On behavior of signs for the heat equation and a diffusion method for data separation

Mi-Ho Giga; Yoshikazu Giga; Takeshi Ohtsuka; Noriaki Umeda

doi:10.3934/cpaa.2013.12.2277

Article Contents

2013, Volume 12, Issue 5: 2277-2296. Doi: 10.3934/cpaa.2013.12.2277

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On behavior of signs for the heat equation and a diffusion method for data separation

1.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914
2.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914
3.
Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-cho, Maebashi, 371-8510
4.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8914

Received: February 2012

Revised: October 2012

Published: September 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K05; Secondary: 35B40.

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