# American Institute of Mathematical Sciences

October  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 and Continuous Dynamical Systems, 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 79, Birkhäuser, (expanded from original Japanese version: M.-H. Giga and Y. Giga, "Nonliear Partial Differential Equations - Asymptotic Behaviour of Solutions and Self-Similar Solutions," Kyoritsu, Tokyo, 1999). [2] Y. Giga, "Surface Evolution Equations. A Level Set Approach," Birkhäuser, Basel, 2006. [3] Y. Giga, Y. Seki and N. Umeda, Blow-up at space infinity for nonlinear heat equations, in "Recent Advances in Nonlinear Analysis" (M. Chipot, C.-S. Lin, D.-H. Tsai, eds.), World Scientific Publishing, Hackensack, NJ, 77-94. (also in EPrint Series of Department of Mathematics, Hokkaido University, 2007.) [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), 1508-1529. 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), 538-555. 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), 9-28; Correction is available in Bol. Soc. Parana. Mat. (3), 24 (2006), 19-24. [7] A. L. Gladkov, The behavior as $x\to \infty$ of solutions of semilinear parabolic equations (Russian), Mat. Zametki, 51 (1992), 29-34, 156; translation in Math. Notes, 51 (1992), 124-128. 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), 183-202. [9] Y. Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion, J. Math. Anal. Appl., 338 (2008), 572-587. 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), 379-405. [11] M. Shimojo, The global profile of blow-up at space infinity in semilinear heat equations, J. Math. Kyoto Univ., 48 (2008), 339-361. [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 79, Birkhäuser, (expanded from original Japanese version: M.-H. Giga and Y. Giga, "Nonliear Partial Differential Equations - Asymptotic Behaviour of Solutions and Self-Similar Solutions," Kyoritsu, Tokyo, 1999). [2] Y. Giga, "Surface Evolution Equations. A Level Set Approach," Birkhäuser, Basel, 2006. [3] Y. Giga, Y. Seki and N. Umeda, Blow-up at space infinity for nonlinear heat equations, in "Recent Advances in Nonlinear Analysis" (M. Chipot, C.-S. Lin, D.-H. Tsai, eds.), World Scientific Publishing, Hackensack, NJ, 77-94. (also in EPrint Series of Department of Mathematics, Hokkaido University, 2007.) [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), 1508-1529. 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), 538-555. 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), 9-28; Correction is available in Bol. Soc. Parana. Mat. (3), 24 (2006), 19-24. [7] A. L. Gladkov, The behavior as $x\to \infty$ of solutions of semilinear parabolic equations (Russian), Mat. Zametki, 51 (1992), 29-34, 156; translation in Math. Notes, 51 (1992), 124-128. 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), 183-202. [9] Y. Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion, J. Math. Anal. Appl., 338 (2008), 572-587. 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), 379-405. [11] M. Shimojo, The global profile of blow-up at space infinity in semilinear heat equations, J. Math. Kyoto Univ., 48 (2008), 339-361. [12] M. Shimojo and N. Umeda, Blow-up at space infinity for solutions of cooperative reaction-diffusion systems,, preprint., ().
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