研究者詳細 - 若狭　徹

論文 - 若狭　徹

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Stability and bifurcation diagram for a shadow Gierer-Meinhardt system in one spatial dimension (共著) 査読有り国際誌

Kaneko Y., Miyamoto Y., Wakasa T.

Nonlinearity 37 ( 5 ) 2024年05月

記述言語：英語掲載種別：研究論文（学術雑誌）

We are concerned with a Neumann problem of a shadow system of the Gierer-Meinhardt model in an interval I = ( 0 , 1 ) . A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each n ⩾ 1 , two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, n ⩾ 2 , are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.

DOI： 10.1088/1361-6544/ad3596

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Partially overlapping travelling waves in a parabolic-hyperbolic system (共著) 招待有り査読有り国際誌

Bertsch M., Izuhara H., Mimura M., Wakasa T.

Discrete and Continuous Dynamical Systems- Series B ( American Institute of Mathematical Sciences ) 28 ( 12 ) 5934 - 5966 2023年12月

記述言語：英語掲載種別：研究論文（学術雑誌）

We study the existence of travelling wave solutions of a one-dimensional parabolic-hyperbolic system for u(x, t) and v(x, t), which arises as a model for contact inhibition of cell growth. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and strongly parameter-dependent. In the present paper we consider a parameter regime where the minimal wave speed is positive. We show that there exists a branch of travelling wave solutions for wave speeds which are larger than the minimal one. But the main result is more surprising: for certain values of the parameters the travelling wave with minimal wave speed is not segregated (a solution is called segregated if the product uv vanishes almost everywhere) and in that case there exists a second branch of “partially overlapping” travelling wave solutions for speeds between the minimal one and that of the (unique) segregated travelling wave.

DOI： 10.3934/dcdsb.2023076

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Asymptotic formulas of the eigenvalues for the linearization of the scalar field equation (共著) 査読有り国際誌

Miyamoto Y., Takemura H., Wakasa T.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( Cambridge University Press ) 155 ( 1 ) 307 - 344 2023年09月

記述言語：英語掲載種別：研究論文（学術雑誌）

We establish asymptotic formulas for all the eigenvalues of the linearization problem of the Neumann problem for the scalar field equation in a finite interval {equation presented} In the previous paper of the third author [T. Wakasa and S. Yotsutani, J. Differ. Equ. 258 (2015), 3960-4006] asymptotic formulas for the Allen-Cahn case {equation presented} were established. In this paper, we apply the method developed in the previous paper to our case. We show that all the eigenvalues can be classified into three categories, i.e., near -3 eigenvalues, near 0 eigenvalues and the other eigenvalues. We see that the number of the near -3 eigenvalues (resp. the near 0 eigenvalues) is equal to the number of the interior and boundary peaks (resp. the interior peaks) of a solution for the nonlinear problem. The main technical tools are various asymptotic formulas for complete elliptic integrals.

DOI： 10.1017/prm.2023.95

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Asymptotic formulas of the eigenvalues for the linearization of a one-dimensional sinh-Poisson equation (共著) 査読有り国際誌

Aizawa S., Miyamoto Y., Wakasa T.

Journal of Elliptic and Parabolic Equations ( Birkhauser ) 9 ( 2 ) 1043 - 1070 2023年07月

記述言語：英語掲載種別：研究論文（学術雑誌）

We are concerned with a Neumann problem of a one-dimensional sinh-Poisson equation {u′′+λsinhu=0for0<x<1,u′(0)=u′(1)=0, where λ> 0 is a parameter. A complete bifurcation diagram of this problem is obtained. We also consider the linearized eigenvalue problem at every nontrivial solution u. We derive exact expressions of all the eigenvalues and eigenfunctions, using Jacobi elliptic functions and complete elliptic integrals. Then, we also derive asymptotic formulas of eigenvalues as λ→ 0 . Exact eigenvalues and eigenfunctions for a Dirichlet problem are presented without proof. The main technical tool is an ODE technique.

DOI： 10.1007/s41808-023-00233-9

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The structure of stationary solutions to a micro-electro mechanical system with fringing field (共著) 査読有り国際誌

Guo J.S., Huang B.C., Wakasa T., Wang C.J., Yu C.Y.

Journal of Differential Equations 269 ( 9 ) 7676 - 7704 2020年10月

記述言語：英語掲載種別：研究論文（学術雑誌）

We study the structure of stationary solutions of a micro-electro mechanical system with fringing field. It is known that there is a positive critical value such that no stationary solutions exist for the applied voltage beyond this critical value, at least two stationary solutions exist for the applied voltage below this critical value and there is a unique stationary solution at this critical value. In this paper, we verify that there are exactly two solutions below this critical value analytically. Moreover, the stability of the smaller stationary solutions is derived.

DOI： 10.1016/j.jde.2020.05.041

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A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic Fisher KPP equation (共著) 査読有り国際誌

Bertsch M., Hilhorst D., Izuhara H., Mimura M., Wakasa T.

Discrete and Continuous Dynamical Systems Series A 40 ( 6 ) 3117 - 3142 2020年01月

記述言語：英語掲載種別：研究論文（学術雑誌）

We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

DOI： 10.3934/dcds.2019226

Kyutacar

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Standing and travelling waves in a parabolic-hyperbolic system (共著) 査読有り

Bertsch M., Izuhara H., Mimura M., Wakasa T.

Discrete and Continuous Dynamical Systems- Series A 39 ( 10 ) 5603 - 5635 2019年10月

記述言語：英語掲載種別：研究論文（学術雑誌）

© 2019 American Institute of Mathematical Sciences. All rights reserved. We consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product uv vanishes.

DOI： 10.3934/dcds.2019246

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Exact eigenvalues and eigenfunctions for a one-dimensional Gel’fand problem (共著) 査読有り

Miyamoto Y., Wakasa T.

Journal of Mathematical Physics 60 ( 2 ) 2019年02月

記述言語：英語掲載種別：研究論文（学術雑誌）

© 2019 Author(s). It is known that every positive solution of a one-dimensional Gel’fand problem can be written explicitly. In this paper, we give exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We study asymptotic behaviors of eigenvalues and eigenfunctions as the L ∞ -norm of the solution goes to infinity. We also study the problem u + λe −u = 0 and the associated linearized problem.

DOI： 10.1063/1.5021825

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Limiting classification on linearized eigenvalue problems for 1-dimensional Allen–Cahn equation II—asymptotic profiles of eigenfunctions (共著) 査読有り国際誌

Tohru Wakasa, Shoji Yotsutani

Journal of Differential Equations 261 ( 10 ) 5465 - 5498 2016年09月

Limiting classification on linearized eigenvalue problems for 1-dimensional Allen–Cahn equation I—asymptotic formulas of eigenvalues (共著) 査読有り

Tohru Wakasa, Shoji Yotsutani

Journal of Differential Equations 258 ( 11 ) 3960 - 4006 2015年06月

Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition (共著) 査読有り国際誌

M.Bertsch, D.Hilhorst, H.Izuhara, M.Mimura, T.Wakasa

European Journal of Applied Mathematics 26 ( 3 ) 297 - 323 2015年06月

The Heun differential equation and the Gauss differential equation related to quantum walks (共著) 査読有り

Norio Konno, Takuya Machida, Tohru Wakasa

Yokohama Mathematical Journal 58 53 - 64 2013年04月

Modeling contact inhibition of growth: traveling waves (共著) 招待有り査読有り国際誌

Michiel Bertsch, Masayasu Mimura, Tohru Wakasa

Networks and Heterogeneous Media 8 ( 1 ) 131 - 147 2013年04月

Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems (共著) 査読有り国際誌

Tohru Wakasa, Shoji Yotsutani

Communications in Pure and Applied Analysis 9 ( 2 ) 539 - 561 2010年03月

Representation and asymptotic formulas for 1-dimensional linearized eigenvalue problems with Dirichlet boundary condition 査読有り

Wakasa T.

Nonlinear Analysis, Theory, Methods and Applications 71 ( 12 ) 2009年12月

担当区分：筆頭著者記述言語：英語掲載種別：研究論文（国際会議プロシーディングス）

In this article nontrivial solutions to a nonlinear Dirichlet boundary value problem with a small parameter and the corresponding linearized eigenvalue problems are investigated. In a typical case of nonlinearity, asymptotic formulas of all eigenvalues and asymptotic profiles of all eigenfunctions are proved. Proofs are based on the method of representation equation and asymptotic results in the corresponding Neumann problem obtained in the recent papers by the author and S. Yotsutani. © 2009 Elsevier Ltd. All rights reserved.

DOI： 10.1016/j.na.2009.06.030

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Representation formulas for some 1-dimensional linearized eigenvalue problems (共著) 査読有り国際誌

Tohru Wakasa, Shoji Yotsutani

Communications in Pure and Applied Analysis 7 ( 4 ) 745 - 763 2008年07月

Generation of interfaces for Lotka-Volterra competition-diffusion system with large interaction rates (共著) 査読有り国際誌

Nakashima K., Wakasa T.

Journal of Differential Equations 235 ( 2 ) 586 - 608 2007年04月

担当区分：責任著者記述言語：英語掲載種別：研究論文（学術雑誌）

We discuss the generation and motion of interfaces for Lotka-Volterra competition-diffusion system with large interaction. An asymptotic analysis of solutions shows that the two competing species are segregated and an interface appears on the common boundary of their habitats. The motion of the interface is governed by a free boundary problem. In this paper we establish a mathematical theory for the formation of interfaces (at the initial stage) by using an upper and lower solutions method. In addition, combining our results and a known result for the motion of interfaces (after the initial stage), we obtain some information on the generation and motion of interfaces for given almost any smooth initial data. © 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jde.2007.01.002

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Exact eigenvalues and eigenfunctions associated with linearization for Chafee-Infante problem 査読有り

Ｔｏｈｒｕ　Ｗａｋａｓａ

Funkcialaj Ekvacioj ４９ ( ２ ) ３２１ - ３３６ 2006年08月

Eigenvalue problems associated with 1-dimensional scalar field equations (Succession and Innovation of Studies on ODEs in Real Domains)

若狭徹

数理解析研究所講究録 ( 京都大学数理解析研究所 ) 2080 94 - 105 2018年08月

線形化固有値問題と大域的分岐問題への応用招待有り

若狭　徹

2016年度応用数学勉強会レクチャーノート 2017年10月

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