# American Institute of Mathematical Sciences

September  2013, 12(5): 2277-2296. doi: 10.3934/cpaa.2013.12.2277

## 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, Japan 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  January 2013

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.
Citation: Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277
##### References:
 [1] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics, 223, Birkhäuser Verlag, 2004. doi: 10.1007/978-3-0348-7928-6. [2] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79. [3] A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, Multiscale Modeling and Simulation, 10 (2012), 1090-1118. doi: 10.1137/11083109X. [4] M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differential Equations, 252 (2012), 4455-4480. doi: 10.1016/j.jde.2012.01.003. [5] A. Briani, A. Chambolle, M. Novaga and G. Orlandi, On the gradient flow of a one-homogeneous functional, Confluentes Mathematici, 3 (2011), 617-635. doi: 10.1142/S1793744211000461. [6] A. Chambolle, An algorithm for mean curvature motion, Interfaces and free boundaries, 6 (2004), 195-218. doi: 10.4171/IFB/97. [7] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E. [8] X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630. doi: 10.1007/s002080050202. [9] X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190. doi: 10.1016/0022-0396(89)90081-8. [10] R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, Proc. Natl. Acad. Sci., 102 (2005), 7426-7431. doi: 10.1073/pnas.0500334102. [11] N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'' Cambridge University Press, 2000. [12] M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Japan J. Indust. Appl. Math., 27(2010), 323-345. doi: 10.1007/s13160-010-0020-y. [13] M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications, 79, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4651-6. [14] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 598 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6. [15] K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow., (to appear in Journal of Evolution Equations), ().  doi: 10.1007/s00028-012-0167-x. [16] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441. [17] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola, Boll. Un. Mat. Ital., (B5) (1977), 285-299. [18] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen, J. Reine Angew. Math., 211 (1962), 78-94. doi: 10.1515/crll.1962.211.78. [19] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Englewood Cliffs, 1967. [20] W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), 799-811. doi: 10.1051/m2an:2000104. [21] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [22] G. Steidl, Supervised learning by support vector machines, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.), vol 3, Springer, 2011, 959-1013. [23] P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122. [24] D. G. Widder, "The Heat Equations,'' Academic Press, New York, 1975.

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##### References:
 [1] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics, 223, Birkhäuser Verlag, 2004. doi: 10.1007/978-3-0348-7928-6. [2] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79. [3] A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, Multiscale Modeling and Simulation, 10 (2012), 1090-1118. doi: 10.1137/11083109X. [4] M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differential Equations, 252 (2012), 4455-4480. doi: 10.1016/j.jde.2012.01.003. [5] A. Briani, A. Chambolle, M. Novaga and G. Orlandi, On the gradient flow of a one-homogeneous functional, Confluentes Mathematici, 3 (2011), 617-635. doi: 10.1142/S1793744211000461. [6] A. Chambolle, An algorithm for mean curvature motion, Interfaces and free boundaries, 6 (2004), 195-218. doi: 10.4171/IFB/97. [7] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E. [8] X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630. doi: 10.1007/s002080050202. [9] X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190. doi: 10.1016/0022-0396(89)90081-8. [10] R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, Proc. Natl. Acad. Sci., 102 (2005), 7426-7431. doi: 10.1073/pnas.0500334102. [11] N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'' Cambridge University Press, 2000. [12] M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Japan J. Indust. Appl. Math., 27(2010), 323-345. doi: 10.1007/s13160-010-0020-y. [13] M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications, 79, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4651-6. [14] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 598 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6. [15] K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow., (to appear in Journal of Evolution Equations), ().  doi: 10.1007/s00028-012-0167-x. [16] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441. [17] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola, Boll. Un. Mat. Ital., (B5) (1977), 285-299. [18] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen, J. Reine Angew. Math., 211 (1962), 78-94. doi: 10.1515/crll.1962.211.78. [19] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Englewood Cliffs, 1967. [20] W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), 799-811. doi: 10.1051/m2an:2000104. [21] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [22] G. Steidl, Supervised learning by support vector machines, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.), vol 3, Springer, 2011, 959-1013. [23] P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122. [24] D. G. Widder, "The Heat Equations,'' Academic Press, New York, 1975.
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