記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：SIAM PUBLICATIONS  

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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：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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記述言語：英語   出版者・発行元：東京大学  

Bifurcation from equilibrium solutions to reaction diffusion systems is considered in a two-dimensional domain. This solution has an internal transition layer that forms a flat interface. If the length of the interface in the tangential direction is small enough, the equilibrium solution is stable, but it is unstable if the length is larger than some critical value. In this paper, it is shown that bifurcation occurs at this critical length. We construct the bifurcating solutions and discuss their stability. Numerical results suggest that the bifurcation is subcritical.

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記述言語：日本語   出版者・発行元：一般社団法人 日本応用数理学会  

DOI： 10.11540/bjsiam.11.1_82

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記述言語：日本語   出版者・発行元：一般社団法人 日本応用数理学会  

DOI： 10.11540/bjsiam.6.4_334

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開催年月日： 2022年3月14日 - 2022年3月18日

記述言語：英語   会議種別：口頭発表（一般）  

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開催年月日： 2019年12月21日 - 2019年12月22日

記述言語：日本語   会議種別：口頭発表（招待・特別）  

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開催年月日： 2008年3月6日 - 2008年3月7日

記述言語：日本語   会議種別：口頭発表（招待・特別）  

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記述言語：日本語   会議種別：口頭発表（一般）  

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記述言語：日本語   会議種別：口頭発表（一般）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：日本語   会議種別：口頭発表（一般）  

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