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CPAA

Consider the solution $u(x,t)$ of the heat equation
with initial data $u_0$.
The diffusive sign $S_D[u_0](x)$ is defined
by the limit of sign of $u(x,t)$ as $t\to 0$.
A sufficient condition for $x\in R^d$
and $u_0$ such that $S_D[u_0](x)$ is well-defined is given.
A few examples of $u_0$ violating and fulfilling
this condition are given.
It turns out that this diffusive sign
is also related to variational problem
whose energy is the Dirichlet energy
with a fidelity term.
If initial data is a difference of characteristic functions
of two disjoint sets, it turns out that the boundary
of the set $S_D[u_0](x) = 1$ (or $-1$) is roughly
an equi-distance hypersurface from $A$ and $B$ and
this gives a separation of two data sets.

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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