A remark on blow-up at space infinity
Yukihiro Seki
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
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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