A remark on blow-up at space infinity
Yukihiro Seki
Conference Publications 2009, 2009(Special): 691-696 doi: 10.3934/proc.2009.2009.691
In this note we discuss blow-up at space infinity for quasilinear parabolic equation $u_t = \Delta u^m + u^{p}$. It is known that if initial data is not a constant and takes its maximum at space infinity in a certain sense, the solution blows up only at space infinity at minimal blow-up time. We show that if $m \ge 1$ and a solution blows up at minimal blow-up time, then it blows up completely at the blow-up time.
keywords: minimal blow-up time complete blow-up Blow-up at space infinity
On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow
Yoshikazu Giga Yukihiro Seki Noriaki Umeda
Discrete & Continuous Dynamical Systems - A 2011, 29(4): 1463-1470 doi: 10.3934/dcds.2011.29.1463
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 $a>0$).
keywords: At space infinity quenching profile axisymmetric mean curvature flow equation decay rate.

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