34 (2009), 1508. doi: doi:10.1080/03605300903296926. [5] Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations,, J. Math. Anal. Appl., 316 (2006), 538. doi: doi:10.1016/j.jmaa.2005.05.007. [6] Y. Giga and N. Umeda, Blow-up directions at space infinity for solutions of semilinear heat equations,, Bol. Soc. Parana. Mat. (3), 23 (2005), 9. [7] A. L. Gladkov, The behavior as $x\to \infty$ of solutions of semilinear parabolic equations (Russian),, Mat. Zametki, 51 (1992), 29. doi: doi:10.1007/BF02102115. [8] A. A. Lacey, The form of blow-up for nonlinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 183. [9] Y. Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion,, J. Math. Anal. Appl., 338 (2008), 572. doi: doi:10.1016/j.jmaa.2007.05.033. [10] Y. Seki, R. Suzuki and N. Umeda, Blow-up directions for quasilinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect., 138 (2008), 379. [11] M. Shimojo, The global profile of blow-up at space infinity in semilinear heat equations,, J. Math. Kyoto Univ., 48 (2008), 339. [12] M. Shimojo and N. Umeda, Blow-up at space infinity for solutions of cooperative reaction-diffusion systems,, preprint., ().

show all references

##### References:
 [1] M.-H. Giga, Y. Giga and J. Saal, "Nonliear Partial Differential Equations - Asymptotic Behaviour of Solutions and Self-Similar Solutions,", Progress in Nonlinear Differential Equations and Their Applications \textbf{79}, 79 (1999). [2] Y. Giga, "Surface Evolution Equations. A Level Set Approach,", Birkhäuser, (2006). [3] Y. Giga, Y. Seki and N. Umeda, Blow-up at space infinity for nonlinear heat equations,, in, (2007), 77. [4] Y. Giga, Y. Seki and N. Umeda, Mean curvature flow closes open sets of noncompact surface of rotation,, Comm. Partial Differential Equations, 34 (2009), 1508. doi: doi:10.1080/03605300903296926. [5] Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations,, J. Math. Anal. Appl., 316 (2006), 538. doi: doi:10.1016/j.jmaa.2005.05.007. [6] Y. Giga and N. Umeda, Blow-up directions at space infinity for solutions of semilinear heat equations,, Bol. Soc. Parana. Mat. (3), 23 (2005), 9. [7] A. L. Gladkov, The behavior as $x\to \infty$ of solutions of semilinear parabolic equations (Russian),, Mat. Zametki, 51 (1992), 29. doi: doi:10.1007/BF02102115. [8] A. A. Lacey, The form of blow-up for nonlinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 183. [9] Y. Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion,, J. Math. Anal. Appl., 338 (2008), 572. doi: doi:10.1016/j.jmaa.2007.05.033. [10] Y. Seki, R. Suzuki and N. Umeda, Blow-up directions for quasilinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect., 138 (2008), 379. [11] M. Shimojo, The global profile of blow-up at space infinity in semilinear heat equations,, J. Math. Kyoto Univ., 48 (2008), 339. [12] M. Shimojo and N. Umeda, Blow-up at space infinity for solutions of cooperative reaction-diffusion systems,, preprint., ().
 [1] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 [2] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [3] Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549 [4] Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9 [5] Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441 [6] Marie Henry, Danielle Hilhorst, Masayasu Mimura. A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 125-154. doi: 10.3934/dcdss.2011.4.125 [7] Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124. [8] Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155 [9] Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367 [10] Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075 [11] Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307 [12] Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231 [13] Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1 [14] Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256 [15] Santiago Cano-Casanova. Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations. Conference Publications, 2011, 2011 (Special) : 240-249. doi: 10.3934/proc.2011.2011.240 [16] Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297 [17] Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165 [18] Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076 [19] Yu-Zhu Wang, Si Chen, Menglong Su. Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$. Evolution Equations & Control Theory, 2017, 6 (4) : 629-645. doi: 10.3934/eect.2017032 [20] Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687

2017 Impact Factor: 1.179