On behavior of signs for the heat equation and a diffusion method for data separation
Mi-Ho Giga Yoshikazu Giga Takeshi Ohtsuka Noriaki Umeda
Communications on Pure & Applied Analysis 2013, 12(5): 2277-2296 doi: 10.3934/cpaa.2013.12.2277
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.
keywords: equi-distance hypersurface Diffusive sign heat equations. sign-changing
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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