Faculty Profiles - TANIGUCHI Masaharu

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TANIGUCHI Masaharu

Organization

Scheduled update Professor

Homepage

http://www.math.okayama-u.ac.jp/~taniguchi/index-j.html

External link

Degree

Doctor
Master of Engineering

Research Areas

Natural Science / Mathematical analysis

Education

The University of Tokyo 大学院数理科学研究科

1991.4 - 1993.9

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Country： Japan

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The University of Tokyo 大学院工学系研究科物理工学専攻

1989.4 - 1991.3

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The University of Tokyo 理学部数学科

1985.4 - 1989.3

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Country： Japan

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Research History

Okayama University Research Institute for Interdisciplinary Science Professor

2016.4

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Okayama University Graduate School of Natural Science and Technology Professor

2013.4 - 2015.3

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Tokyo Institute of Technology 大学院情報理工学研究科 Associate Professor

2007.4 - 2012.3

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Tokyo Institute of Technology Graduate School of Information Science and Engineering Associate Professor (as old post name)

2001.3 - 2007.4

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Tokyo Institute of Technology Graduate School of Information Science and Engineering Lecturer

1996.10 - 2001.2

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Kyoto University 数理解析研究所 Research Assistant

1993.10 - 1996.9

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Professional Memberships

日本応用数理学会

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日本数学会

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The Japan Society of Industrial and Applied Mathematics

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Mathematical Society of Japan

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Committee Memberships

Research Institute for Interdisciplinary Science Chairman of Division of Quantum Univrerse

2022.4 - 2024.3

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An Editor of "Journal of Differential Equations"

2021.4

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Department of Mathematics Chairman

2020.4 - 2021.3

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Division of Interdisciplinary Science Chairman

2019.4 - 2021.3

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大学入試センター教科科目第一委員会委員

2016.4 - 2018.3

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日本数学会全国区代議員（評議員）

2016.3 - 2017.2

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Committee type：Academic society

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An Editor of "Mathematical Journal of Okayama University"

2013.4

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An Editor of "Discrete and Continuous Dynamical Systems"

2008.10

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日本数学会代議員

2008

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Committee type：Academic society

日本数学会

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日本応用数理学会日本応用数理学会誌「応用数理」編集委員

2006.4 - 2012.3

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Committee type：Academic society

日本応用数理学会

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Papers

Polyhedral entire solutions in reaction-diffusion equations Reviewed

Masaharu Taniguchi

Journal of Differential Equations 429 529 - 565 2025.6

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Authorship：Lead author,　Corresponding author Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier BV

DOI： 10.1016/j.jde.2025.02.034

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Traveling front solutions of dimension $n$ generate entire solutions of dimension $(n-1)$ reaction-diffusion equations a the speeds go to infinity Reviewed

Hirokazu Ninomiya, Masaharu Taniguchi

Archive for Rational Mechanics and Analysis 249 ( 13 ) 2025.1

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Authorship：Corresponding author Language：English Publishing type：Research paper (scientific journal)

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Entire solutions with and without radial symmetry in balanced bistable reaction–diffusion equations Reviewed

Masaharu Taniguchi

Mathematische Annalen 390 ( 3 ) 3931 - 3967 2024.4

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Authorship：Lead author,　Corresponding author Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC

Abstract

Let $$n\ge 2$$ be a given integer. In this paper, we assert that an n-dimensional traveling front converges to an $$(n-1)$$-dimensional entire solution as the speed goes to infinity in a balanced bistable reaction–diffusion equation. As the speed of an n-dimensional axially symmetric or asymmetric traveling front goes to infinity, it converges to an $$(n-1)$$-dimensional radially symmetric or asymmetric entire solution in a balanced bistable reaction–diffusion equation, respectively. We conjecture that the radially asymmetric entire solutions obtained in this paper are associated with the ancient solutions called the Angenent ovals in the mean curvature flows.

DOI： 10.1007/s00208-024-02844-6

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Other Link： https://link.springer.com/article/10.1007/s00208-024-02844-6/fulltext.html
Traveling front solutions for perturbed reaction-diffusion equations Reviewed

Wah Wah, Masaharu Taniguchi

Mathematical Journal of Okayama University 65 125 - 143 2023.1

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Language：English Publishing type：Research paper (scientific journal)

File： wahwah-taniguchi-MJOU2023.pdf

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Traveling fronts in balanced bistable reaction-diffusion equations Invited Reviewed

Masaharu Taniguchi

Advanced Studies in Pure Mathematics 85 417 - 428 2020.12

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Authorship：Lead author Language：English Publishing type：Research paper (international conference proceedings)

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Existence and stability of stationary solutions to the Allen--Cahn equation discretized in space and time Reviewed

Amy Poh Ai Ling, Masaharu Taniguchi

62 197 - 210 2020.1

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Language：English Publishing type：Research paper (scientific journal)

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Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited Reviewed

Masaharu Taniguchi

Discrete and Continuous Dynamical Systems. Series A 40 ( 6 ) 3981 - 3995 2020

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Authorship：Lead author Language：English Publishing type：Research paper (scientific journal)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Reviewed

Masaharu Taniguchi

Annales de l'Institut Henri Poincare C, Analyse Non Lineaire 36 ( 7 ) 1791 - 1816 2019

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1016/j.anihpc.2019.05.001

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Convex compact sets in $\mathbb{R}^{N-1}$ give traveling fronts of cooperation-diffusion systems in $\mathbb{R}^{N}$ Reviewed

Masaharu Taniguchi

JOURNAL OF DIFFERENTIAL EQUATIONS 260 ( 5 ) 4301 - 4338 2016.3

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Language：English Publishing type：Research paper (scientific journal) Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE

This paper studies traveling fronts to cooperation diffusion systems in R-N for N >= 3. We consider (N - 2)-dimensional smooth surfaces as boundaries of strictly convex compact sets in RN-1, and define an equivalence relation between them. We prove that there exists a traveling front associated with a given surface and show its stability. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation. (C) 2015 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jde.2015.11.010

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AN (N-1)-DIMENSIONAL CONVEX COMPACT SET GIVES AN N-DIMENSIONAL TRAVELING FRONT IN THE ALLEN-CAHN EQUATION Reviewed

Masaharu Taniguchi

SIAM JOURNAL ON MATHEMATICAL ANALYSIS 47 ( 1 ) 455 - 476 2015

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Language：English Publishing type：Research paper (scientific journal) Publisher：SIAM PUBLICATIONS

This paper studies traveling fronts to the Allen-Cahn equation in RN for N >= 3. Let (N - 2)-dimensional smooth surfaces be the boundaries of compact sets in RN-1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.

DOI： 10.1137/130945041

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NON-EXISTENCE OF LOCALIZED TRAVELLING WAVES WITH NON-ZERO SPEED IN SINGLE REACTION-DIFFUSION EQUATIONS Reviewed

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 33 ( 8 ) 3707 - 3718 2013.8

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. If [lim inf(vertical bar x vertical bar -> infinity) u(x), lim sup(vertical bar x vertical bar -> infinity) u(x)] is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile u has to be identically zero.

DOI： 10.3934/dcds.2013.33.3707

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TRAVELING FRONTS OF PYRAMIDAL SHAPES IN COMPETITION-DIFFUSION SYSTEMS Reviewed

Wei-Ming Ni, Masaharu Taniguchi

NETWORKS AND HETEROGENEOUS MEDIA 8 ( 1 ) 379 - 395 2013.3

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

It is well known that a competition-diffusion system has a one-dimensional traveling front. This paper studies traveling front solutions of pyramidal shapes in a competition-diffusion system in R-N with N >= 2. By using a multi-scale method, we construct a suitable pair of a supersolution and a subsolution, and find a pyramidal traveling front solution between them.

DOI： 10.3934/nhm.2013.8.379

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MULTI-DIMENSIONAL TRAVELING FRONTS IN BISTABLE REACTION-DIFFUSION EQUATIONS Reviewed

Masaharu Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 32 ( 3 ) 1011 - 1046 2012.3

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given g is an element of C-infinity (S-1) with min(0 <=theta <= 2 pi) g(theta) = 0.

DOI： 10.3934/dcds.2012.32.1011

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TRAVELING FRONTS IN PERTURBED MULTISTABLE REACTION-DIFFUSION EQUATIONS Reviewed

Masaharu Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 31 1368 - 1377 2011.9

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES-AIMS

In this paper we study the existence and non-existence of traveling front solutions in multistable reaction-diffusion equations. If this equation has a traveling front solution, a perturbed equation also has a traveling front solution. We study how the speed and the traveling profile depend on nonlinear terms.

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Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations Reviewed

Yu Kurokawa, Masaharu Taniguchi

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 141 1031 - 1054 2011

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Language：English Publishing type：Research paper (scientific journal) Publisher：ROYAL SOC EDINBURGH

We study travelling-front solutions of pyramidal shapes in the Allen-Cahn equation in R(N) with N >= 3. It is well known that two-dimensional V-form travelling fronts and three-dimensional pyramidal travelling fronts exist and are stable. The aim of this paper is to show that for N >= 4 there exist N-dimensional pyramidal travelling fronts. We construct a supersolution and a subsolution, and find a pyramidal travelling-front solution between them. For the construction of a supersolution we use a multi-scale method.

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The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations Reviewed

Masaharu Taniguchi

JOURNAL OF DIFFERENTIAL EQUATIONS 246 ( 5 ) 2103 - 2130 2009.3

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Language：English Publishing type：Research paper (scientific journal) Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE

This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front. (c) 2008 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jde.2008.06.037

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Stability of Planar Waves in the Allen-Cahn Equation Reviewed

Hiroshi Matano, Mitsunori Nara, Masaharu Taniguchi

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 34 ( 9 ) 976 - 1002 2009

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Language：English Publishing type：Research paper (scientific journal) Publisher：TAYLOR & FRANCIS INC

We study the asymptotic stability of planar waves for the Allen-Cahn equation on n, where n epsilon 2. Our first result states that planar waves are asymptotically stable under anypossibly largeinitial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t. The convergence is uniform in n. Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.

DOI： 10.1080/03605300902963500

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The condition on the stability of stationary lines in a curvature flow in the whole plane

Mitsunori Nara, Masaharu Taniguchi

JOURNAL OF DIFFERENTIAL EQUATIONS 237 ( 1 ) 61 - 76 2007.6

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Language：English Publishing type：Research paper (scientific journal) Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O (t(-1/2)) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane. (C) 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jde.2007.02.012

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Traveling fronts of pyramidal shapes in the Allen-Cahn equations

Masaharu Taniguchi

SIAM JOURNAL ON MATHEMATICAL ANALYSIS 39 ( 1 ) 319 - 344 2007

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This paper studies pyramidal traveling fronts in the Allen-Cahn equation or in the Nagumo equation. For the nonlinearity we are concerned mainly with the bistable reaction term with unbalanced energy density. Two-dimensional V-form waves and cylindrically symmetric waves in higher dimensions have been recently studied. Our aim in this paper is to construct truly three-dimensional traveling waves. For a pyramid that satisfies a condition, we construct a traveling front for which the contour line has a pyramidal shape. We also construct generalized pyramidal fronts and traveling waves of a hybrid type between pyramidal waves and planar V-form waves. We use the comparison principles and construct traveling fronts between supersolutions and subsolutions.

DOI： 10.1137/060661788

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Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations

M Nara, M Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 16 ( 1 ) 137 - 156 2006.9

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

This paper is concerned with the long time behavior for evolution of a curve governed by a curvature flow with constant driving force in the two-dimensional space. This problem has two types of traveling waves: traveling lines and V-shaped fronts, except for stationary circles. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. In this paper, we consider the uniform convergence of curves to the V-shaped fronts. Convergence results for a class of spatially non-decaying initial perturbations are established. Our results hold true with no assumptions on the smallness of given perturbations.

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Global stability of traveling curved fronts in the Allen-Cahn equations

H Ninomiya, M Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 15 ( 3 ) 819 - 832 2006.7

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

This paper is concerned with the global stability of a traveling curved front in the Allen-Cahn equation. The existence of such a front is recently proved by constructing supersolutions and subsolutions. In this paper, we introduce a method to construct new subsolutions and prove the asymptotic stability of traveling curved fronts globally in space.

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Stability of a traveling wave in curvature flows for spatially non-decaying initial perturbations

M Nara, M Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 14 ( 1 ) 203 - 220 2006.1

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

This paper is concerned with the long time behavior for the evolution of a curve governed by the curvature flow with constant driving force in two-dimensional space. Especially, the asymptotic stability of a traveling wave whose shape is a line is studied. We deal with moving curves represented by the entire graphs on the x-axis. By studying the Cauchy problem, the asymptotic stability of traveling waves with spatially decaying initial perturbations and the convergence rate are obtained. Moreover we establish the stability result where initial perturbations do not decay to zero but oscillate at infinity. In this case, we prove that one of the sufficient conditions for asymptotic stability is that a given perturbation is asymptotic to an almost periodic function in the sense of Bohr at infinity. Our results hold true with no assumptions on the smallness of given perturbations, and include the curve shortening flow problem as a special case.

DOI： 10.3934/dcds.2006.14.203

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Existence and global stability of traveling curved fronts in the Allen-Cahn equations

H Ninomiya, M Taniguchi

JOURNAL OF DIFFERENTIAL EQUATIONS 213 ( 1 ) 204 - 233 2005.6

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Language：English Publishing type：Research paper (scientific journal) Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE

This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front. (c) 2004 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jde.2004.06.011

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Instability of planar traveling waves in bistable reaction-diffusion systems

M Taniguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 3 ( 1 ) 21 - 44 2003.2

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES

This paper is concerned with the stability of a planar traveling wave in a cylindrical domain. The equation describes activator-inhibitor systems in chemistry or biology. The wave has a thin transition layer and is constructed by singular perturbation methods. Let epsilon be the width of the layer. We show that, if the cross section of the domain is narrow enough, the traveling wave is asymptotically stable, while: it is unstable if the cross section is wide enough by studying the linearized eigenvalue problem.. For the latter case, we study the wavelength associated with an eigenvalue with the largest real part, which is called the fastest growing wavelength. We prove that this wavelength is O(epsilon(1/3)) as epsilon goes to zero mathematically rigorously. This fact shows that, if unstable planar waves are perturbed randomly, this fastest growing wavelength is selectively amplified with as time goes on. For this analysis, we use a new uniform convergence theorem for some inverse operator and carry out. the Lyapunov-Schmidt reduction.

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A uniform convergence theorem for singular limit eigenvalue problems

Masaharu Taniguchi

Advances in Differential Equations 8 ( 1 ) 29 - 54 2003

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Stability of traveling curved fronts in a curvature flow with driving force

H. Ninomiya, M. Taniguchi

Methods and Applications of Analysis 8 ( 3 ) 429--450 - 449 2001

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Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem

M Taniguchi

QUARTERLY OF APPLIED MATHEMATICS 58 ( 2 ) 283 - 302 2000.6

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Language：English Publishing type：Research paper (scientific journal) Publisher：AMER MATHEMATICAL SOC

This paper studies construction and linear stability of spherical interfaces in an equilibrium state in a two-phase boundary problem arising in activator-inhibitor models in chemistry. By studying the linearized eigenvalue problem near a given equilibrium ball, we show that the eigenvalues with nonnegative real parts are all real, and that they are characterized as values of a strictly convex function for specific discrete values of its argument. The stability is determined by the location of the zero points of this convex function. Using this fact, we present a criterion of stability in a useful form. We show examples and illustrate that stable equilibrium balls and unstable ones coexist near saddle-node bifurcation points in the bifurcation diagram, and a given equilibrium ball located far from bifurcation points is unstable and the eigenfunction associated with the largest eigenvalue consists of spherically harmonic functions of high degrees.

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Modified SLEP method by uniform convergence theorems for linearized eigenvalue problems

M. Taniguchi

Proceeding of International Conference on: Free Boundary Problems: Theory and Applications I, Gakkotosho 13 369 - 384 2000

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Traveling curved fronts of a mean curvature flow with constant driving force

H. Ninomiya, M. Taniguchi

Proceeding of International Conference on: Free Boundary Problems: Theory and Applications I, Mathematical Sciences and Applications, Gakkotosho 13 206 - 221 2000

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Instability of spherical interfaces in a nonlinear free boundary problem

X. Chen, M. Taniguchi

Advances in Differential Equations 5 ( 4-6 ) 747 - 772 2000

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Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor

M Taniguchi, Y Nishiura

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 126 117 - 145 1996

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Language：English Publishing type：Research paper (scientific journal) Publisher：ROYAL SOC EDINBURGH

A characteristic wavelength and its parametric dependency are studied for planar interfaces of activator-inhibitor systems as well as their stability in two-dimensional space. When an unstable planar interface is slightly perturbed in a random way, it develops with a characteristic wavelength, that is, the fastest-growing one. A natural question is to ask under what conditions this characteristic wavelength remains finite and approaches a positive definite value as the width of interface, say epsilon, tends to zero. In this paper, we show that the fastest-growing wavelength has a positive limit value as epsilon tends to zero for the system:
u(t) = Delta u + epsilon(-2) f(u, v), v(t) = epsilon(-1) Delta v + g(u, v).
This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.

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A remark on singular perturbation methods via the Lyapunov-Schmidt reduction

M Taniguchi

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 31 ( 6 ) 1001 - 1010 1995.12

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Language：English Publishing type：Research paper (scientific journal) Publisher：KYOTO UNIV

For some reaction-diffusion equations, Lyapunov-Schmidt reduction was shown to be applicable to construct singularly perturbed equilibrium solutions. For this application, it is indispensable to show that some inverse operator are uniformly bounded. In this paper, we give an elementary proof of this fact.

DOI： 10.2977/prims/1195163593

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INSTABILITY OF PLANAR INTERFACES IN REACTION-DIFFUSION SYSTEMS

M TANIGUCHI, Y NISHIURA

SIAM JOURNAL ON MATHEMATICAL ANALYSIS 25 ( 1 ) 99 - 134 1994.1

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Language：English Publishing type：Research paper (scientific journal) Publisher：SIAM PUBLICATIONS

Instability of planar front solutions to reaction-diffusion systems in two space dimensions is studied. Let epsilon denote the width of interface. Then the planar front solution-or a solution having an internal transition layer which is flat-loses its stability when the length of interface along the tangential direction exceeds O(epsilon(1/2)). The wavelength of the fastest growth is of O(epsilon(1/3)) which is inherent in the system and determined by the nonlinearity and diffusion coefficients. Complete asymptotic characterization of these quantities as a epsilon --> 0 is given by the analysis of what is called the singular dispersion relation derived from the linearized eigenvalue problem. The numerical computations also confirm that the theoretically predicted fastest growth wavy pattern actually arises from a randomly perturbed planar front.

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Bifurcation from flat-layered solutions to reaction diffusion systems in two space dimensions

M. Taniguchi

Journal of Mathematical Sciences The University of Tokyo 1 ( 2 ) 339 - 367 1994

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Language：English Publisher：The University of Tokyo

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Books

Traveling front solutions in reaction-diffusion equations

Masaharu Taniguchi

Mathematical Society of Japan 2021 （ ISBN:9784864970976 ）

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Total pages：xiii, 170 p. Language：English

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現代数理科学事典

広中, 平祐（ Role： Contributor , 「特異摂動論」（XI-9-2，Page 1283--1289）を執筆）

丸善 2009.12 （ ISBN:9784621081259 ）

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Total pages：xvi, 1454p Language：Japanese

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数学の言葉と論理

渡辺治, 北野晃朗, 木村泰紀, 谷口雅治

朝倉書店 2008 （ ISBN:9784254117516 ）

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MISC

An $(N-2)$-dimensional surface with positive principal curvatures gives an $N$-dimensional traveling front in bistable reaction-diffusion equations (Mathematical Analysis of Pattern Formation Arising in Nonlinear Phenomena)

Taniguchi Masaharu

RIMS Kokyuroku 1924 1 - 10 2014.11

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Language：English Publisher：Kyoto University

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Encyclopedia of Modern Mathematical Science, The 2nd Edition

Takada Toshikazu, Taniguchi Masaharu

Bulletin of the Japan Society for Industrial and Applied Mathematics 20 ( 4 ) 350 - 351 2010

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Language：Japanese Publisher：The Japan Society for Industrial and Applied Mathematics

DOI： 10.11540/bjsiam.20.4_350

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Pyramidal traveling fronts in the Allen-Cahn equations (Viscosity Solutions of Differential Equations and Related Topics)

Taniguchi Masaharu

RIMS Kokyuroku 1651 92 - 109 2009.5

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Language：English Publisher：Kyoto University

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Other Link： http://hdl.handle.net/2433/140785
Equadiff 2003

Taniguchi Masaharu

Bulletin of the Japan Society for Industrial and Applied Mathematics 14 ( 1 ) 94 - 94 2004

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Language：Japanese Publisher：The Japan Society for Industrial and Applied Mathematics

DOI： 10.11540/bjsiam.14.1_94

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Motion of forced curvature flow on the strip domain (Viscosity Solutions of Differential Equations and Related Topics)

Ninomiya Hirokazu, Taniguchi Masaharu

RIMS Kokyuroku 1323 27 - 32 2003.5

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Language：English Publisher：Kyoto University

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Other Link： http://hdl.handle.net/2433/43127
SIAM Pacific Rim Dynamical Systems Conference(Conference Reports)

Taniguchi Masaharu

Bulletin of the Japan Society for Industrial and Applied Mathematics 11 ( 1 ) 82 - 83 2001

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Language：Japanese Publisher：The Japan Society for Industrial and Applied Mathematics

DOI： 10.11540/bjsiam.11.1_82

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Introduction to Research institute for Mathematical Sciences, Departments of Industrial and Applied Mathematics, Kyoto university

Taniguchi Masaharu

Bulletin of the Japan Society for Industrial and Applied Mathematics 6 ( 4 ) 334 - 336 1996

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Language：Japanese Publisher：The Japan Society for Industrial and Applied Mathematics

DOI： 10.11540/bjsiam.6.4_334

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Presentations

Entire solutions with and without radial symmetry in balanced bistable reaction-diffusion equations

Masaharu Taniguchi

2024.3.20

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Event date： 2024.3.17 - 2024.3.20

Language：Japanese Presentation type：Oral presentation (general)

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Traveling front solutions of dimension $n$ generate entire solutions of dimension $(n-1)$ in reaction-diffusion equations

Masaharu Taniguchi, Hirokazu Ninomiya

2024.3.18

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Event date： 2024.3.17 - 2024.3.20

Language：Japanese Presentation type：Oral presentation (general)

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等エネルギー型反応拡散方程式における与えられた長軸と短軸をもつ凸図形を切断面とする進行波

谷口雅治

日本数学会2023年度秋季総合分科会 2023.9.22

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Event date： 2023.9.20 - 2023.9.23

Language：Japanese Presentation type：Oral presentation (general)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited

Masaharu Taniguchi

The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications 2023.5.31

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Event date： 2023.5.31 - 2023.6.4

Language：English Presentation type：Oral presentation (invited, special)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited

Masaharu Taniguchi

The 2022 Pacific Rim Mathematical Association Congress 2022.12.8

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Event date： 2022.12.4 - 2022.12.9

Language：English Presentation type：Oral presentation (invited, special)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited

Masaharu Taniguchi

International Conference on Nonlinear Partial Differential Equations 2022 2022.10.19

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Event date： 2022.10.19 - 2022.10.21

Language：English Presentation type：Oral presentation (general)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited

Masaharu Taniguchi

BIRS workshop (22w5165) "Interfacial Phenomena in Reaction-Diffusion Systems" 2022.8.2

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Event date： 2022.7.31 - 2022.8.5

Language：English Presentation type：Oral presentation (general)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited

Masaharu Taniguchi

SIAM 2022 Conference on Analysis of Partial Differential Equations 2022.3.15

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Event date： 2022.3.14 - 2022.3.18

Language：English Presentation type：Oral presentation (general)

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等エネルギー型反応拡散方程式における軸非対称進行波 Invited

谷口雅治

日本数学会函数方程式論分科会「微分方程式の総合的研究」 2019.12.22

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Event date： 2019.12.21 - 2019.12.22

Language：Japanese Presentation type：Oral presentation (invited, special)

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Allen-Cahn方程式における角錐型進行波 Invited

谷口雅治

非線形現象の数値シミュレーションと解析2008 2008.3.6

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Event date： 2008.3.6 - 2008.3.7

Language：Japanese Presentation type：Oral presentation (invited, special)

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Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations

Masaharu Taniguchi

2019.9.18

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Language：Japanese Presentation type：Oral presentation (general)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited

Masaharu Taniguchi

2019.9.10

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Language：Japanese Presentation type：Oral presentation (general)

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Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations Invited International conference

Masaharu Taniguchi

Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type, MATRIX Research Centre, Melbourne, Australia 2018.11.12

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Language：English Presentation type：Oral presentation (invited, special)

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Axially non-symmetric traveling fronts in balanced bistable reaction-diffusion equations

Masaharu Taniguchi

2018.9.25

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Language：Japanese Presentation type：Oral presentation (general)

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Pyramidal traveling fronts in the Allen-Cahn equations

PDE Seminar, School of Mathematics 2008

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Pyramidal traveling fronts in the Allen-Cahn equations

PDE Seminar, School of Mathematics 2008

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Allen-Cahn方程式における角錐型進行波の一意性と安定性

日本数学会2008年度秋期総合分科会 2008

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Allen-Cahn方程式における角錐型進行波とその応用

関数方程式セミナー 2008

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Stability of pyramidal traveling fronts in the Allen-Cahn equation

Workshop on Singularities Arising in Nonlinear Problems 2007 2007

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Stability of pyramidal traveling fronts in the Allen-Cahn equation

Workshop on Singularities Arising in Nonlinear Problems 2007 2007

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Allen-Cahn 方程式における多次元進行波

日本数学会年会 2006

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Research Projects

Is it possible to mathematically formulate origami for materials with the property of stretching and shrinking?

Grant number：22K03288 2022.04 - 2027.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

近藤慶, 谷口雅治, 物部治徳

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Grant amount：\3900000 （ Direct expense: \3000000 、 Indirect expense：\900000 ）

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Traveling fronts whose cross sections are convex shapes with major axes and minor axes in balanced bistable reaction-diffusion equations

Grant number：20K03702 2020.04 - 2025.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)

谷口雅治, 二宮広和

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Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

等エネルギー型反応拡散方程式において，「長軸と短軸をもつ凸図形を切断面とする進行波」の存在を証明することが本研究の目的である。反応項が等エネルギー型である場合，1次元進行波は速度ゼロの Standing Frontとなる。この場合，2次元以上の空間において進行波は進行軸にたいして軸対称なもの存在することが Chen, Guo, Ninomiya, Hamel and Roquejoffre (ANIHPC 2007)により証明された。進行軸にたいして非対称な形状をもつ進行波が存在するかどうかは未解決であった。本研究では，等高面の切断面が「長軸と短軸をもつ凸図形をなす進行波」が存在することを証明した。この成果は Taniguchi (ANIHPC 2019)に掲載された。ある高さでの等高面の切断面に対して，長軸と短軸の比を任意に設定した場合，そのような進行波が存在することを示したものである。一方，長軸と短軸の数値を任意に設定した場合，高さを適当に定めることにより，そのような凸図形を切断面とする進行波が存在するかどうかという問題が新規に得られた。この問題に対して肯定的な解答が本研究で得られた。
また，上記の進行波解について，等高面の切断面の形状が，漸近的にどのようになっているかという課題は未解決であり，現在この課題に取り組んでいる。
2022年3月にドイツのベルリンにおいて SIAM Conference on Analysis of Partial Differential Equations (PD22) が開催され，私は招待講演を行った。2022年7月においては，カナダのBanff International Research Stationにおいて研究集会が開催される。この研究集会に参加し，国内外の研究者と情報交換とディスカッションを行う予定である。また2022年12月にカナダのバンクーバーで開催される研究集会PRIMA2022において招待講演をおこなう予定である。今後も国内外の研究者と情報交換とディスカッションを行いつつ，本研究を推進する。

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Mathematical analysis of pattern dynamics of reaction-diffusion systems and their singular limit problems

Grant number：20H01816 2020.04 - 2024.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B) Grant-in-Aid for Scientific Research (B)

二宮広和, 飯田雅人, 谷口雅治, 三竹大寿, 物部治徳

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Grant amount：\17420000 （ Direct expense: \13400000 、 Indirect expense：\4020000 ）

非線形放物型偏微分方程式の解のダイナミクスを決定することは，非線形放物型偏微分方程式の理論的研究における重要な問題のひとつである．しかし，比較的簡単と考えられる反応拡散系でさえ，解のダイナミクスを決定できていないのが現状である．本研究課題では，反応拡散系の解のダイナミクスを決定するための解析手法の開発と普遍的な数理構造の抽出を行うことを目的としている．反応拡散系の解の普遍的数理構造を抽出するにあたっては，未知変数の数，反応項（非線形項），空間の次元，領域が主要なパラメータとなる．解のダイナミクスを決定するアトラクターを調べることは，その要素である全域解の特徴付けることに対応しているので，前述の主要なパラメータを変えることで，次の３つのテーマを扱う．
（１）単独反応拡散系の全域解の特徴付け
（２）特異極限系の適切性・収束性・全域解の特徴付け
（３）複雑領域におけるパターンダイナミクスの数理解析
（１）では，多次元双安定単独反応拡散系の進行波解の解析を行った．また，全域解と進行波解の関係を調べた．（２）では，反応拡散系の特異極限系である反応界面系の解の挙動に関する研究を行い，1次元の場合にその挙動を分類することに成功した．また，Hamilton-Jacobi方程式の全域解の特徴付けについて研究を進めた．また，反応拡散系から非線形波動方程式を導出する手法の開発を行った．（３）の準備として，多次元界面方程式を扱うために，多層界面方程式を考察する手法を開発し論文に発表した．
また，明治非線型数理セミナーおよび明治非線型数理セミナー・秋の学校（2020年 11月22日(日) ～ 11月24日(火)）を開催し，情報収集とともに研究成果の公表していった．

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An (N-2)-dimensional surface with positive principal curvatures gives an N-dimensional traveling front in bistable reaction-diffusion equations

Grant number：26400169 2014.04 - 2019.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)

Masaharu Taniguchi

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Grant amount：\4810000 （ Direct expense: \3700000 、 Indirect expense：\1110000 ）

In this project, I consider a parabolic equation with a bistable nonlinear term. This equation is called the Allen--Cahn equation or the Nagumo equation.The aim of this project is to search unknown traveling fronts. The result is as follows. For every given compact convex set in the (N-1)-Euclidean space, I proved the existence
of an N-dimensional traveling front solution associated with this set. Moreover, I proved that this traveling front solution is asymtotically stable if the given perturbation decays at infinity. These results were published by SIAM J. Math. Anal. 2015 and by J. Differential Equations 2016.

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Pattern dynamics of reaction-diffusion systems and free boundary problems

Grant number：26287024 2014.04 - 2018.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B) Grant-in-Aid for Scientific Research (B)

Ninomiya Hirokazu, MONOBE Harunori

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Grant amount：\16510000 （ Direct expense: \12700000 、 Indirect expense：\3810000 ）

To study the spatial patterns of solutions of partial differential equations, such as reaction-diffusion systems, we introduce a reaction-interface system, which consists of the interface equation and an equation in the whole space. This is derived as a singular limit of some reaction-diffusion systems. We studied the multidimensional traveling wave solution and the pulse dynamics of the reaction-interface system. Moreover, for the curvature flow with the anisotropic external force, we study the influence of the anisotropy to the compact traveling wave solutions. We also introduce the layered system to analyze the spatial profiles of solutions in multidimensional space.

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Systematic studies on the profile and behavior of solutions of partial differential equations

Grant number：24244012 2012.05 - 2017.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

Yanagida Fiji, Polacik Peter, Fila Marek, Shi Jun-Ping, Htoo Khin Phyu Phyu, Hui Kin-Ming, Chern Jann-Long, Chen Chiun-Chuan, Marta Garcia-Huidobro, Lou Yuan

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Grant amount：\44720000 （ Direct expense: \34400000 、 Indirect expense：\10320000 ）

We carried out systematic studies on the profile and behavior of solutions of elliptic and parabolic partial differential equations, and obtained the following results among others.
For elliptic equations, we showed a Liouville-type theorem for an equation transformed by a scaling method. We also obtained results on the existence and no-existence of singular solutions and the structure of positive solutions for equations on the sphere, and equations with space-dependent exponent of nonlinearities.
For parabolic equations, we made clear the structure of solutions of a linear heat equation with inverse-square potential. We also studied the Fujita equation about the boundedness of time-global solutions, existence of singular solutions and their stability, and Fisher's equation about the dynamics of interfaces.

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The Study of Nonlinear Functional Analysis and Nonlinear Problems Based on Fixed Point Theory and Convex Analysis

Grant number：23540188 2011.04 - 2015.03

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

TAKAHASHI Wataru, TANIGUCHI Masaharu, KIMURA Yasunori

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Grant amount：\5070000 （ Direct expense: \3900000 、 Indirect expense：\1170000 ）

In this research, we studied nonlinear functional analysis and nonlinear problems by using fixed point theory and convex analysis. At first, we introduced the concept of attractive points of nonlinear mappings in Hilbert spaces and Banach spaces. Then we proved the existence of atrractive points and nonlinear mean convergence theorems. Furthermore, we proved weak and strong convergence theorems for semigroups of not necessarily continuous mappings in Hilbert spaces and Banach spaces. Using these theorems, we solved nonlinear problems which are important in many areas of applied mathematics.

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Three-dimensional cylindrically non-symmetric traveling fronts in reaction-diffusion equations

Grant number：23540235 2011 - 2013

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)

TANIGUCHI Masaharu

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Grant amount：\5070000 （ Direct expense: \3900000 、 Indirect expense：\1170000 ）

The results are as follows. (1) We proved N-dimensional pyramidal traveling fronts in the Allen-Cahn (Nagumo) equation.(2) We consider the Allen-Cahn (Nagumo) equation in the three-dimensional space, and proved the existence and stability of cylindrically non-symmetric traveling fronts. The cross sections of these traveling fronts are smooth convex shapes, say, ellipses. (3) We prove the existence of N-dimensional pyramidal traveling fronts in competition-diffusion systems. (4) We prove the non-existence of localized traveling spots with non-zero speed in a single reaction-diffusion equation under some condition.

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The Study of Nonlinear Functional Analysis and Convex Analysis and its Applications Based on Optimization Theory and Fixed Point Theory

Grant number：19540167 2007 - 2010

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

TAKAHASHI Wataru, TANIGUCHI Masaharu, KIMURA Yasunori, KOMIYA Hidetoshi, KIDO Kazuo

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Grant amount：\4550000 （ Direct expense: \3500000 、 Indirect expense：\1050000 ）

In this study, we obtain many new and important theorems for nonlinear problems in nonlinear functional analysis and convex analysis by using optimization theory and fixed point theory. For example, we solved an open problem in geometry of Banach spaces which has been posed by Ray in 1980 by proving that every nonspreading mapping of a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space into itself has a fixed point if and only if the set is bounded.

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Global stability of multi-dimensional traveling fronts

Grant number：18540208 2006 - 2009

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)

TANIGUCHI Masaharu

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Grant amount：\4150000 （ Direct expense: \3400000 、 Indirect expense：\750000 ）

I studied the Allen-Cahn equation (Nagumo equation) in three-dimensional Euclidean space, and constructed pyramidal traveling front solutions and convex polyhedral traveling front solutions. I also proved that they are stable for given fluctuations.

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Interface motion and blow-up phenomena in nonlinear partial differential equations

Grant number：17340044 2005 - 2007

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

MATANO Hiroshi, FUNAKI Tadahisa, WEISS Georg, TANIGUCHI Masaharu, MIZUMACHI Tetsu, NAKAMURA Ken-Ichi

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Grant amount：\15620000 （ Direct expense: \14300000 、 Indirect expense：\1320000 ）

The aim of this research project is to make a theoretical study of various nonlinear problems related to interfacial motions and blow-up phenomena, by developing asymptotic methods based on the theory of infinite dimensional dynamical systems and stochastic methods, and also performing numerical simulations if necessary. We have obtained the following results.
(1) We have given an optimal estimate concerning the singular limit of Allen-Cahn type nonlinear diffusion equations, that has not been known previously. We also obtained similar results for FitzHugh-Nagumo systems and Lotka-Volterra competition systems.
(2) We have clarified the global dynamics of blow-up solutions in nonlinear diffusion equations. This was made possible by extending the existing theory on global attractors to the case where blow-up occurs.
(3) In nonlinear heat equations with power nonlinearity, the blow-up of solutions is classified into type I and type II, the latter being much more difficult to analyze than the former. By using a topological method based on the braid group theory, we have succeeded in determining all possible type II blow-up rates.
(4) We studied the stationary problem for the Allen-Cahn equation on the 2-dimensional space having lattice periodicity by using variational methods. We have shown that a necessary and sufficient condition for the existence of multi-layered stationary solution is that the set of single layered solutions has a gap somewhere ; in other words, this set should not be a continuum (foliation).
(5) We considered periodic traveling waves in a two-dimensional infinite strip whose boundaries are saw-tooth shaped. We determined the homogenization limit of such traveling waves as one lets the boundary undulation finer and finer.
(6) Various partial differential equations are derived from microscopic models via hydrodynamic limit. Those equations include the Stefan problem and some stochastic differential equations.
(7) Regularity of free boundaries arising in various elliptic equations such as the combustion model has bee established.
(8) Stability of V-shaped traveling wave in the equation of curvature-dependent motion has been established.

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Nonlinear functional analysis and nonlinear problems by using fixed point theory

Grant number：15540157 2003 - 2006

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

TAKAHASHI Wataru, TANIGUCHI Masaharu, KIMURA Yasunori

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Grant amount：\3600000 （ Direct expense: \3600000 ）

We studied nonlinear functional analysis and some nonlinear problems by using fixed point theory. We first studied the existence of zero points of maximal monotone operators in Banach spaces. We proved existence theorems for maximal monotone operators by a new boundary condition. Using one of them, we obtained a generalization of Kakutani's fixed point theorem for multivalued mappings. Then we considered iteration schemes of finding zero points of maximal monotone operators in Banach spaces. We found two new resolvents for maximal monotone operators and then obtained weak and strong convergence theorems for resolvents of maximal monotone operators in Banach spaces with generalized projections. In particular, we obtained two generalizations of Solodov and Svaiter's theorem by using generalized projections and metric projections. Further, we introduced the notion of relatively nonexpansive mappings in a Banach space which generalize nonexpansive mappings in a Hilbert space. Then, we obtained two convergence theorems for relatively nonexpansive mappings in Banach spaces. One of them is a generalization of Nakajo and Takahash's theorem. We also obtained important examples of sunny generalized nonexpansive retractions which are related to one of new resolvents. Next, we introduced an iteration scheme of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an inverse-strongly-monotone operator in a Hilbert space. Further, we extended this iteration scheme to that of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space. Then we proved some weak and strong convergence theorems. Using these results, we improved and extended Wittmann' strong convergence theorem, Reich's weak convergence theorem and Baillon's nonlinear ergodic theorem which are well known in this field.

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Allen-Cahn方程式におけるV字型進行曲面波

Grant number：15740102 2003 - 2005

日本学術振興会科学研究費助成事業若手研究(B)

谷口雅治

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Grant amount：\2400000 （ Direct expense: \2400000 ）

Allen-Cahn方程式を含む相安定な反応拡散方程式において,V字型進行曲面波の存在をしめし,その(局所)安定性を証明するという目的はH.Ninomiya and M.Taniguchi(J.Difrerential Equations,213,No 1005),204-233)において達成されたことを報告する.
この研究の過程において新たな課題が発生した.以下の課題である.
(1)1次元進行波をもつ双安定な非線形項はどのようなものがあるか?
(2)V字型進行曲面波の安定性は空間大域的であるか?
Allen-Cahn方程式にあらわれる非線形項は3次式であるが,双安定な非線形項はこれに限られない.しかしながら1次元進行波をもたない双安定な非線形項も知られている.課題(1)および(2)にたいする部分的な回答を,Ninomiya and Taniguchi(Discrete and Continuous Dynamical Systems,掲載受理)において行った.1次元進行波をもつ双安定な非線形項の例を出し,その場合の進行波の具体的な表現式を与えた.Allen-Cahn方程式および,それらのより一般の非線形項をもつ反応拡散方程式において,初期擾乱が無限遠方で減衰するならば,V字型進行曲面波が漸近安定であることを証明した.
無限遠方で減衰しない初期摂動にたいして,V字型進行曲面波の漸近安定性は,未解決の課題である.Allen-Cahn方程式でなく,ある意味でその極限形と考えられる曲率流方程式において,Nara and Taniguchi(Discrete and Continuous Dynamical Systems,掲載受理)により,直線およびV字型進行曲面波が漸近安定となる十分条件を与えた.また,漸近安定とならない有界な初期擾乱の例も与えた.
以上を報告する.

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Qualitative theory and asymptotic analysis of nonlinear partial differential equations

Grant number：13440028 2001 - 2004

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

MATANO Hiroshi, FUNAKI Tadahisa, YAMAMOTO Masahiro, WEISS Georg, EI Shin-Ichiro, TANIGUCHI Masaharu

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Grant amount：\16100000 （ Direct expense: \16100000 ）

We have considered the behavior of solutions after the blow-up time for nonlinear heat equations with a power nonlinearity and those with an exponential nonlinearity. It is shown that singularities that apper at the blow-up time disappear and the solutions become smooth immediately (Matano, SIAM J.Math.Anal., in press). We have also studied the blow-up rate for nonlinear heat equations with a power nonlinearity and proved that the blow-up is always type 2 so far as the power is in the intermediate supercritical range (Matano, Comm.Pure Appl.Math., 2004).
Yamamoto has studied an inverse problem of determining two unknown convection terms in a two-dimensional elliptic equation, and proved that those terms can be determined by the so-called Dirichlet-Neumann map (Inverse Problems, 2004).
Weiss has considered a singular limit problem for parabolic equations that are applicable to the double obstacle problem. Using a new monotonicity formula, he has succeeded in exstimating the Hausdorff dimension of the free boundary (Calc.Var.PDE, 2003).
Ei has studied a reaction-diffusion system on a two-dimensional cylinder and analysed the behavior of solutions having a pulse-like profile. He derived an equation governing the motion of slow pulses and proved that traveling pulses are mutually repelling (DCDS, Ser.A, in press).
Taniguchi has considered the so-called "singular limit eigenvalue problem method" (SLEP method), which is a powerful tool in analyzing the stability of stationary solution of the singular limit problem. He has generalized this method so that it applies to problems in unbounded domains. Using this result, he has proved the stability of planar traveling waves in a bistable reaction-diffusion system (DCDS, Ser.B, 2003).

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V字型の進行曲面波の漸近安定性

Grant number：13740113 2001 - 2002

日本学術振興会科学研究費助成事業若手研究(B)

谷口雅治

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Grant amount：\2100000 （ Direct expense: \2100000 ）

等速成長効果のある平均曲率流方程式においてV字型進行曲面波の安定性について得られた結果を報告いたします.方程式υ=H+kを全平面R^2で考える.ここで,υは曲面の法線方向の速度を表し,Hは曲率を表し,kは与えられた正の定数で,等速成長効果を表す.曲面がy=u(x, t)とグラフで表される場合に方程式はu_t=(1+u^2_x)^<-1>u_<xx>+k(1+u^2_x)^<1/2>となる.任意のc∈(k,+∞)に対しcを速度とするV字型進行曲面波とよばれる進行波のワンパラメータ族が存在することが従来より知られていた.この進行曲面波がどのような与えられた摂動(擾乱)に対し,漸近安定であるか,またそうでないかについて次の結果が得られた.比較定理により摂動が増大しないという意味での安定性については直ちにしたがう.摂動が時間とともに減衰するのかという漸近安定性については未知であった.本研究では,優解と劣解を構成することにより,与えられた摂動が遠方で減衰する場合に対して,V字型進行曲面波の漸近安定性を証明した.この優解は任意に大きく,また劣解は任意に小さくとることができるので,空間大域的に漸近安定であることがわかる.すなわち,どのような大きい摂動であってもそれが遠方で減衰するならば,V字型進行曲面波は元の形に復元される.また,与えられた摂動が減衰しない場合においては,適当な比較関数を構成することにより,漸近安定でない例を構成した.
この結果は,学術雑誌Methods and Applications of Analysisにおいて発表いたしました.

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Nonlinear functional analysis and convex analysis problem by using fixed point theory

Grant number：12640157 2000 - 2002

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

TAKAHASHI Wataru, KIUCHI Hirobumi, TANIGUCHI Masahara, KOJIMA Masakazu

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Grant amount：\3500000 （ Direct expense: \3500000 ）

We studied some problems concerning nonlinear functional analysis and convex analysis by using fixed point theory. We first considered iteration schemes given by an infinite family of nonexpansive mappings in Hilbert spaces or Banach spaces and then proved strong convergence theorems for the family of nonexpansive mappings. Using these results, we also considered the feasibility problem of finding a common fixed point of infinite nonexpansive mappings. Next, we introduced two proximal point algorithms suggested by the iterative schemes introduced by Solodov and Svaiter in order to find a solution of $v \in T^∧{-1}0$, where $T$ is a maximal monotone operator. Main results were established by using metric projections and generalized projections in the case of the strong convergence. We also applied these results to find a minimizer of a lower semicontinuous convex function in a Banach space. Finally, we introduced iteration schemes of finding a common element of the set of fixed points of nonexpansive mappings and the set of solutions of the variational inequality for inverse-strongly-monotone mappings. Using these results, we considered the problem of finding a common element of the set of zeros of a maximal monotone mapping and the set of zeros of an inverse-strongly-monotone mapping.

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等速成長効果のある平均曲率流方程式における進行曲面波

Grant number：12740100 2000

日本学術振興会科学研究費助成事業奨励研究(A)

谷口雅治

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Grant amount：\800000 （ Direct expense: \800000 ）

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自由境界問題における定常球の多重存在とその安定性

Grant number：10740083 1998 - 1999

日本学術振興会科学研究費助成事業奨励研究(A)

谷口雅治

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Grant amount：\1400000 （ Direct expense: \1400000 ）

活性因子・抑制因子モデルとよばれるモデルに現れる境界面を記述するステファン型の発展方程式について、定常状態にある球面の存在と安定性について研究を行ないました。得られた結果は以下の通りです。まず球面状の境界面をもつ定常状態は一般に複数個存在することがわかりました。その安定性を調べた結果、それら複数個の定常球は、4種類に分類できることを証明しました。第1のタイプは安定な場合であり、第2のタイプは球対称なある摂動にのみ不安定であるものです。第3のタイプは最不安定なある非対称モードをもつもので、第4のタイプは2重のゼロ固有値をものものです。この結果は化学反応において核の生成と成長に対して知見と示唆を与えるものと思われます。

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Study of singularities arising in nonlinear partial differential differential equations and asymptotic methods

Grant number：09304019 1997 - 1999

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

MATANO Hiroshi, YAMAMOTO Masahiro, YANAGIDA Eiji, FUNAKI Tadahisa, TANIGUCHI Masaharu, MIMURA Masayasu

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Grant amount：\21500000 （ Direct expense: \21500000 ）

(1) Dynamics of blow-up solutions Some blow-up solutions of a nonlinear heat equation can be continued beyond the blow-up time in a certain weak sense. Matano studied the dynamics of such solutions from the point of view of dynamical systems.
(2) Motion of interfaces with random deviation In a class of diffusion equations involving a small parameter, say ε, solutions develop sharp transition layers, or interfaces, as ε→0. Funaki considered the case where the equation involves a random deviation term.
(3) Estimate of blow-up time in a nonlinear heat equation Yanagida sutdied blow-up phenomena in a nonlinear heat equation and extended the classical results of Fujita and others.
(4) Motion of interface in competition systems Mimura studied the behavior of interfaces that arise in the singular limit of a three-species competition-diffusion system.
(5) Order-preserving systems in the presence of symmetry Matano extended the existing theory on order-preserving dynamical systems in the presence of symmetry. He then applied the general theory to the stability analysis of traveling waves and other problems.

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An Over-all Mathematical Study of the Nonlinear Boltzmann Equation and Fluid Dynamical Equations

Grant number：09440051 1997 - 1999

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

UKAI Siji, HIRANO Norimihi, TAKANO Seiji, KITADA Yasuhiko, SHIOJI Naoki, KONNO Norio

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Grant amount：\6500000 （ Direct expense: \6500000 ）

Boltzmann equation :
(1) A simple existence proof of the Boltzmann-Grad limit by means of the Cauchy-Kowalevskaya theorem and the establishement of an asymptotic relation between the Boltzmann hierachy and the macroscopic fluid equation (commpressible Euler equation) by the same theorem.
(2) An existence theorem of travering (shock) wave solutions and a solvability condition for the stationary problem in the half space, both for the discrete velocity model, which are expected to make an contribution to the study of boundary and shock layer structures of the Boltzmann equation.
(3) An existence theorem of time-periodic solutions of the Boltzmann equation, being a first analysis of the nonlinear acoustics of that equation.
Macroscopic fluid dynamical equation :
(1) Non-relativistic limits of solutions of the relativisitic Euler equation. In the case of the 1D flat Minkowski space-time, time-global weak solutions are shown to converge globally in time stongly in LィイD11ィエD1, as the speed of light tens to infinity, and similary for the case of the 3D non-flat space-time, but the convergence is time-local.
(2) A time-global existence theorem of weak and srong solutions to the Stokes approximation equation for the storngly viscous commonpressible fluid flow. While the equation has a strong nonlinearity, initial data can be arbitrarily large.
(3) An existence proof of the stationary solution to the heat covection equation without the unphysical condition of the zero outflow on each component of boundaries of the domain.

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Mathematical Analysis of Infinite Dimensional Stochastic Models

Grant number：09640246 1997 - 1998

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

SHIGA Tokuzo, TANIGUCHI Masaharu, TAKAOKA Koichiro, NINOMIYA Hirokazu, MORITA Takehiko, UCHIYAMA Kohei

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Grant amount：\3000000 （ Direct expense: \3000000 ）

Performing the reseach based on the project plan we obtained the following reseach results.
1. Fleming-Viot processes play an important role in population genetics, for which we obtained two significant results.
First, we considered the model with mutation and unbounded selectionas genetic factors. In this case it has not proved even the well-posedness of the diffusion processes, which we settled together with the uniqueness problem of the stationary distributions. This work was caried out jointly with S.N.Ethier (USA). Furthermore we solved the problem of diffusion approximation from discrete time Markov chain models.
Second, we solved a reversibility problem for the Fleming-Viot processes with mutation and selection, that is to characterize the mutation operator for the process to have a reversible distribution. This work was done with Z.H.Li (China) and L.Yau (USA). (Shiga)
2. We considered a suvival probability problem of random walker in temporarily and spatially varing random environment, and obtained a precise asymprotics of the suvival probability for small parameter rigion. To prove it we developed a detailed analysis of linear stochastic partial differential equations which are dual objects of the random walk model. This result appeared as ajoint work with T.Furuoya.
Directed polymer model is a closely related with this problem in mathematical context, and we get some significant results on asymptotical behaviorof the random partition function in low dimensional case, which is harder than higher dimensional case. (Shiga)
3. For a mechanical many particle system Uchiyama established the hydrodynamic limit and identified its hydrodynamic equation, that is a diffusion equation in this situation.
4. For a dynamical system in cofinite Fuchsian groups which can be regarded as a Markov system, Morita developed a perterbational analysis of the transfer operator and solved some ergodic problem that is related to number theory.
5. Motivated by mathematical finance Takaoka obtained a neccesary and suffucient condition for a continuous local martingale to be uniformly integrable.

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Nonlinear problem by using fixed point theory

Grant number：09640160 1997 - 1998

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

TAKAHASHI Wataru, KIUCHI Hirobumi, TANIGUCHI Masaharu, UKAI Seiji

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Grant amount：\2900000 （ Direct expense: \2900000 ）

We studied some nonlinear problems concerning nonlinear evolution equation, mathematical economics, mathematical programming and image recovery by using nonlinear functional analysis and fixed point theory. We first proved some fixed point theorems for families of nonexpansive mappings in a Banach space. Next, we proved nonlinear ergodic theorems of Baillon's type for nonlinear semigroups of nonexpansive mappings. In particular, we gave an answer to the open problem posed during the Second World Congress on Nonlinear Analysts, Athens, Greece, 1996, by extending Takahashi's result and Rode's result to a Banach space for an amenable semigroup of nonexpansive mappings. Further, we established weak convergence theorems of Mann's type for families of nonexpansive mappings. We also proved strong convergence theorems of Halpern's type for families of nonexpansive mappings. Finally, using these results, we discussed the problem of image recovery by convex combinations of nonexpansive retractions, the problem of finding a common fixed point of a commuting family of nonexpansive mappings, the convex minimization problem and so on.

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平面状進行波の不安定性

Grant number：08740100 1996

日本学術振興会科学研究費助成事業奨励研究(A)

谷口雅治

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Grant amount：\800000 （ Direct expense: \800000 ）

非線形の反応項をもつ放物型偏微分方程式の解として,平面状の境界面をもつ進行波の安定性を論じた.化学において系の状態がある境界面を境に二相に分かれ,その間にうすい遷移層が存在し,それが一定の速度で伝わっていく現象が観測される.これらの現象では通常,境界面は最初に平面状であっても次第に変形し,複雑な形となる.このときに変形の第一段階としての一定の波長の変形が観測される.本研究ではこの現象を理論的に調べた.
平面状進行波について,線形化固有値問題で安定性をしらべた結果として最初につぎのことがわかった.エッセンシャルスペクトラムはすべて実部が負である.すなわち,あとは固有値のみを調べればよい.エヴァンス関数を使う方法によって固有値の分布を調べたところ,固有値はすべて実数であり,上に狭義凸なある関数の上に離散的に分布していることがわかった.この関数はある区間で正の値をもつ.このため,正の固有値が存在し,平面状進行波は不安定であることが理論的に証明された.
またさらに,この関数の性質をくわしく調べることによって,最大の固有値をあたえる固有関数の性質が得られた.この固有関数の形により,平面状進行波に微小な外乱がくわえられた直後に生じる一定の波長が得られた.この波長は,活性因子の拡散係数と抑制因子のそれとの積の1/3乗に比例している.この結果は,“Instabiliy of planar traveling fronts in bistable reaction-diffusion systems"として現在,投稿中である.

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反応拡散方程式による直線状内部遷移層解からの解の分岐構造の研究

Grant number：07740102 1995

日本学術振興会科学研究費助成事業奨励研究(A)

谷口雅治

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Grant amount：\1100000 （ Direct expense: \1100000 ）

研究実績は以下の通りです。反応拡散方程式の定常解で直線状界面をもつ定常解の安定性を詳しく調べパラメーターをかえて安定性が変わるときの新たな定常解が分岐してくる構造を詳しく調べました。この成果は研究発表の項の3番目の論文として発表しております。
内容をより詳しく述べますと以下の通りです。直線状界面の長さをlとし、これをパラメータとして動かします。遷移層の厚さをεとおくと、あるクリティカルな長さl_c(ε)があり、lがl_c(ε)より長いときには不安定、短いときには安定となっています。すなわちlをしだいに小さくしていくとl=l_c(ε)で安定性がかわります。これをCrandall-Rabinowitzらの分岐理論にのせて分岐解をさがすためにはゼロ固有値が“algelraically simple"であることを示す必要があります。これに対し若干の新しい工夫をなすごとによりその証明を行うことができました。さらに数値計算により、この分岐がサブクリティカルな分岐ではないかという示唆を得ています。

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非線形特異摂動現象の数学解析と数値計算

Grant number：07454032 1995

日本学術振興会科学研究費助成事業一般研究(B)

岡本久, 谷口雅治, 岩田覚, 竹井義次, 室田一雄, 河合隆裕

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Grant amount：\6100000 （ Direct expense: \6100000 ）

特異摂動理論のNavier-Stokes方程式への応用に関し、岡本久は同方程式の新しい厳密解を発見し,その流体力学的な性質を解明した.そのひとつは,TamadaとDorrepaalによる「斜めにぶつかる淀み点流」の3次元版であり,もうひとつはLerayによる相似解のスキームの解の発見である.
流体の数値シミュレーションに関しては,渦法による数値実験を岡本久が行った.せん断流と渦層の相互作用,あるいは2枚の渦層の相互作用による複雑な運動,特に特異点の発生とその構造に関して,これまで漠然と想像されてきたこととは違うメカニズムが存在することがわかった.
河合隆裕と竹井義次は代数解析の手法を用いて特異摂動理論を研究し,パンルベ方程式との興味る関係を導いた.
室田一雄と岩田覚は数値線形代数理論の研究を通じて本研究に貢献した.特異摂動問題を離散化すると条件数の大きな行列あるいは独特の構造を持つ行列が現れる.こういった行列に関する連立方程式の解法には離散数学の手法が役に立つ.室田は精度の改良や高速化に関する多くの方法を提唱し,その有用性をしめした.
谷口雅治は反応拡散方程式系において特異摂動状態にある定常解の安定性を論じた.これまで知られていた「解の存在証明法」を大幅に改良し,すっきりした理論体系を構築した.また2次元遷移層の安定性を特異摂動理論の枠組で厳密に取り扱うことが可能であることを示し,そのスケーリング指数を決定することに成功した.

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非線形反応拡散方程式による直線状界面の解析

Grant number：06740113 1994

日本学術振興会科学研究費助成事業奨励研究(A)

谷口雅治

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Grant amount：\700000 （ Direct expense: \700000 ）

この研究は放物型で二変数の反応拡散方程式を詳しく調べることで二相問題の境界面現象を解明しようとしたものです。方程式としては化学や生物学に現れる活性因子と抑制因子の混合系を対象とし、界面の形としてはもっとも単純な直線状のものを考えます。
従来の研究で界面が不安定である場合に、界面の厚さをあたえる方程式の微小パラメーターと最不安定波長との関係があたえられてきました。ここで最不安定波長とは、不安定な界面に外部から微小摂動が加えられたときに現れる特徴的な波長をいいます。この研究では方程式への微小パラメーターの入り方を変えることという工夫をすることにより、次の事実が新たな知見として得られました。
1.直線状界面の安定性の判定条件は方程式の微小パラメーターと本質的に独立にあたえられ得ることがわかった。
2.1の工夫のもとで界面の不安定である場合の最不安定波長もこの微小パラメーターと独立にあたえられることがわかった。
すなわち、界面の安定性というものは、界面の厚さ(または、それをあたえる方程式の微小パラメーター)と独立に定義できるものであることをすくなくとも直線状界面に関しては確かめたといえます。これにより、厚さゼロの境界面を扱うには方程式はどうあるべきかについて一つの示唆が得られました。

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Exercises in Calculus IIIa （2021academic year） 1st semester - 月5,月6
Calculus IIIb （2021academic year） Second semester - 月3,月4
Exercises in Calculus IIIb （2021academic year） Second semester - 月5,月6
Exercises in Calculus III （2021academic year） 1st and 2nd semester - 月5,月6
Group Study on Mathematical Sciences （2021academic year） special - その他
Group Study on Mathematical Sciences （2021academic year） special - その他
Seminar in Mathematical Analysis （2021academic year） Year-round - その他
Philosophy and ethics in the science （2021academic year） special - その他
Advanced Analysis II （2021academic year） 3rd and 4th semester - 月7,月8
Advanced Analysis IIa （2021academic year） Third semester - 月7,月8
Advanced Analysis IIb （2021academic year） Fourth semester - 月7,月8
Seminar in Mathematics （2021academic year） special - その他
Seminar in Mathematics （2021academic year） special - その他
Mathematical Theory on Traveling Waves （2021academic year） Late - その他
Discrete Mathematics II （2021academic year） 3rd and 4th semester - 月5,月6
Discrete Mathematics IIa （2021academic year） Third semester - 月5,月6
Discrete Mathematics IIb （2021academic year） Fourth semester - 月5,月6
Advanced practice in scientific presentation （2020academic year） Year-round - その他
Partial Differential Equations （2020academic year） Prophase - 火1,火2
Internship for advanced research （2020academic year） Year-round - その他
Advanced Experiment （2020academic year） special - その他
Practice in Scientific Presentation 1 （2020academic year） Summer concentration - その他
Practice in Scientific Presentation 2 （2020academic year） Late - その他
Global seminar in interdisciplinary science （2020academic year） Year-round - その他
Introduction to Interdisciplinary Science 1 （2020academic year） Prophase - 水1,水2
Introduction to Interdisciplinary Science 2 （2020academic year） Prophase - 水1,水2
Calculus III （2020academic year） 1st and 2nd semester - 月3,月4
Calculus IIIa （2020academic year） 1st semester - 月3,月4
Exercises in Calculus IIIa （2020academic year） 1st semester - 月5,月6
Calculus IIIb （2020academic year） Second semester - 月3,月4
Exercises in Calculus IIIb （2020academic year） Second semester - 月5,月6
Exercises in Calculus III （2020academic year） 1st and 2nd semester - 月5,月6
Advanced lectures in Mathematics （2020academic year） Summer concentration - その他
Seminars in Mathematics and Physics （2020academic year） Other - その他
Seminars in Mathematics and Physics （2020academic year） Year-round - その他
Advanced Study in Mathematics and Physics （2020academic year） Other - その他
Advanced Study in Mathematics and Physics （2020academic year） Year-round - その他
Glance at Mathematical Science C （2020academic year） Third semester - 金5,金6
Group Study on Mathematical Sciences （2020academic year） special - その他
Group Study on Mathematical Sciences （2020academic year） special - その他
Seminar in Mathematical Analysis （2020academic year） Prophase
Seminar in Mathematical Analysis （2020academic year） Year-round - その他
Seminar in Mathematical Analysis （2020academic year） Other - その他
Philosophy and ethics in the science （2020academic year） Year-round - その他
Advanced Analysis II （2020academic year） 3rd and 4th semester - 月7,月8
Advanced Analysis IIa （2020academic year） Third semester - 月7,月8
Advanced Analysis IIb （2020academic year） Fourth semester - 月7,月8
Seminar in Mathematics （2020academic year） special - その他
Seminar in Mathematics （2020academic year） 1st and 2nd semester - その他
Seminar in Mathematics （2020academic year） special - その他
Mathematical Theory on Traveling Waves （2020academic year） Prophase - その他

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Social Activities

振り子の周期について

Role(s)：Lecturer

岡山大学異分野基礎科学研究所岡山大学異分野基礎科学研究所公開講座 2023.7.23

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Type：Lecture

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教員免許状更新講習会

Role(s)：Lecturer

岡山大学 2020.8.19

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Type：Certification seminar

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Academic Activities

Okayama Workshop on Partial Differential Equations

Role(s)：Planning, management, etc.

2023.11.18

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Type：Academic society, research group, etc.

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The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications

Role(s)：Planning, management, etc.

American Institute of Mathematical Sciences 2023.5.31 - 2023.6.4

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Type：Competition, symposium, etc.

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日本数学会応用数学研究奨励賞口頭発表評価委員

Role(s)：Review, evaluation

2022年度応用数学合同研究集会 2022.12.15 - 2022.12.17

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Type：Academic society, research group, etc.

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Okayama Workshop on Partial Differential Equations 主催

Role(s)：Planning, management, etc.

隠居良行，谷口雅治，物部治徳，下條昌彦，瓜屋航太，宮崎隼人 2022.10.29

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Type：Academic society, research group, etc.

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