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PROC

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.

DCDS

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 $a>0$).

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