Details of a Researcher - YAMADA, Yoshio

写真a

YAMADA, Yoshio

Affiliation

Faculty of Science and Engineering

Job title

Professor Emeritus

Degree

Doctor of Science ( 1984.03 名古屋大学 )

Homepage URL

http://www.f.waseda.jp/yamada/

Research Experience

2019.04

-

　

Waseda University Faculty of Science and Engineering Professor Emeritus
2006.04

-

2019.03

Professor, Waseda University, Faculty of Science and Engineering
1992.04

-

2006.03

Professor, Waseda University, School of Science and Engineering
1987.04

-

1992.03

Assistant Professor, Waseda University, School of Science and - Engineering
1977.02

-

1987.03

Research Associate, Nagoya University, Faculty of Science

Professional Memberships

　

　

　

Mathematical Society of Japan

Research Areas

Basic analysis / Mathematical analysis

Research Interests

Basic Analysis,Larger-area Analysis

Papers

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions III: General case

Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada

Discrete and Continuous Dynamical Systems, Series S 17 ( 2 ) 742 - 761 2024.02 [Refereed] [International journal]

DOI
A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic profiles of solutions and radial terrace solution

Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada

Journal de Mathematiques Pures et Appliquees 178 1 - 48 2023.08 [Refereed]

DOI
A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior

Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada

Discrete and Continuous Dynamical Systems 42 ( 6 ) 2719 - 2719 2022.06 [Refereed] [International journal]

　View Summary

<p lang="fr"><p style='text-indent:20px;'>We study a free boundary problem of a reaction-diffusion equation <inline-formula><tex-math id="M1">\begin{document}$ u_t = \Delta u+f(u) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ t&gt;0,\ |x|&lt;h(t) $\end{document}</tex-math></inline-formula> under a radially symmetric environment in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The reaction term <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive bistable nonlinearity, which satisfies <inline-formula><tex-math id="M5">\begin{document}$ f(0) = 0 $\end{document}</tex-math></inline-formula> and allows two positive stable equilibrium states and a positive unstable equilibrium state. The problem models the spread of a biological species, where the free boundary represents the spreading front and is governed by a one-phase Stefan condition. We show multiple spreading phenomena in high space dimensions. More precisely the asymptotic behaviors of solutions are classified into four cases: big spreading, small spreading, transition and vanishing, and sufficient conditions for each dynamical behavior are also given. We determine the spreading speed of the spherical surface <inline-formula><tex-math id="M6">\begin{document}$ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $\end{document}</tex-math></inline-formula>, which expands to infinity as <inline-formula><tex-math id="M7">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>, even when the corresponding semi-wave problem does not admit solutions.</p></p>

DOI
Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity

Maho Endo, Yuki Kaneko, Yoshio Yamada

Discrete & Continuous Dynamical Systems - A 40 ( 6 ) 3375 - 3394 2020 [Refereed]

DOI
Asymptotic Profiles of Solutions and Propagating Terrace for a Free Boundary Problem of Nonlinear Diffusion Equation with Positive Bistable Nonlinearity

Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada

SIAM Journal on Mathematical Analysis 52 ( 1 ) 65 - 103 2020.01 [Refereed]

DOI
Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions

Yuki Kaneko, Yoshio Yamada

Journal of Mathematical Analysis and Applications 465 ( 2 ) 1159 - 1175 2018.09 [Refereed]

　View Summary

We discuss a free boundary problem for a reaction–diffusion equation with Dirichlet boundary conditions on both fixed and free boundaries of a one-dimensional interval. The problem was proposed by Du and Lin (2010) to model the spreading of an invasive or new species by putting Neumann boundary condition on the fixed boundary. Asymptotic properties of spreading solutions for such problems have been investigated in detail by Du and Lou (2015) and Du, Matsuzawa and Zhou (2014). The authors (2011) studied a free boundary problem with Dirichlet boundary condition. In this paper we will derive sharp asymptotic properties of spreading solutions to the free boundary problem in the Dirichlet case under general conditions on f. It will be shown that the spreading speed is asymptotically constant and determined by a semi-wave problem and that the solution converges to a semi-wave near the spreading front as t→∞ provided that the semi-wave problem has a unique solution.

DOI
Nonlinear diffusion equations with cross-diffusion: Reaction-diffusion equations appearing in mathematical ecology

Yoshio Yamada

Sugaku Expositions 2016.12
Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity

Yusuke Kawai, Yoshio Yamada

JOURNAL OF DIFFERENTIAL EQUATIONS 261 ( 1 ) 538 - 572 2016.07 [Refereed]

　View Summary

This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem. (C) 2016 Elsevier Inc. All rights reserved.

DOI
GLOBAL-IN-TIME BEHAVIOR OF THE SOLUTION TO A GIERER-MEINHARDT SYSTEM

Georgia Karali, Takashi Suzuki, Yoshio Yamada

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 33 ( 7 ) 2885 - 2900 2013.07 [Refereed]

　View Summary

Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, tau = s+1/p-1. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.

DOI
ON LIMIT SYSTEMS FOR SOME POPULATION MODELS WITH CROSS-DIFFUSION

Kousuke Kuto, Yoshio Yamada

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 17 ( 8 ) 2745 - 2769 2012.11 [Refereed]

　View Summary

This paper deals with the following reaction-diffusion system
(SP) {Delta[(1 + alpha nu)u] + u(a - u - cv) = 0
Delta[(1 + beta u)v] + v(b - du - v) = 0
in a bounded domain of R-N with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients alpha and beta. For this purpose, our strategy is to study limiting behavior of positive solutions when alpha or beta goes to infinity and derive the corresponding limit systems. We will obtain a priori estimates of u and v independently of beta (resp. alpha) with small alpha >= 0 (resp. beta >=> 0) in case 1 <= N <= 3 under Neumann boundary conditions, while we will obtain a priori estimates of u and v independently of alpha and beta in case 1 <= N <= 5 under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When alpha = 0 and beta -> infinity, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.

DOI
A free boundary problem for a reaction-diffusion equation appearing in ecology

Yuki Kaneko, Yoshio Yamada

Advances in Mathematical Sciences and Applications 21 ( 2 ) 467 - 492 2011.12 [Refereed]
Coexistence problem for a prey-predator model with density-dependent diffusion

Kousuke Kuto, Yoshio Yamada

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 71 ( 12 ) E2223 - E2232 2009.12 [Refereed]

　View Summary

We study a prey-predator model with nonlinear diffusions. In a case when the spatial dimension is less than 5, a universal bound for coexistence steady-states is found. By using the bound and the bifurcation theory, we obtain the bounded continuum of coexistence steady-states. (C) 2009 Elsevier Ltd. All rights reserved.

DOI
ON THE LONG-TIME LIMIT OF POSITIVE SOLUTIONS TO THE DEGENERATE LOGISTIC EQUATION

Yihong Du, Yoshio Yamada

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 25 ( 1 ) 123 - 132 2009.09 [Refereed] [International journal]

　View Summary

We study the long-time behavior of positive solutions to the problem
u(t) - Delta u = au - b(x)u(p) in (0, infinity) x Omega, Bu = 0 on (0, infinity) x partial derivative Omega,
where a is a real parameter, b >= 0 is in C(mu)((Omega) over bar) and p > 1 is a constant, Omega is a C(2+mu) bounded domain in R(N) (N >= 2), the boundary operator B is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that (Omega) over bar (0) := {x is an element of Omega : b(x) = 0} has non-empty interior, is connected, has smooth boundary and is contained in Omega, it is shown in [8] that when a >= lambda(D)(1) (Omega(0)), for any fixed x is an element of (Omega) over bar (0),
lim(t ->infinity) u(t, x) = 1, and for any fixed x is an element of (Omega) over bar\(Omega) over bar (0), (lim) over bar (t ->infinity) u(t, x) <= (U) over bar (a) (x), (lim) under bar (t ->infinity) u(t, x) >= (U) over bar (a) (x), where (U) under bar (a) and (U) under bar (a) denote respectively the minimal and maximal positive solutions of the boundary blow-up problem
-Delta u = au - b(x)u(p) in Omega\(Omega) over bar (0), Bu = 0 on partial derivative Omega, u = infinity on partial derivative Omega(0).
The main purpose of this paper is to show that, under the above assumptions,
(t ->infinity)lim u(t, x) = (U) under bar (a)(x), for all x is an element of (Omega) over bar\(Omega) over bar (0).
This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

DOI
Limiting characterization of stationary solutions of a prey-predator model with nonlinear diffusion of fractional type

Kousuke Kuto, Yoshio Yamada

Differential and Integral Equations Vol. 22 ( 7 - 8 ) 725 - 752 2009.09 [Refereed]
Chapter 6 Positive solutions for Lotka-Volterra systems with cross-diffusion

Yoshio Yamada

Handbook of Differential Equations: Stationary Partial Differential Equations 6 ( Chap. 6 ) 411 - 501 2008 [Refereed]

　View Summary

This article is concerned with reaction-diffusion systems with nonlinear diffusion effects, which describe competition models and prey-predator models of Lotka-Volterra type in population biology. The system consists of two nonlinear diffusion equations where two unknown functions denote population densities of two interacting species. The main purpose is to discuss the existence and nonexistence of positive steady state solutions to such systems. Here a positive solution corresponds to a coexistence state in population models. We will derive a priori estimates of positive solutions by maximum principle for elliptic equations and employ the degree theory on a positive cone to show the existence of a positive solution. The existence results can be reconsidered from the view-point of bifurcation theory. We will give some information on the direction of bifurcation of positive solutions and their stability properties in terms of some biological coefficients. Moreover, we will also study the existence of multiple positive solutions for a certain class of prey-predator systems with nonlinear diffusion by making one of cross-diffusion coefficients sufficiently large. © 2008 Elsevier B.V. All rights reserved.

DOI
Multiple coexistence states for a prey-predator system with cross-diffusion

K Kuto, Y Yamada

JOURNAL OF DIFFERENTIAL EQUATIONS 197 ( 2 ) 315 - 348 2004.03 [Refereed]

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We study the multiple existence of positive solutions for the following strongly coupled elliptic system:
Delta[(1 + alphaupsilon)u] + u(a - u - cupsilon) = 0 in Omega,
Delta[(1 + betau)upsilon] + upsilon(b + du - upsilon) = 0 in Omega,
u = v = 0 on Omega,
where alpha, beta, a, b, c, d are positive constants and Omega is a bounded domain in R-N. This is the steadystate problem associated with a prey-predator model with cross-diffusion effects and u (resp. upsilon) denotes the population density of preys (resp. predators). In particular, the presence of beta represents the tendency of predators to move away from a large group of preys. Assuming that a is small and that beta is large, we show that the system admits a branch of positive solutions, which is S or D shaped with respect to a bifurcation parameter. So that the system has two or three positive solutions for suitable range of parameters. Our method of analysis uses the idea developed by Du-Lou (J. Differential Equations 144 (1998) 390) and is based on the bifurcation theory and the Lyapunov-Schmidt procedure. (C) 2003 Elsevier Inc. All rights reserved.

DOI
Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion

Y. S. Choi, Roger Lui, Yoshio Yamada

Discrete & Continuous Dynamical Systems - A 10 ( 3 ) 719 - 730 2004 [Refereed]

DOI
Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with-weak cross-diffusion

YS Choi, R Lui, Y Yamada

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 9 ( 5 ) 1193 - 1200 2003.09 [Refereed]

　View Summary

The Shigesada-Kawasaki-Teramoto model is a generalization of the classical Lotka-Volterra competition model for which the competing species undergo both diffusion, self-diffusion and cross-diffusion. Very few mathematical results are known for this model, especially in higher space dimensions. In this paper, we shall prove global existence of strong solutions in any space dimension for this model when the cross-diffusion coefficient in the first species is sufficiently small and when there is no self-diffusion or cross-diffusion in the second species.

DOI
Solvability and smoothing effect for semilinear parabolic equations

Hiroki Hoshino, Yoshio Yamada

Funkcialaj Ekvaioj 34 ( 3 ) 475 - 494 1991 [Refereed] [International journal]

CiNii
Stability of Steady States for Prey–Predator Diffusion Equations with Homogeneous Dirichlet Conditions

Yoshio Yamada

SIAM Journal on Mathematical Analysis 21 ( 2 ) 327 - 345 1990.03 [Refereed]

DOI CiNii
A free boundary problem in ecology

Masayasu Mimura, Yoshio Yamada, Shoji Yotsutani

Japan Journal of Applied Mathematics 2 ( 1 ) 151 - 186 1985.06 [Refereed]

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This article is concerned with a free boundary problem for semilinear parabolic equations, which describes the habitat segregation phenomenon in population ecology. The main purpose is to show the global existence, uniqueness, regularity and asymptotic behavior of solutions for the problem. The asymptotic stability or instability of each solution is completely determined using the comparison theorem. © 1985 JJAM Publishing Committee.

DOI
Asymptotic stability for some systems of semilinear Volterra diffusion equations

Yoshio Yamada

Journal of Differential Equations 52 ( 3 ) 295 - 326 1984 [Refereed]

DOI
On evolution equations generated by subdifferential operators

Yoshio Yamada

Journal of the Faculty of Science, the University of Tokyo 23 ( 3 ) 491 - 515 1976 [Refereed] [International journal]

DOI
Global existence of weak solutions to forest kinematic model with nonlinear degenerate diffusion

Mitsuki Kobayashi, Yoshio Yamada

Advances in Mathematical Sciences and Applications 29 ( 1 ) 187 - 209 2020.11 [Refereed] [International journal]
Asymptotic propertied of a free boundary problem for a reaction-diffusion equation with multi-stable nonlinearity

Yoshio Yamada

52 65 - 89 2020.11 [Refereed] [International journal]

DOI
Lotka-Volterra Systems with Periodic Orbits

Manami Kobayashi, Takashi Suzuki, Yoshio Yamada

Funkcialaj Ekvacioj 62 ( 1 ) 129 - 155 2019.05 [Refereed]

DOI
Dissipative reaction diffusion systems with quadratic growth

Michel Pierre, Takashi Suzuki, Yoshio Yamada

Indiana University Mathematics Journal 68 ( 1 ) 291 - 322 2019 [Refereed] [International journal]

DOI
Effect of cross-diffusion in the diffusion prey-predator model with a protection zone II

Shanbing Li, Yoshio Yamada

Journal of Mathematical Analysis and Applications 461 ( 1 ) 971 - 992 2018.05 [Refereed]

　View Summary

In the current paper, we continue the mathematical analysis studied in Li and Wu (2017) [15] and Oeda (2011) [22], and further study the effect of cross-diffusion for the predator on the stationary problem. The existence of positive solutions is first established by the bifurcation theory. We next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species tends to infinity. Moreover, as the prevention coefficient tends to infinity, we obtain two shadow systems and give the complete limiting characterization of positive solutions.

DOI
Analysis of forest kinematic model with nonlinear degenerate diffusion

Rei Yamamoto, Yoshio Yamada

Advances in Mathematical Sciences and Applications 25 ( 1 ) 307 - 320 2016.09 [Refereed] [International journal]
Asymptotic Behavior of Solutions for Semilinear Volterra Diffusion Equations with Spatial Inhomogeneity and Advection

Yusuke Yoshida, Yoshio Yamada

TOKYO JOURNAL OF MATHEMATICS 39 ( 1 ) 271 - 292 2016.06 [Refereed]

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This paper is concerned with semilinear Volterra diffusion equations with spatial inhomogeneity and advection. We intend to study the effects of interaction among diffusion, advection and Volterra integral under spatially inhomogeneous environments. Since the existence and uniqueness result of global-in-time solutions can be proved in the standard manner, our main interest is to study their asymptotic behavior as t -> infinity. For this purpose, we study the related stationary problem by the monotone method and establish some sufficient conditions on the existence of a unique positive solution. Its global attractivity is also studied with use of a suitable Lyapunov functional.

DOI
On logistic diffusion equations with nonlocal interaction terms

Yoshio Yamada

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 118 51 - 62 2015.05 [Refereed]

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This paper is concerned with logistic diffusion equations with nonlocal interaction terms appearing in population biology. We intend to study effects of nonlocal terms and discuss the similarity and difference between local problems and nonlocal problems. Mainly, the stationary problem is investigated for a certain class of nonlocal terms. A constructive approach is proposed to look for positive stationary solutions and the unique existence of such a positive solution is established. The analysis of stationary solutions depends on the spectrum for the linearized operator around the stationary solution. However, the linearized operator contains a nonlocal term which makes the spectral analysis delicate and difficult. Putting some additional assumptions we will derive the asymptotic stability of the unique positive solution and, furthermore, its global attractivity. Finally, it will be seen that some arguments are valid to show the unique existence of a positive stationary solution for a considerably general class of nonlocal terms. (C) 2015 Elsevier Ltd. All rights reserved.

DOI
Global-In-Time Behavior of Lotka-Volterra System with Diffusion: Skew-Symmetric Case

Takashi Suzuki, Yoshio Yamada

INDIANA UNIVERSITY MATHEMATICS JOURNAL 64 ( 1 ) 181 - 216 2015 [Refereed]

　View Summary

We study the global-in-time behavior of the Lotka-Volterra system with diffusion. In the first category, the interaction matrix is skew-symmetric and the linear terms are non-increasing. There, the solution exists globally in time with compact orbit, provided that n <= 2, where n denotes the space dimension. Under the presence of entropy, its omega-limit set is composed of a spatially homogeneous orbit. Furthermore, any spatially homogeneous solution is periodic in time, provided with constant entropy. In the second category, the interaction matrix exhibits a dissipative profile. There, the solution exists globally in time with compact orbit if n <= 3. Its omega-limit set, furthermore, is contained in spatially homogeneous stationary states. In particular, no periodic-in-time solution is admitted.

DOI
Remarks on Spreading and Vanishing for Free Boundary Problems of Some Reaction-Diffusion Equations

Yuki Kaneko, Kazuhiro Oeda, Yoshio Yamada

FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA 57 ( 3 ) 449 - 465 2014.12 [Refereed]

　View Summary

We discuss a free boundary problem for a diffusion equation in a one-dimensional interval which models the spreading of invasive or new species. Moreover, the free boundary represents a spreading front of the species and its dynamical behavior is determined by a Stefan-like condition. This problem has been proposed by Du and Lin (2010) and, recently, Kaneko and Yamada have studied a free boundary problem for a general reaction-diffusion equation under Dirichlet boundary conditions. The main purpose of this paper is to define "spreading" and "vanishing" of species for a free boundary problem with general nonlinearity and study the underlying principle to determine the spreading or vanishing behavior as time tends to infinity. It will be proved that vanishing occurs if and only if the free boundary stays in a bounded interval, and that, when vanishing occurs, the population decreases exponentially to zero in large time.

DOI CiNii
Nonlinear diffusion equations with cross-diffusion -- Reaction-diffusion equations appearing in mathematical ecology-- (in Japanese)

Yoshio Yamada

"Sugaku," Mathematical Society of Japan 64 ( 4 ) 384 - 406 2012.10 [Refereed] [Invited]

DOI CiNii
Transient and asymptotic dynamics of a prey-predator system with diffusion

Evangelos Latos, Takashi Suzuki, Yoshio Yamada

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 35 ( 9 ) 1101 - 1109 2012.06 [Refereed]

　View Summary

In this paper, we study a preypredator system associated with the classical LotkaVolterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t -> infinity. Next, we take the shadow system to approximate the original system as D -> infinity. The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D -> infinity. Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any delta > 0 admits D-0 > 0 such that if T, the time period of the ODE, satisfies (T)over-cap > delta, then the solution to the original system with D >= D-0 cannot approach the stationary state. Copyright (C) 2012 John Wiley & Sons, Ltd.

DOI
Positive solutions for Lotka-Volterra competition systems with large cross-diffusion

Kousuke Kuto, Yoshio Yamada

Applicable Analysis 89 ( 7 ) 1037 - 1066 2010.07 [Refereed]

　View Summary

This paper discusses the stationary problem for the Lotka-Volterra competition systems with cross-diffusion under homogeneous Dirichlet boundary conditions. Although some sufficient conditions for the existence of positive solutions are known, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we study the effects of large cross-diffusion on the structure of positive solutions and focus on the limiting behaviour of positive solutions by letting one of the cross-diffusion coefficients to infinity. Especially, it will be shown that positive solutions of the competition system converge to a positive solution of a suitable limiting system. We will also derive some satisfactory results on positive solutions for this limiting system. These results give us important information on the structure of positive solutions for the competition system when one of the cross-diffusion coefficients is sufficiently large. © 2010 Taylor &amp
Francis.

DOI
Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion

Yoshio Yamada

Recent Progress on ReactionーDiffusion Systems and Viscosity Solutions, edited by Yihong Du, Hitoshi Ishii and Wei-Yueh Lin 282 - 299 2009.04 [Refereed] [International journal]
Transition layers and spikes for a reaction-diffusion equation with bistable

Michio Urano, Kimie Nakashima, Yoshio Yamada

Discrete and Continuous Dynamical Systems Suppl. Vol. 868 - 877 2005.11
Coexistence states for a prey-predator model with cross-diffusion

Kousuke Kuto, Yoshio Yamada

Discrete and Continuous Dynamical Systems Suppl. Vol. 2005 536 - 545 2005.11
Steady-states with transition layers and spikes for a bistable reaction-diffusion equation

Michio Urano, Kimie Nakashima, Yoshio Yamada

Mathematical Approach to Nonlinear Phenomena; Modeling, Analysis andGakuto International Series, Math. Sci. Appl Simulations, Vol. 23 267 - 279 2005.11
Transition layers and spikes for a bistable reaction-diffusion equation

Michio Urano, Kimie Nakashima, Yoshio Yamada

Advances in Mathematical Sciences and Applications 15 ( 2 ) 683 - 707 2005 [Refereed]
Multiple existence and stability of steady-states for a prey-predator system with cross diffusion

Kousuke Kuto, Yoshio Yamada

Banach Center Publications 66 199 - 210 2004.12 [Refereed]
One-phase Stefan problems for sublinear heat equations: Asymptotic behavior of solutions

T. Aiki, H. Imai, N. Ishimura, Y. Yamada

Communications in Applied Analysus /Dynamic Publishers 8 1 - 15 2004.01
Well-posedness of one-phase Stefan problems for sublinear heat equations

Toyohiko Aiki, Hitoshi Imai, Naoyuki Ishimura, Yoshio Yamada

Nonlinear Analysis, Theory, Methods and Applications 51 ( 4 ) 587 - 606 2002.11 [Refereed]

　View Summary

Well-posedness of one-phase Stefan problems for sublinear heat equations was studied. Nonnegative solutions were considered because the uniqueness theorem held only for them. The global existence and uniqueness of solutions of Stefan problems were established. Results indicated that the large-time behavior of solutions of the problem was similar to that of the initial boundary value problem.

DOI
Multiple existence of positive solutions of competing species equations with diffusion and large interactions

Takefumi Hirose, Yoshio Yamada

Advances in Mathematical Sciences and Applications/学校図書 12 ( 1 ) 435 - 453 2002.06 [Refereed]
Positive solutions for Lotka-Volterra competition system with diffusion

Y. Yamada

Nonlinear Analysis, Theory, Methods and Applications 47 ( 9 ) 6085 - 6096 2001.08 [Refereed]

　View Summary

Positive stationary solutions of Lotka-Volterra competition system with diffusion were discussed. Sufficient conditions on the multiple existence of positive solutions were presented for two cases. The two cases were analyzed using the degree or local bifurcation theory and the theory of Dancer and Du.

DOI
Some remarks on global solutions to quasilinear parabolic system with cross- diffusion

Tatsuo Ichikawa, Yoshio Yamada

Funkcialaj Ekvakioj/日本数学会 43 ( 2 ) 285 - 301 2000.08

CiNii
Coexistence states for Lotka-Volterra systems with cross-diffusion

Yoshio Yamada

Fields Institute Communications Vol. 23 551 - 564 2000
Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian

Shingo Takeuchi, Yoshio Yamada

Nonlinear Analysis, Theory, Methods and Applications 42 ( 1 ) 41 - 61 2000 [Refereed]

　View Summary

A reaction-diffusion equation referred to as P, and a stationary problem associated with P (denoted here as SP) are studied in the general case p&gt
2, q≤2 and r&gt
0. SP in particular results from the p-Laplacian. The kind of results produced by the interaction between the p-Laplacian and the reaction term are highlighted.

DOI
Global attractivity of coexistence states for a certain class of reaction diffusion systems with 3x3 cooperative matrices

Atsushi Yoshida, Yoshio Yamada

Advances in Mathematical Sciences and Applications/学校図書 9 ( 2 ) 695 - 706 1999.12
Coexistence states for some population models with nonlinear cross-diffusion

Yoshio Yamada

Forma/KTK Scientific Publishers 12 ( 2 ) 153 - 166 1997.06 [Refereed]

CiNii
Positive steady states for prey-predator models with cross-diffusion

Kimie Nakashima, Yoshio Yamada

Advances in Differential Equations/Khayyam Publishing Company 1 ( 6 ) 1099 - 1122 1996.11

J-GLOBAL
On positive steady-states for some reaction-diffusion system

Kimie Nakashima, Yoshio Yamada

Advances in Mathematical Sciences and Applications 6 ( 1 ) 279 - 289 1996.05 [Refereed]
A certain class of reaction-diffusion systems with feedback effects

Yoshio Yamada

Advances in Mathematical Sciences and Applications 5 ( 2 ) 477 - 485 1995.10 [Refereed]
On the convergence rates for solutions of some chemical interfacial reaction problems

M Iida, Y Yamada, S Yotsutani

OSAKA JOURNAL OF MATHEMATICS 32 ( 2 ) 373 - 396 1995.06 [Refereed]

CiNii
Global solutions for quasilinear parabolic systems with cross-diffusion effects

Yoshio Yamada

Nonlinear Analysis: Theory, Methods & Applications 24 ( 9 ) 1395 - 1412 1995.05 [Refereed] [International journal]

DOI
ASYMPTOTIC-BEHAVIOR OF GLOBAL-SOLUTIONS FOR SOME REACTION-DIFFUSION SYSTEMS

H HOSHINO, Y YAMADA

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 23 ( 5 ) 639 - 650 1994.09 [Refereed]

DOI
Exponential convergence of solutions for a mathematical model on chemical interfacial reactions

M. Iida, Y. Yamada, S. Yotsutani, E. Yanagida

Advances in Matheamtical Sciences and Applications 3 ( 1 ) 335 - 352 1994
ASYMPTOTIC-BEHAVIOR OF SOLUTIONS FOR SEMILINEAR VOLTERRA DIFFUSION-EQUATIONS

Y YAMADA

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 21 ( 3 ) 227 - 239 1993.08

DOI
Convergence of solutions of a chemical interfacial reaction model(共著)

M. Iida, Y.Yamada, E.Yanagida, S.Yotsutani

Funkcialaj Ekvacioj 36 ( 2 ) 311 - 328 1993 [Refereed] [International journal]
ASYMPTOTIC-BEHAVIOR OF SOLUTIONS FOR A MATHEMATICAL-MODEL ON CHEMICAL INTERFACIAL REACTIONS

M IIDA, Y YAMADA, S YOTSUTANI

OSAKA JOURNAL OF MATHEMATICS 29 ( 3 ) 483 - 495 1992.09 [Refereed] [International journal]
Global solutions for the heat convection equations in an exterior domain

Toshiaki Hishida, Yoshio Yamada

Tokyo Journal of Mathematics 15 ( 1 ) 135 - 151 1992 [Refereed] [International journal]

　View Summary

A nonstationary problem of convection in the exterior domain to a heated sphere is studied. In the Boussinesq approximation, convection phenomena are governed by the system of the Navier-Stokes and heat equations. We find sufficient conditions on boundary and initial data to ensure the global existence of Lp-solutions for this system. © 1992, International Academic Printing Co. Ltd., All rights reserved.

DOI
On some nonlinear wave equations Ⅱ: global existence and energy decay of solutions

Masanori Hosoya, Yoshio Yamada

J. Fac. Sci. Univ. Tokyo Sec.IA 38 ( 2 ) 239 - 250 1991 [Refereed] [International journal]

DOI CiNii
On some nonlinear wave equations Ⅰ: local existence and regularity of solutions

Masanor Hosoya, Yoshio Yamada

J. Fac. Sci. Univ. Tokyo Sec.IA 38 225 - 238 1991
A mathematical model on chemical interfacial reactions

Yoshio Yamada, Shoji Yotsutani

Japan Journal of Applied Mathematics 7 ( 3 ) 369 - 398 1990.10 [Refereed]

　View Summary

This paper discusses a sort of parabolic system with nonlinear boundary conditions, which comes from the chemical interfacial models. The results obtained here are the uniqueness and the existence of the global solutions. © 1990 JJAM Publishing Committee.

DOI
On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms

Kenji Nishihara, Yoshio Yamada

Funkcialaj Ekvacioj 33 ( 1 ) 151 - 159 1990 [Refereed] [International journal]

CiNii
SOME NONLINEAR DEGENERATE WAVE-EQUATIONS

Y YAMADA

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 11 ( 10 ) 1155 - 1168 1987.10 [Refereed]

DOI
Free boundary problems for some reaction-diffusion equations

Masayasu Mimura, Yoshio Yamada, Shoji Yotsutani

Hiroshima Mathematical Journal 17 ( 2 ) 241 - 280 1987 [Refereed]

DOI
Note on chemical interfacial reaction models

Yoshio Yamada, Shoji Yotsutani

Proceedings of the Japan Academy, Series A, Mathematical Sciences 62 ( 10 ) 379 - 381 1986 [Refereed]

DOI
Stability analysis for free boundary problems in ecology

Masayasu Mimura, Yoshio Yamada, Shoji Yotsutani

Hiroshima Mathematical Journal 16 ( 3 ) 477 - 498 1986 [Refereed]

DOI
Bifurcation of periodic solutions for nonlinear parabolic equations with infinite delays

Yasuo Niikura, Yoshio Yamada

Functional Equation 29 ( 3 ) 309 - 333 1986 [Refereed] [International journal]
Stability and instability for semilinear parabolic equations with free boundary conditions

Yoshiaki Hashimoto, Yasuo Niikura, Yoshio Yamada

Nonlinear Analysis: Theory, Methods & Applications 8 ( 6 ) 683 - 694 1984.06 [Refereed]

DOI
Parabolic equations with free boundary conditions

Yoshikazu Giga, Yoshiaki Hashimoto, Yoshio Yamada

Funkcialaj Ekvacioj 26 ( 3 ) 263 - 279 1983 [Refereed]
ON A CERTAIN CLASS OF SEMI-LINEAR VOLTERRA DIFFUSION-EQUATIONS

Y YAMADA

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 88 ( 2 ) 433 - 451 1982 [Refereed]

DOI
ON SOME QUASILINEAR WAVE-EQUATIONS WITH DISSIPATIVE TERMS

Y YAMADA

NAGOYA MATHEMATICAL JOURNAL 87 ( NOV ) 17 - 39 1982 [Refereed]

DOI
QUASILINEAR WAVE-EQUATIONS AND RELATED NON-LINEAR EVOLUTION-EQUATIONS

Y YAMADA

NAGOYA MATHEMATICAL JOURNAL 84 ( DEC ) 31 - 83 1981

DOI
Some remarks on the equation $y_{tt}-\sigma(y_x)y_{xx}-y_{xtx}=f$

Yoshio Yamada

Osaka Mathematical Journal 17 ( 2 ) 303 - 323 1980 [International journal]
On the decay of solutions for some nonlinear evolution equations of second order

Yoshio Yamada

Nagoya Mathematical Journal 73 69 - 98 1979.03 [Refereed]

　View Summary

In this paper we consider nonlinear evolution equations of the form

(E) <italic>u″(t) + Au(t) + B(t)u′(t) = f(t),</italic> 0 ≦ <italic>t</italic> < ∞,

(<italic>u′(t) = d²u(t)/dt², u′(t) = du(t)/dt</italic>), where <italic>A</italic> and <italic>B(t</italic>) are (possibly) nonlinear operators. Various examples of equations of type (E) arise in physics; for instance, if <italic>Au</italic> = –<italic>Δu</italic> and <italic>B(t)u′</italic> = | <italic>u′</italic> | <italic>u′</italic>, the equation represents a classical vibrating membrane with the resistance proportional to the velocity.

DOI
Note on certain nonlinear evolution equations of second order

Yoshio Yamada

Proceedings of the Japan Academy, Series A, Mathematical Sciences 55 ( 5 ) 167 - 171 1979 [Refereed]

DOI
Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries

Yoshio Yamada

Nagoya Mathematical Journal 70 111 - 123 1978.07 [Refereed]

DOI
On the Navier-Stokes equations in non-cylindrical domains: An approach by the subdifferential operator theory

Mitsuharu Otani, Yoshio Yamada

Journal of the Faculty of Science, the University of Tokyo, Sect. IA Math. 25 ( 2 ) 185 - 204 1978 [Refereed] [International journal]

DOI

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Books and Other Publications

非線形楕円型微分方程式の解析

山田義雄( Part： Sole author)

大学院GP数学レクチャーノートシリーズ，東北大学大学院理学研究科 2010.03
Handbook of Differential Equations:Stationary Partial Differential Equations, Vol. 6

Yoshio Yamada, Michel Chioot

Elsevier 2008.06 ISBN: 9780444532411
理工系のための「微分積分Ⅱ」

鈴木武, 山田義雄, 柴田良弘, 田中和永

内田老鶴圃 2007.11 ISBN: 9784753601837
理工系のための「微分積分Ⅰ」

鈴木武, 山田義雄, 柴田良弘, 田中和永

内田老鶴圃 2007.04 ISBN: 9784753601813

Presentations

反応拡散方程式の自由境界問題とテラス型伝播解

山田義雄 [Invited]

徳島偏微分方程式小研究集会 (徳島大学常三島地区)

Presentation date： 2022.12

Event date：
2022.12

　

　
Positive bistable 項を伴う反応拡散方程式の自由境界問題に対する球対称解の漸近挙動

兼子裕大, 松澤寛, 山田義雄

日本数学会秋季総合分科会 (北海道大学) 日本数学会

Presentation date： 2022.09

Event date：
2022.09

　

　
数理生態学に現れる Stefan 型自由境界問題について

山田義雄 [Invited]

東北大学解析セミナー (東北大学) 東北大学数学教室

Presentation date： 2020.01
交差拡散を伴う数理生態学モデル

山田義雄 [Invited]

四ツ谷晶二先生退職記念研究集会龍谷大学理工学部

Presentation date： 2019.03
Positive bistable 型非線形項をもつ反応拡散方程式の自由境界問題における解の漸近的形状について

兼子裕大, 松澤寛, 山田義雄

日本数学会年会 (東工大大岡山キャンパス) 東京工業大学

Presentation date： 2019.03
A free boundary problem arising in mathematical ecology and asymptotic estimates of solutions

Yoshio Yamada [Invited]

Matsuyama Analysis Seminar 2019 (Ehime University, Matsuyama) Department of Mathematics, Ehime University

Presentation date： 2019.02

Event date：
2019.02

　

　
数理生態学におけるStefan 型自由境界問題に関わる話題

山田義雄 [Invited]

早稲田大学応用解析研究会 (早稲田大学西早稲田キャンパス) 早稲田大学理工学術院

Presentation date： 2018.10
A free boundary problem for reaction diffusion equation with positive bistable nonlinearity

Maho Endo, Yuki Kaneko, Yoshio Yamada [Invited]

Presentation date： 2018.10
反応拡散方程式の自由境界問題における spreading 解の漸近形状と漸近速度

兼子裕大, 山田義雄

日本数学会秋季総合分科会 (岡山大学) 岡山大学

Presentation date： 2018.09
Free boundary problems appearing in mathematical biology

Yoshio Yamada [Invited]

Nonlinear Evolutionary PDEs and Their Equilibrium states II (Nishi-Waseda Campus, Waseda University) Waseda University

Presentation date： 2018.09

Event date：
2018.09

　

　
Asymptotic estimates of solutions for a certain class of one-dimensional free boundary problems

Yoshio Yamada [Invited]

The 12th AIMS Conference in Dynamical Systems, Differential Equations and Applications (National Taiwan University, Taipei) American Institute of Mathematical Sciences/National Taiwan University

Presentation date： 2018.07

Event date：
2018.07

　

　
On free boundary problems appearing in mathematical ecology

Yoshio Yamada [Invited]

Let's enjoy mathematical sciences (Shibaura Institute of Technology, Ohmiya Campus)

Presentation date： 2017.09
反応拡散方程式に対する解の漸近的性質

山田義雄

2017秋の偏微分方程式セミナー (大阪大学) 大阪大学待兼山会館

Presentation date： 2017.09
Asymptotic behavior of solutons of free boundary problems for reaction-diffusion equations

Yoshio Yamada [Invited]

The 149th Kagurazaka Analysis Seminar (Tokyo University of Science)

Presentation date： 2017.06
Free boundary problems for reaction-diffusion equations

Yoshio Yamada [Invited]

The 5th Lecture Series on PDEs (Fukuoka Institute of Technology) Fukuoka Institute of Technology

Presentation date： 2017.05

Event date：
2017.05

　

　
Spreading, vanishing and singularity for radially symmetric solutions for a Stefan-type free boundary problem

Yuki Kaneko, Yoshio Yamada [Invited]

RIMS研究集会「非線形現象の解析への応用としての発展方程式論の展開」 (京都大学数理解析研究所) 数理解析研究所

Presentation date： 2016.10
１．軌道を持つLotka-Volterra 系， 2．2次増大度の散逸的反応拡散系

鈴木貴, 山田義雄

日本数学会秋季総合分科会 (関西大学千里山キャンパス) 関西大学

Presentation date： 2016.09
Asymptotic profiles of solutions for free boundary problems appearing in mathematical ecology

Yoshio Yamada

PDE Seminar in summer, 2016 (Osaka University, Suita campus)

Presentation date： 2016.08
Multiple spreading phenomena for a free boundary problem of diffusion equations with bistable nonlinarity

Yoshio Yamada [Invited]

11th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Orlando, Florida) American Institute for Mathematical Sciences, Orlando, Florida

Presentation date： 2016.07

Event date：
2016.07

　

　
Asymptotic profiles of solutions for some free boundary problems in ecology

Yoshio Yamada [Invited]

International Conference on Reaction-Diffusion Equations and Their Applications to the Life, Social and Physical Sciences (Beijing) Renmin University of China, Institute for Mathematical Sciences

Presentation date： 2016.05

Event date：
2016.05

　

　
Multiple spreading phenomena of a free boundary problem for reaction-diffusion equation with positive bistable nonlinearity

Yoshio Yamada

2015 Fall PDE Seminar -Workshop on evolution equations and related topics- (Osaka University)

Presentation date： 2015.09
Multiple spreading phenomena for a free boundary problem in ecology

Yoshio Yamada [Invited]

Conference "Asymptotic Problem, Elliptic and Parabolic Issues," (Vilnus, Lithuania) University of Vilnius/ University of Zurichs, Vilnus, Lithuania

Presentation date： 2015.06

Event date：
2015.06

　

　
Multiple spreading phenomena for a certain class of free boudnary problems of reaction-diffusion equations

Yoshio Yamada [Invited]

The 11th Japanese-German International Workshop on Mathmatical Fluid Dynamics (Waseda University, Tokyo) Waseda University, Tokyo

Presentation date： 2015.03

Event date：
2015.03

　

　
Reaction-diffusion equations with spatial inhomogeneity

The 3rd Seminar on Mathematical Modelling, Atami, Shizuoka

Presentation date： 2015.01
Logistic diffusion equations with nonlocal effects

2014 Summer seminar on PDEs --Workshop on evolution equations and related topics--1,Osaka University

Presentation date： 2014.08
Logisitc diffusion equations with nonlocal effects in population biology

Yosho Yamada [Invited]

10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Madrid) American Institute of Mathematical Sciences, Madrid

Presentation date： 2014.07

Event date：
2014.07

　

　
Spreading and vanishing dichotomy for some free boundary problems in population biology

[Invited]

International Sympojium on Applied Analysis, University of Zurich, Zurich

Presentation date： 2014.06

Event date：
2014.06

　

　
Reaction-diffusion equations and related topics

The 2nd Semnar on Mathematical Modelling, Ito, shizuoka

Presentation date： 2014.01
Population model and free boudnary problems

[Invited]

Gifu Seminar on Mathematical Sciences

Presentation date： 2013.12
On logisitic equations with diffusion and nonlocal terms

[Invited]

RIMS workshop "Progress in Qualitative Theory of Ordinary Differential Equations" (Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 2013.11

Event date：
2013.11

　

　
On a population model with a free boundary and related elliptic problems

Yuki Kaneko, Yoshio Yamada [Invited]

RIMS研究集会「常微分方程式の定性的理論の新展開」 (京都大学数理解析研究所) 数理解析研究所

Presentation date： 2013.11

Event date：
2013.11

　

　
Asymtotic behavior of solutions for some free boundary problems in population biology

Yoshio Yamada [Invited]

Workshop on New Mathematical Developments Arising from Ecology, Epidemiology and Environmental Science (Pekin University, Beijing) Beijin International Center for Mathematical Researches, University

Presentation date： 2013.10

Event date：
2013.10

　

　
On limit systems for some Lotka-Volterra models with cross-diffusion

Yoshio Yamada [Invited]

Capital Normal University, Special talk (Capital Normal University, Beijing) Capital Normal University

Presentation date： 2013.10
Spreading and vanishing for some free boundary problems in population dynamics

Yoshio Yamada [Invited]

Workshop on Nonlinear Equations in Population Biology (East China Normal University, Shanhai) Center for Partial Differential Equations, East China Normal University

Presentation date： 2013.05

Event date：
2013.05

　

　
Remarks on logistic diffusion equations with nonlocal effects

Yoshio Yamada [Invited]

PDE Seminar, Tonji University (Tonji University, Shanhai) Tonji University, Shanhai

Presentation date： 2013.05
Global-in-time behavior of Lotka-Volterra systems

鈴木貴, 山田義雄

日本数学会年会 (京都大学吉田キャンパス) 京都大学

Presentation date： 2013.03
数理生態学モデルに現れる spreading と vanishing

兼子裕大; 大枝和浩; 山田義雄

日本数学会年会 (京都大学吉田キャンパス) 京都大学

Presentation date： 2013.03
On free boundary problems modelling biological invasion

[Invited]

Workshop on Mathematical Scieces at Fujita Health University (Fujita Health University, Toyoake)

Presentation date： 2013.02
On logistic diffusion equations with nonlocal effects

[Invited]

Functional Analysis and Applications-Evolution Equations and Control Theory, Kobe University

Presentation date： 2013.02
Reaction-diifusion equations and bifurcation

Yoshio Yamada [Invited]

The 5th Tohoku Symposium on Elliptic and Parabplic Differential Equations, Tohoku University (Tohoku University, Sendai) Department of Mathematics, Tohoku University

Presentation date： 2013.01

Event date：
2013.01

　

　
Free boundary problems appearing in mathematical ecology

[Invited]

Seminar on Fluid Mathemtatics, Waseda University

Presentation date： 2013.01
Spreading and vanishing dichotomy for some free boundary problems in ecology

Yosio Yamada [Invited]

Swiss-Japanese Seminar (Zuruch, Swiss) University of Zurich, Zurich

Presentation date： 2012.12

Event date：
2012.12

　

　
On logistic equations with diffusion and nonlocal effects

Yoahio Yamada [Invited]

Workshop on Nonlocal Problems--in the framework of the EU Programme FIRST (Zurich, Swiss) EU Programm FIRST, University of Zurich

Presentation date： 2012.12

Event date：
2012.12

　

　
1. Basic results on a free boundary problem in ecology, 2. Asymptotic behaviors of solutions for a free boundary problem in ecology

Yoshio Yamada [Invited]

Intensive lectures at the University of Zurich (Zurich) University of Zurich,

Presentation date： 2012.12

Event date：
2012.12

　

　
Free boundary problems for reaction-diffusion equations in ecology

[Invited]

Seminar of Applied Mathematics, Univ. Compultense de Madrid, Madrid

Presentation date： 2012.12
Spreading and vanishing for free boundary problems in ecology

Yoshio Yamada [Invited]

5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Sciences, Modellings, Theory and Simulations Kansai Seminar House, Kyoto

Presentation date： 2012.11

Event date：
2012.11

　

　
多次元円環領域における数理生態学モデルの自由境界問題について

兼子裕大, 山田義雄

日本数学会秋季総合分科会 (九州大学伊都キャンパス) 九州大学

Presentation date： 2012.09
Gierer-Meinhardt 系の時間大域挙動

鈴木貴, 山田義雄

日本数学会年会 (東京理科大学神楽坂キャンパス) 東京理科大学

Presentation date： 2012.03
Spreading and vanishing for free boundary problems arising in mathematical biology

Yoshio Yamada [Invited]

Conference on Evolution Equations, Related Topics and Applications, JSPS-DFG Seminar (早稲田大学、東京) JSPS-DFG, Waseda University

Presentation date： 2012.03

Event date：
2012.03

　

　
Population models with nonlinear diffusion

Yoshio Yamada [Invited]

PDE Seminar, University of New England School of Mathematics and Computer Science, University of New England, Armidale, Australia

Presentation date： 2012.02
On mathematical ecology model with cross-diffusion

Yoshio Yamada

RIMS Workshop ``Mathematical Scinence on Nonlinear Diffusion" (Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 2012.02

Event date：
2012.02

　

　
A free boundary problem modellin the invasion of species

兼子裕大、山田義雄 [Invited]

RIMS研究集会「非平衡非線形現象の解析ー発展方程式の立場からー」 (京都大学数理解析研究所) 数理解析研究所

Presentation date： 2011.10

Event date：
2011.10

　

　
On a free boundary problem for reaction-diffusion equations appearing in mathematical ecology

Fall Meeting of the Mathematical Society of Japan

Presentation date： 2011.09
Coexistence states for the SKT-model with with large cross-diffusion

久藤衡介, 山田義雄

日本数学会秋季総合分科会 (信州大学) 信州大学理学部

Presentation date： 2011.09
Free boundary problems for some population models with diffusion

Yoshio Yamada [Invited]

One Forum, Two Cities, Aspect of Nonlinear PDEs (Taipei) TAida Institute for Mathematical Sciences, National Taiwan University

Presentation date： 2011.08

Event date：
2011.08

-

2011.09
反応方程式に対する力学系理論

山田義雄

「現象の数理」研究会 (静岡県伊東市) 伊東市「山喜旅館」

Presentation date： 2011.02
On limit systems for some population models with cross-diffusion

Yoshio Yamada [Invited]

Workshop on PDE Models fo Biological Processes (Hsinchu, TAiwan) National Center of Theoretical Sciences (NCTS), Hsinchu, Taiwan

Presentation date： 2010.12

Event date：
2010.12

　

　
Lotka-Volterra competition systems with nonlinear diffusion and related problems

Yoshio Yamada [Invited]

Nonlinear Evolutionary PDEs and Their Equilibrium States Waseda University, Tokyo

Presentation date： 2010.06

Event date：
2010.06

　

　
On Lotla-Volterra competition model with nonlinear diffusion

Yoshio Yamada [Invited]

Workshop on Nonlinear Evolution Equations and Their Related Topics Kobe University

Presentation date： 2010.03
Mathematical analysis of SKT model in population biology

Yosho Yamada [Invited]

The 2nd Nagoya Workshop on Differential Equations, Nagoya University-in Honor of the Retirement of Professor Masatake Miyake--, Nagoya University, Nagoya

Presentation date： 2010.03

Event date：
2010.03

　

　
On Lotka-Volterra competition model with cross-diffusion

Yoshio Yamada [Invited]

RDS Seminar (Meiji University, Tokyo) Meiji Institute for Advanced Study of Mathematical Sciences

Presentation date： 2009.11
Limiting behavior of positive steady-state solutions for the Lotka-Volterra competition with large cross-diffusion

Yoshio Yamada [Invited]

Conference on Evolution Equatons, Related Topics and Applications (Helmholtz Center, Munich,) JSPS-DFG, Helmholtz Center, Munich, Germany

Presentation date： 2009.09

Event date：
2009.09

　

　
On a certain class of population models with nonlinear diffusion

Yoshio Yamada [Invited]

6th European Conference on Elliptic and Parabolic Problems Gaeta, Italy

Presentation date： 2009.05

Event date：
2009.05

　

　
Positive solutions for Lotka-Volterra competition sytems with large cross-diffusion

Yoshio Yamada [Invited]

Matheamtical fluid Dynamics Launching Workshop (Waseda University, Tokyo) German-Japanese Program, Waseda University/Technische Universitat Darmstadt

Presentation date： 2009.04
Limiting characterization for coexistence states to a Lotka-Volterra model with nonlinear diffusion of fractional type

久藤衡介, 山田義雄

日本数学会秋季総合分科会 (東工大) 東京工業大学

Presentation date： 2008.09
Limiting behavior of solutions for some reaction-diffusion systems with nonlinear diffusion

[Invited]

5th World Congress of Nonlinear Analysts,Orlando, Florida,USA (East China Normal University, Shanhai)

Presentation date： 2008.07

Event date：
2008.07

　

　
Limiting characterization of stationary solutions for some Lotka-Volterra models with nonlinear diffusion

Yoshio Yamada [Invited]

Workshop on Recent Advances on Nonlinear Parabolic and Elliptic Differential Equations (Ryukoku University, Ohtsu) Ryukiku University

Presentation date： 2007.12

Event date：
2007.12

　

　
Population models of interacting species with cross and self diffusion

Yoshio Yamada [Invited]

Eco Summit 2007, Symposium 8: The Mathematics of Spatial Ecology (Beijing) Ecological Society of China

Presentation date： 2007.05

Event date：
2007.05

　

　
数理生態学におけるSKTモデルと関連する話題

山田義雄 [Invited]

東北大学談話会東北大学数学教室

Presentation date： 2006.11
反応拡散方程式とパターン

山田義雄

早稲田大学数学応用数学研究所コロキウム早稲田大学理工学部

Presentation date： 2006.04
Transition layers and spikes for a class of bistable reaction-diffusion equations

Yoshio Yamada [Invited]

Workshop on Computor-Aided Analysis and Reaction-Diffusion Systems, National Taiwan Normal Univ., Taiwan (Taipei) National Taiwan Normal University/ National Taiwan University

Presentation date： 2005.09

Event date：
2005.09

　

　
Transition layers and spikes for inhomogeneous reaction-diffusion equations with bistable nonlinearity

Yoshio Yamda [Invited]

International Conference on Nonlinear Partial Differential Equations Chufu Normal University, Shandon Prov., China

Presentation date： 2005.07

Event date：
2005.07

　

　
双安定型反応拡散方程式の遷移層やスパイクを持つ解の安定性

中島主恵, 浦野道雄, 山田義雄

日本数学会年会 (日本大学) 日本大学

Presentation date： 2005.03
Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with Cross-Diffusion

Yoshio Yamada [Invited]

Workshop on Evolution Equations and Asymptotic Analysis of Solutions (Research Institute for Mathematical Sciences, Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 2003.10

Event date：
2004.10

　

　
Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity

Michio Urano [Invited]

The 5th AIMS Conference on Dynamical Systems and Differential Equations California State Polytechnic University, Pomona, CA

Presentation date： 2004.06
ある双安定型反応拡散方程式の解に対する遷移層とスパイクについて

中島主恵, 浦野三著, 山田義雄

日本数学会年会 (筑波大学) 筑波大学

Presentation date： 2004.03
Stationary problem for reaction-diffusion equations with cross-diffusion term and related topics

Yoshio Yamada [Invited]

The 41th Real Analysis and Functional Analysis Joint Symposium (Nara University of Education, Nara) Mathematical Society of Japan, Real Analysis Section/Functional Analysis Section

Presentation date： 2002.07
Multiple positive solutions for prey-predator systems with cross-diffusion

Yoshio Yamda [Invited]

Mathematics Seminar, Worcester Polytechnic Institute Worcester Polytechnic Institute, Worcester, MA

Presentation date： 2002.05
Multiple existence of positive solutions for prey-predator systems with cross-diffusion

Yoshio Yamada [Invited]

The 4th International Congress of Dynamical Systems and Differential Equations, University of North Carolina, Wilmington, USA University of North Carolina, Wilmington

Presentation date： 2002.05

Event date：
2002.02

　

　
Multiple existence of steady-states for a prey-predator system with cross diffusion

久藤衡介;山田義雄

日本数学会秋季総合分科会 (九州大学) 九州大学

Presentation date： 2001.10
Global solutions for a certain class of competition model with cross-diffusion

[Invited]

PDE Seminar, School of Mathematics and Statistics, University of Sydney, Australia

Presentation date： 2001.09
Positive solutions of diffusion problems with concave-convex nonlinearities

[Invited]

Mathematics Seminar, School of Mathematics and Computer Science, University of New England, Australia

Presentation date： 2001.09
Positive solutions for Lotka-Volterra system with cross-diffusion

山田義雄

盛岡における夏の微分方程式セミナー岩手大学，盛岡

Presentation date： 2001.08
1. Lotka-Volterra competition models with linear and nonlinear diffusion, 2. Positive solutions of Lotka-Volterra diffusion equations with large interactions

Yoshio Yamada [Invited]

Intensive Lectures at Shanxi Normal University Shanxi Normal University, Xian

Presentation date： 2001.05

Event date：
2001.05

　

　
Positive solutions for Lotka-volterra systems with diffusion

Yoshio Yamada [Invited]

The 18th Kyushu Symposium of Partial Differential Equations Kyushu University

Presentation date： 2001.02

Event date：
2001.01

-

2001.02
拡散項を含む Lotka-Volterra 型競合モデルについて

山田義雄 [Invited]

都立大学談話会都立大学

Presentation date： 2000.11
Multiple positive solutions for Lotka-Volterra competition system with diffusion

yosho Yamada [Invited]

The First East Asia Symposium on Nonlinear PDE (International Institute for Advanced Studies, Kidu, Kyoto) International Institute for Advanced Studies, Kidu, Kyoto

Presentation date： 2000.09

Event date：
2000.09

　

　
Lotka-Volterra 型競合モデルの正値解の多重性と安定性の解析について

山田義雄; 中口悦史

夏の偏微分方程式セミナー東海大学山中湖セミナーハウス

Presentation date： 2000.08

Event date：
2000.08

　

　
Positive solutions for Lotka-Volterra competition system with diffusion

Yoshio Yamada [Invited]

Th3 3rd World Congress of Nonlinear Analysts (Catania, Sicily) , July 19-26, Catania, Italy

Presentation date： 2000.07

Event date：
2000.07

　

　
Multiple coexistence states for Lotka-Volterra competition model with diffusion

Yoshio Yamada [Invited]

RIMS Symposium on Nonlinear Diffusion Systems- Dynamics and Asymptotic Analysis (Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 2000.06

Event date：
2000.05

-

2000.06
Sublinear term をもつ拡散方程式の定常解とその安定性について

久藤衡介;山田義雄

日本数学会年会, 早稲田大学理工学部

Presentation date： 2000.03
Lotka-Volterra 型反応拡散方程式系の正値定常解について

山田義雄 [Invited]

応用解析セミナー (東京大学駒場キャンパス) 東京大学大学院数理科学研究科

Presentation date： 2000.01
Uniqueness of solutions for parabolic differential equations with sublinear term

Yoshio Yamada [Invited]

Workshop on Non Linear Evolution Equation and Its Applications (Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 1999.10

Event date：
1999.10

　

　
反応拡散方程式系の正値定常解の一意性と非一意性について

山田義雄 [Invited]

広島大学談話会広島大学理学部

Presentation date： 1999.07
Lotka-Volterra competition models with cross-diffusion effects

Yosho Yamada [Invited]

Hiroshima Workshop '99 “Nonlinear Diffusion Equations and Related Topics" Hiroshima University

Presentation date： 1999.01
Coexistence states for Lotka-Volterra systems with cross-diffusion

[Invited]

International Conference on Operator Theory and Its Applications to Scientific and Industrial Problems, Minisymposia on Biomathematics, Winnipeg, Canada (University of Manitoba/Delta Hotel, Winnipeg) University of Manitoba

Presentation date： 1998.10

Event date：
1998.10

　

　
Asymptotic behavior of a reaction-diffusion equation with $p-$Laplacian

Yoshio Yamada [Invited]

Groupe de Travail, Equations Elliptiques et Paraboliques Non Lineaires Universite Paris-Sud, Paris

Presentation date： 1998.07
Asymptotic properties of a reaction-diffusion equation with $p-$Laplacian

Yoshio Yamada [Invited]

Seminar on Applied Mathematics University of Paderborn, Paderborn, Germany

Presentation date： 1998.07
退化型準線形放物型方程式とflat hat について

山田義雄 [Invited]

名古屋市立大学談話会

Presentation date： 1998.01
非線形拡散モデルに対する定常問題の解析

山田義雄 [Invited]

蛯原幸義教授追悼研究集会福岡大学セミナーハウス

Presentation date： 1997.07
$p$-Laplacian を拡散項にもつ Chafee-Infante 問題

竹内慎吾, 山田義雄

日本数学会年会 (信州大学理学部) 信州大学理学部

Presentation date： 1997.04
uniqueness and non-uniqueness of positive solutions for reaction-diffusion systems of Lotka-Volterra type

Yoshio Yamada [Invited]

Annual Meeting of Japanese Society for Mathematical Sciences (Osaka Prefecture University) Osaka Prefecture University

Presentation date： 1996.09
Coexistence states for prey-predator systems with cross-diffusion

Yosho Yamada [Invited]

Workshop on Cross-Diffusion and Related Topics (University of Minnesota) School of Mathematics, University of Minnesota

Presentation date： 1996.05

Event date：
1996.05

　

　
Positive coexistence solutions for population models with cross-diffusion effects

Yoshio Yamada [Invited]

6th Conference on Mathematical Biology (Komaba Campus, University of Tokyo) Graduate School of Mathematical Sciences, University of Tokyo

Presentation date： 1995.11
Positive steady-states for prey-predator models with cross-diffusion

Yoshio Yamada [Invited]

Seta Summer Seminar '95 (Ryukjoku University) Ryukoku University

Presentation date： 1995.08
Positive steady-states for prey-predator models with cross-diffusion

山田義雄 [Invited]

東京大学NLPM セミナー (東京大学駒場キャンパス) 東京大学大学院数理科学研究科

Presentation date： 1995.05
Positive steady-states for prey-predator models with cross-diffusion

中島主恵、山田義雄

日本数学会年会 (立命館大学理工学部) 立命館大学理工学部

Presentation date： 1995.03
非線形現象の数理

山田義雄 [Invited]

早稲田大学理工総合研究センター創設１周年記念シンポジウム早稲田大学理工学部

Presentation date： 1994.11
Positive solutions for some reaction-diffusion systems

Yoshio Yamada [Invited]

The 2nd Japanese-Sino Joint Seminar on Nonlinear Partial Differential Equations, (Ryukjoku University) Ryukoku University

Presentation date： 1994.07
Reaction diffusion systems with feedback effects

Yoshio Yamada [Invited]

The 3rd Workshop on Nonlinear Partial Differential Equations (Chuo University) Chuo University

Presentation date： 1994.01

Event date：
1994.01

　

　
フィードバック効果を持つ反応拡散方程式系

山田義雄

第１３回発展方程式研究会 (高知市) 高知市

Presentation date： 1993.11

Event date：
1993.11

　

　
Global solutions for a certain class of quasilinear parabolic systems

Yoshio Yamada [Invited]

Symposium on Partial Differential Equations at Tohoku University (Tohoku University) Department of Mathematics, Tooku University

Presentation date： 1993.01

Event date：
1993.01

　

　
準線型拡散方程式系の有界大域解

山田義雄

第１２回発展方程式研究会 (岐阜市「阜山荘」) 岐阜市「阜山荘」

Presentation date： 1992.12
反応拡散方程式の振動現象について

山田義雄 [Invited]

東海大学談話会 (東海大学) 東海大学

Presentation date： 1992.11
Global solutions for quasilinear parabolic systems with cross-diffusion effects

山田義雄

応用解析系研究ー非線形解析学における最近の発展，早稲田大学理工系シンポジウム (早稲田大学理子学部) 早稲田大学理工学部

Presentation date： 1992.11
Global solutions of quasilinear parabolic systems with cross diffusion effects

Yoshio Yamada [Invited]

PDE Seminar, University of Minnesota

Presentation date： 1992.05
Asymptotic behaviors of solutions to semilinear diffusion equations of Volterra type

Yoshio Yamada [Invited]

The 15th Sapporo Symposium on Partial Differential Equations (Hokkaido University) Department of Mathematics, Hokkaido University

Presentation date： 1991.02

Event date：
1991.02

　

　
Asymptotic behavior of solutions for some reaction-diffusion systems

Yoshio Yamada [Invited]

Sino-Japanese Japanese Joint Seminar on Nonlinear Partial Differential Equations --with Emphasis on Reaction-Diffusion Aspects-- (Academia Sinica, Taipei) Academia Sinica, Taipei

Presentation date： 1990.12

Event date：
1990.12

　

　
界面における化学反応のモデル：解の漸近挙動について

飯田雅人, 山田義雄, 四ツ谷晶二 [Invited]

「発展方程式の非線形問題への応用」研究集会 (京都大学数理解析研究所) 京都大学数理解析研究所

Presentation date： 1990.10
Evolution equations and their applications to partial differential equations （特別講演）

山田義雄 [Invited]

第１１回発展方程式若手セミナー (和歌山県高野山町「巴陵院」) 和歌山県高野山町「巴陵院」

Presentation date： 1989.08

Event date：
1989.08

　

　
On a non-stationary heat convection equation

Yoshio Yamada [Invited]

The 6th Kyushu Symposium of Partial Differential Equations (Kumamoto Univesity) Department of Mathematics, Kumamoto University

Presentation date： 1989.02

Event date：
1989.02

　

　
ある種の半線型放物型発展方程式の可解性とその応用

山田義雄

第８回発展方程式研究会 (高松市「さぬき荘」) 高松市「さぬき荘」

Presentation date： 1988.12
半線型高階放物型方程式の大域解について

星野弘喜, 山田義雄 [Invited]

「発展方程式とその応用」研究集会 (京都大学数理解析研究所) 京都大学数理解析研究所

Presentation date： 1988.10

Event date：
1988.10

　

　
退化型のある準線形波動方程式について

西原健二;山田義雄

日本数学会年会立教大学

Presentation date： 1988.04
界面での化学反応モデル

山田義雄;四ツ谷晶二

日本数学会秋季総合分科会千葉大学

Presentation date： 1986.09
界面における化学反応の方程式について

山田義雄;四ツ谷晶二

第5回発展方程式研究会 (赤穂市) 兵庫勤労福祉センター「赤穂ハイツ」

Presentation date： 1986.01
反応拡散系の自由境界問題

山田義雄 [Invited]

研究集会「線形および非線形の諸問題」信州大学理学部数学教室

Presentation date： 1985.12

Event date：
1985.12

　

　
ある種の退化準線型波動方程式についての注意

山田義雄 [Invited]

Seminar on Nonlinear Problems (広島大学理学部) 広島大学理学部

Presentation date： 1985.11
Free boundary problems for some reaction-diffusion equations

Yoshio Yamada [Invited]

Hiroshima Workshop on Dynamical Biological Systems (Hiroshima Sun Plaza) Department of Mathematics, Hiroshima University

Presentation date： 1985.11

Event date：
1985.11

　

　
$L^p-L^q$ 評価とその応用について（特別講演）

山田義雄 [Invited]

第７回発展方程式若手セミナー (和歌山県高野山町「巴陵院」) 和歌山県高野山町「巴陵院」

Presentation date： 1985.08

Event date：
1985.08

　

　
$L^p-L^q$ 評価とその応用

山田義雄 [Invited]

偏微分方程式セミナーー中津川サマーセミナー (中津川研修センター) 東海地区国立大学共同中津川研修センター

Presentation date： 1985.07

Event date：
1985.07

　

　
自由境界問題と分岐

山田義雄 [Invited]

第２５回宮崎PDEセミナー (宮崎大学) 宮崎大学

Presentation date： 1985.03
A free boundary problem and bifurcation

Yoshio Yamada [Invited]

Symposium on Nonlinear Partial Differential Equations (Kyoto University) Department of Mathematics, Kyoto University

Presentation date： 1985.02

Event date：
1985.02

　

　
すみわけ現象のす学モデル

山田義雄; 三村昌康; 四ツ谷晶二

第４回発展方程式研究会 (福井市「竜川荘」) 福井市「竜川荘」

Presentation date： 1984.12
生態学に現れる自由境界問題

山田義雄

東大NAセミナー東京大学理学部

Presentation date： 1984.07
生物モデルに関連した自由境界問題

山田義雄;三村昌康; 四ツ谷晶二

日本数学会年会 (大阪大学吹田キャンパス) 大阪大学

Presentation date： 1984.04
生物モデルに関連した自由境界問題

四ツ谷晶二, 三村昌泰, 山田義雄 [Invited]

Nonlinear Partial Differential Equations (東北大学) 東北大学理学部

Presentation date： 1984.02

Event date：
1984.02

　

　
生物学に現れる自由境界問題

山田義雄; 四ツ谷晶二

第３回発展方程式研究会

Presentation date： 1984.01
Study of free boundary problems arising in ecology

山田義雄

第５回発展方程式若手セミナー (神奈川県箱根町「箱根青雲荘」) 神奈川県箱根町「箱根青雲荘」

Presentation date： 1983.08

Event date：
1983.08

　

　
半線型放物型方程式に対するある種の自由境界値問題について

山田義雄, 橋本佳明, 儀我美一

日本数学会年会広島大学

Presentation date： 1983.04
放物型方程式に対するある種の自由境界値問題

山田義雄

第２回発展方程式研究会 (兵兵庫県竜野市「あかとんぼ荘」) 兵庫県竜野市「あかとんぼ荘」

Presentation date： 1983.01
Stability and bifurcation of periodic solutions for semilinear Volterra diffusion equations

Yoshio Yamada [Invited]

The 8th Sapporo Symposium on Partial Differential Equations (Hokkaido University) Department of Mathematics, Hokkaido University

Presentation date： 1982.08

Event date：
1982.08

　

　
ある種の自由境界条件をもつ放物型方程式

山田義雄

第４回発展方程式若手セミナー (長野県上松町「ねざめホテル」) 長野県上松町「ねざめホテル」

Presentation date： 1982.08

Event date：
1982.08

　

　
Stability and bifurcation of periodic solutions for semilinear diffusion equations with delay

Yoshio Yamada [Invited]

Mathematics of Oscillatory Phenomena '82 (Hiroshima University) Department of Mathematics, Hiroshima Univesity

Presentation date： 1982.01

Event date：
1982.01

　

　
Volterra 型拡散方程式の安定性

山田義雄

第１回発展方程式研究会 (鳥取市) 鳥取市

Presentation date： 1981.12
On some semilinear Volterra diffusion equations and their applications

Yoshio Yamada [Invited]

Equadiff 5: The 5th Chechoslovak Conference on Differential Equations and Their Applications (Comenius University, Bratislava) Comenius University, Bratislava

Presentation date： 1981.08

Event date：
1981.08

　

　
Volterraの方程式について

山田義雄 [Invited]

中部地区関数方程式セミナー (信州大学松本キャンパス) 信州大学理学部数学教室

Presentation date： 1981.08
Volterra 型半線型拡散方程式について

山田義雄

東大理学部ＮＡセミナー (東京大学本郷キャンパス) 東京大学理学部

Presentation date： 1981.02
履歴項を持つ prey-predator モデルについて

山田義雄 [Invited]

偏微分方程式阪大シンポジウム (大阪大学豊中キャンパス) 大阪大学理学部

Presentation date： 1981.02

Event date：
1981.02

　

　
On semilinear Volterra diffusion equations

Yoshio Yamada [Invited]

Workshop on Mathematical Topics in Biology, '80 Biology (Research Institute for Mathematical Sciences, Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 1980.12

Event date：
1980.12

　

　
Volterra 型半線型拡散方程式について

山田義雄

広島大学セミナー (広島大学理学部) 広島大学理学部

Presentation date： 1980.09
積分項をもつ半線型拡散方程式について

山田義雄

第2回発展方程式若手セミナー (長野県望月町「望月荘」) 長野県望月町「望月荘」

Presentation date： 1980.07

Event date：
1980.07

　

　
Volterra 型半線型拡散方程式について

山田義雄 [Invited]

数理研短期共同研究「波動方程式における Dissipative 項の影響」 (京都大学数理解析研究所) 京都大学数理解析研究所

Presentation date： 1980.07

Event date：
1980.07

　

　
積分項を持つ非線形放物型偏微分方程式系について

山田義雄 [Invited]

東大基礎科セミナー (東京大学教養学部) 東京大学教養学部基礎科学科

Presentation date： 1980.05
On quasilinear wave equations

Yoshio Yamada [Invited]

Workshop on Applied Analysis for Partial Differential Equations in Applied Sciences (Research Institute for Mathematical Sciences, Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 1980.02

Event date：
1980.02

　

　
準線型波動方程式について

山田義雄 [Invited]

名古屋大学数学教室談話会 (名古屋大学理学部) 名古屋大学理学部

Presentation date： 1979.11
Relationship between quasilinear wave equations and nonlinear evolution equations

Yoshio Yamada [Invited]

Sapporo Symposium on Partial Differential Equatins (Hokkaido University) Department of Mathematics, Hokkaido University

Presentation date： 1979.10

Event date：
1979.10

　

　
1．準線型波動方程式と強消散項をもつ発展方程式， 2．減衰項をもつ準線型波動方程式の大域解の存在について

山田義雄

日本数学会秋季総合分科会 (京都大学理学部) 日本数学会，京都大学理学部

Presentation date： 1979.10
準線型波動方程式について

山田義雄

第1回発展方程式若手セミナー新潟県妙高高原「妙高山荘」

Presentation date： 1979.07

Event date：
1979.07

　

　
Nash-Moser-Schwartz の陰関数定理とその応用

山田義雄 [Invited]

研究集会”非線型現象の数学” (名古屋工業大学) 名古屋工業大学数学教室

Presentation date： 1979.03

Event date：
1979.03

　

　
1．ある種の二階非線型発展方程式の解の存在について， 2. 減衰項をもつ二階非線型発展方程式の解の減衰について

山田義雄

日本数学会年会 (名古屋大学教養部) 日本数学会，名古屋大学

Presentation date： 1978.04
二階非線型発展方程式とエネルギー保存則について

山田義雄 [Invited]

偏微分方程式阪大シンポジウム (大阪大学理学部) 大阪大学理学部

Presentation date： 1977.12

Event date：
1977.12

　

　
放物型変分不等式について

山田義雄 [Invited]

名古屋大学談話会 (名古屋大学理学部) 名古屋大学理学部

Presentation date： 1977.04
Non-cylindrical domainiにおける非線型放物型方程式について

山田義雄 [Invited]

東大解析セミナー (東京大学理学部) 東京大学理学部数学教室

Presentation date： 1976.12
On evolution equations generated by subdifferentials

Yoshio Yamada [Invited]

Computattion and Analysis

Presentation date： 1976.07

Event date：
1976.07

　

　
On nonlinear parabolic equations in non-cylindrical domains

Yoshio Yamada [Invited]

Sapporo Symposium on Partial Differential Equatins (Hokkaido University) Department of Mathematics, Hokkaido University

Presentation date： 1976.07

Event date：
1976.07

　

　
On evolution equations associated with subdifferentials

Yoshio Yamada [Invited]

Workshop on Evolutional Systems and the Free Boundary Problems (Research Institute for Mathematical Sciences, Kyoto University) Research Institute for Mathematical Sciences, Kyoto University

Presentation date： 1975.11
Subdifferential の発展方程式について

山田義雄

日本数学会秋季総合分科会 (東京大学教養学部) 日本数学会，東京大学教養学部

Presentation date： 1975.10

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Research Projects

Study on free boundary problems arising in mathematical ecology and related nonlinear diffusion equations

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2019.04

-

2023.03
数理生態学に現れる自由境界問題と反応拡散方程式の研究

Project Year :

2016.04

-

2019.03

　View Summary

2017年度は反応拡散方程式に対する自由境界問題に取り組んだ。この問題は、数理生態学における外来種の侵入や、生物種の移動をモデルとしている。空間変数を x, 時間変数を t とし、生物種の生息領域を [0.h(t)], 個体数密度を u(x,t) で表わし、u(t,x) は反応拡散方程式で記述されるとする。また、自由境界 x=h(t) の運動はStefan型境界条件で支配されるとする。(u,h) を未知関数とする、このようなタイプの自由境界問題は2010年に Du-Lin により提起されて以来、活発に研究されてきた。こ自由境界が時間とともに無限に拡がり、生物種が新領域に展開する状態に対応する解を spreading 解と呼ぶ。現在、spreading 解について、自由境界 h(t) の展開速度や u(x,t) の漸近的形状について、詳しい評価を求めることが重要なテーマである。spreading 解 (u,h) について、時間とともに自由境界の速度 h’(t) が一定値 c に近づき、自由境界の近傍において u(x,t) が q(h(t)-x) の形に表されるとすると、(q,c) が満たす sem
Variational study of nonlinear elliptic problems

Project Year :

2013.04

-

2017.03
Study on reaction-diffusion equations and related free boundary problems

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2012.04

-

2015.03

YAMADA Yoshio, OTANI Mitsuharu, TANAKA Kazunaga, HIROSE Munemitsu, NAKASHIMA Kimie, TAKEUCHI Shingo, KUTO Kousuke, WAKASA Tohru, OEDA Kazuhiro, KANEKO Yuki

　View Summary

This research is concerned with a free boundary problem for reaction-diffusion equations in mathematical ecology. This problem models the invasion or migration of a certain biological species. Our main interest is to study the evolution of the population density and habitat of the species. The population density is described by a reaction-diffusion equation and the boundary (or a part of the boundary) of the habitat is controlled by a free boundary condition of Stefan type. We could obtain theoretical understanding on asymptotic behaviors of solutions for free boundary problems of various types: whether the species vanishes eventually or the species persists with spreading free boundary. Moreover, we got precise results on the spreading speed of the free boundary
Synthetic study of nonlinear evolution equation and its related topics

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2009.04

-

2013.03

OTANI Mitsuharu, YAMADA Yoshio, TANAKA Kazunaga, NISHIHARA Kenji, ISHII Hitoshi, OZAWA Tohru, OGAWA Takayoshi, KENMOCHI Nobuyuki, KOIKE Shigeaki, SAKAGUCHI Shigeru, SUZUKI Takashi, HAYASHI Nakao, IDOGAWA Tomoyuki, ISHIWATA Michinori, AKAGI Gorou

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Various types of nonlinear PDEs (nonlinear elliptic equations, nonlinear diffusion equations, nonlinear wave equations, nonlinear Schrodinger equations) arising in physics and engineering were synthetically studied from the viewpoint of the theory of nonlinear evolution equations by using the techniques from the theory of nonlinear functional analysis, the theory of functions of a real variable, the theory of ordinary differential equations and the calculus of variations
Analysis for nonlinear critical phenomena described by mean field equations

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2008.04

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2013.03

SUZUKI Takashi, MISAWA Masashi, TAKAHASHI Futoshi, NAITO Yuki, OHTSUKA Hiroshi, SUGIYAMA Yoshie, ISHIWATA Michinori, KOBAYASHI Takayuki, NAWA Hayato, WATANABE Kazuo, SATO Tomohiko, MATSUMURA Akitaka, YAGI Atsushi, MISAWA Masashi, TAKAHASHI Futoshi, NAITO Yuki, OHTSUKA Hiroshi, SUGIYAMA Yoshie, KOBAYASHI Takayuki, NAWA Hayato, WATANABE Kazuo, SATO Tomohiko, MATSUMURA Akitaka, YAGI Atsushi, YOSHIKAWA Shuji, KUROKIBA Masaki, FURIHATA Daisuke, KUWATA Kazuhiro, TANAKA Mieko, YAMADA Yoshio, SAITO Norikazu, TAKAHASHI Ryo, TASAKI Sohei, RICCIARDI Tonia, STEVENS Angela, KAVALLARIS Nikos, PAWLOW Irena, CHAPLAIN Mark, QUARANTA Vito, CHAVANIS Pierre-henri

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Several common mathematical structures are found in the models describing the motion of mean field of many particles, point vortices, and so on. In this project we focused on a series of nonlinear partial differential equations provides with scaling invariance, and also variational structure associated with a variable regarded as field. We rigorously proved several phenomena emerged from the model, such as concentration or homeostasis, and at the same time developed new mathematical tools to approach interactions between species and global dynamics under strong nonlinearitie
A comprehensive study of nonlinear problems via variational approaches

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2008

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2011

TANAKA Kazunaga, OZAWA Tohru, OTANI Mitsuharu, NISHIDA Takaaki, YAMAZAKI Masao, YAMADA Yoshio, YANAGIDA Eiji, KURATA Kazuhiro, ADACHI Shinji, HIRATA Jun, SEKIGUCHI Masayoshi

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We study nonlinear problems via variational approaches. Especially (1) we study singular perturbation problems for nonlinear Schrodinger equations and systems. We introduce a new purely variational method which enables us to construct concentrating solutions in a very general setting. (2) We study nonlinear elliptic equations and systems in various settings. We give a new variational construction of radially symmetric ground states. We also study stability and instability of solutions. (3) We also study highly oscillatory solutions in 1-dimensional singular perturbation problems. We give characterization and existence result.
Construction of Theory of Digital Analysis

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2008

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2010

YAMADA Yoshio, TAKAHASHI Daisuke, MATSUSHIMA Toshiyasu, KASHIWAGI Masahide, NISHIDA Takaaki, OISHI Shin'ichi

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Our research group is composed of scholars working in the areas of discrete mathematics, nonlinear differential equations, information theory and numerical computation. We have organized "Seminar on Digital Analysis" so that members can hold common understanding and insight on the fundamental theories and ideas of digital mathematics. As speakers of this seminar, we have invited 16 researchers who are highly active in the areas of discrete mathematics, mathematical modeling, information theory and numerical computation. We have succeeded in getting common understanding on digital analysis through exciting discussions in each lecture of the seminar.
Variational study of nonlinear problems

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2005

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2007

TANAKA Kazunaga, OTANI Mitsuharu, YAMAZAKI Masao, YAMADA Yoshio, SHIBATA Tetsutaro, KURATA Kazuhiro

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We study nonlinear elliptic partial differential equations and Hamiltonian systems via variational meth-ods. We put emphasis on singular perturbation problems.
1. We study the existence of high frequency solutions-families of solutions whose numbers of spikes or layers increase to ∞ as the singular perturbation parameter ε goes to 0. We give the existence and the characterization of such families for 1 dimensional elliptic problems including nonlinear Schrodinger equations, Allen-Cahn equations, Fisher equations and Girerer-Meinhardt systems. Especially for Girerer-Meinhardt systems, we introduce and analyze a limit equation using adiabatic invariants. We also give a precise estimate of the number of positive solutions of nonlinear Schrodinger equations.
2. We also study a singular perturbation problem for -ε^2△μ+V(χ)μ =g(μ) in R^N. Under very general conditions on g(μ), which is related to the work of Berestycki, Gallouet-Kavian, we prove the existence of a concentrating solution for N=1,2.
3. We also study the prescribed energy problem for singular first order Hamiltonian systems. We suc-ceed to obtain the existence of periodic orbit under conditions which generalize t
MATHEMATICAL ANALYSIS AND NUMERICAL ANALYSIS OF SEVERAL KINDS OF DIFFERENTIAL EQUATIONS.

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

1994

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1996

MUROYA Yoshiaki, TANAKA Kazunaga, TSUTSUMI Masayoshi, KORI Toshiaki, OTANI Mitsuharu, YAMADA Yoshio

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To solve non-symmetric linear systems derived from the discretization of singular pertur-bation problems, we propose a generalized SOR method with multiple relaxation parameters, that is the improved SOR method with orderings and study its theory and practical use.
In the case of tridiagonal matrices, optimal choices of the parameters are examined : It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which are computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.
We apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiafonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected and used with this method for an n * n block tridiagonal matrix whose block matrices are all n * n matrices, then this iterative method converges at most n^2 iterations.
We also proposed the improved SSOR method with orderings, whic
非線形偏微分方程式系の総合的研究

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1994

　

　

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研究代表者及び理工学部所属の解析学分野分担者を中心とした、外部にも開かれた定期セミナーを早稲田大学理工学部内において週一回(計21回)開催した。この会(応用解析研究会)には、研究分担者のみならず、東京近郊の若手研究者が多く参加し、研究課題関連の話題について活発な討論、意見交換がおこなわれ、研究を遂行する上で非常に有意義であった。また研究経過発表会を数回おこなった。具体的成果については、個々の単独(非線形楕円形、放物型、双曲型、分散型)方程式に関する多くの成果のほかに、Davey-Stewartson(完全流体の表面波)方程式系に対し、弱解の存在と一意性及びその漸近挙動(時間とともに解のある種ノルムが零に近づく)が解明された。
また、界面で化学反応を起こしている拡散方程式系、伝染病をモデル化した反応拡散系について、大域解の存在を示しその漸近挙動を決定した。更に、熱対流と非圧縮性粘性流との混合方程式系に対しては、流体の占める領域の
非線型放物型方程式系及び楕円型方程式系にたいする解集合の研究

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1993

　

　

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非線形放物型方程式系と関連する楕円型方程式について研究を進めてきたなかで、今年度中に成果が得られた研究テーマは、[1]フイードバック効果をもつ反応拡散方程式系についての解集合の構造、[2]数理生態学にあらわれる準線系放物型方程式系について、の二つである。[1]の研究においては、数値実験の結果から、フイードバックのメカニズムは反応拡散方程式系の解にたいして振動現象をもたらすことが観測されている。ノイマン境界条件の下で拡散方程式系の解が、振動しながらも終局的には定常解に収束するのはどんな場合かを解析した。その結果、定常解が大域的に漸近安定となる条件、および、局所的に漸近安定となる条件をわかりやすい形で導くことができた。さらに適当な定数をパラメーターとみなして変化させるとき、定常解が不安定となる状況が起きる。このときには定常解から周期解が分岐することが証明され、分岐した周期解の軌道安定性を調べることができた
関数方程式の総合的研究

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1992

　

　

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当該年度において、上記研究課題について数理物理学に表される各種の偏微分方程式に対して研究等で著るしい発展があった。まず、山田義雄は、熱対流方程式の外部領域における混合問題の大域解の存在性、一環性および漸近挙動について調べた。類似の問題は、従来ナビア-ストークス方程式等については知られていたが、熱対流方程式に関しては始めての結果であると思われる。(Tokyo J.Math,Vol15)さらに山田は、界面における化学反応方程式の漸近的挙動に対しても新しい結果を得ている。(Osaka J.Math Vol29)
また鈴木武は、統計的仮設検定問題において、ベイズリスクと統計的十分性の関係に注目して、ベイズ危険により漸近十分性を特徴ずけている(Statistics & Resisions Vol15)また、非エルゴード的確率過程モデルにおいて、最大確率推定量を定義し、その特質を論じている(Bull.Sui.& Eng.Lal.Waseda.Uniu)
さらに上野喜三雄は、非コンパクト量子群SU_q(I,1)の球関数に対するブランシェレルの公式をカシ
代数群上の保型函数論とP-進離散群の研究

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1989

　

　

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1.ユニタリ-群の類数の研究。虚2次体Kを係数とする正値ユニタリ-形式の類数を求める問題は代数群の数論に於ける一つの基本問題であるが、本研究ではこれをSelbergの跡公式を応用する事により求める事を試みた。この方法は既に四元数環や二次形式の類数の計算で応用され多くの成果が上げられているが、ユニタリ-群(正値エルミ-ト形式)ではこれが最初の成果である。まず一般階数のユニタリ-群の共役類の分類研究をし、次に跡公式の主要項であるMass formulaの具体的表示の初等的証明を与えた。また階数が2、及び3の場合に、unimodular latticeを含む種(genus)の類数に対する具体的公式を与えた。
2.P-進離散群のSelberg-Ihara型ゼ-タ関数の研究。伊原氏はLie群の離散群に対するSelbergゼ-タ関数の類似をP-進体K上の二次特殊線形群(PSL(2、K)の離散群に対して考察し著しい結果を得た。本研究では伊原の結果をK-rankが1の線形代数群に一般化する事を試み、所期の成果を収めたものである。主要な成果はゼ
非線型放物型方程式系及び関連する楕円型方程式系の解析

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1988

　

　

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当研究課題に関する今年度の研究成果は、半線型放物型方程式に対する結果と準線型放物型方程式に対する結果の二つに大別される。
1.半線型放物型方程式の研究:研究の中心テーマにした反応拡散方程式系に関するものと流体問題に関するものについて述べる。
(1)反応拡散方程式系、2個の未知関数が相互作用を及ぼし合う反応拡散方程式系を考える。解の大域的存在、一意性、正則性についてはよく知られているから、重要な問題となるのは、時間変数が無限に大きくなったときの解の漸近挙動と、関連する定常問題の解の安定性である。比較原理、スペクトル解析、分岐理論をうまく組合せることによって、解の漸近挙動が非常に詳しく理解されるようになり、既存の結果も整理統合された。この結果より、適当な物理量をパラメーターにとると、定常解の分岐や安定性の交代の様子が、分岐図のなかにきれいに描くことができる。特に、難問であった2重固有値からの分岐についても
非線形偏微分方程式の総合的研究

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1988

　

　

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当研究課題の下で得られた主要な研究成果のうちの幾つかを下記に列挙する。
1.非線形シュレディンガー方程式。(1)初期値問題に関して、非線形項が局所的であるとは限らない非常に一般的な条件のもとで、解の平滑化及び局所化の性質を証明した。これは従来知られていた結果を著しく一般化し、統一的展望を与えたものである。(2)爆発項をもつ方程式で、非線形の増大度がクリティカルの場合に、解の爆発点近傍での振舞を部分的にではあるが決定した。これらの研究においては、方程式の持つ対称性とそのリー環の無限小生成作用素が有力な手段である。この手法は他の非線形方程式にも拡張されつつある。(3)初期値境界値問題の滑らかな大域解の存在を示すのは、2次元以上の場合困難で、唯一ブレジス・ガルエの結果があるだけであるが、その非線形項の増大度に関する条件が最善ではないことを、新しい評価上の工夫により示した。
2.非線形放物型方程式(1)p-ラプラシアンを含む
数論的多様体の幾何学的研究

科学研究費助成事業(早稲田大学) 科学研究費助成事業(一般研究(C))

Project Year :

1987

　

　

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(1)曖昧であった擬ベクトルと擬スカラーの概念を明快にし, ベクトル場, 擬ベクトル場の回転rot, 発散divの座標系と独立な定義を与えた. それらを用いて真空の電磁場と回転運動を記述した.
(2)Fermat予測の第1の場合に対し, 新しい合同式を見つけた.
(3)弾性をもつ絃の振動をモデルとするような退化型準線型波動方程式の初期値境界値問題について, 局所解を構成する手法を確立するとともに, 解の一意性を示した.
(4)界面における化学反応を記述する楕円型一放物型方程式系についての初期値境界値問題に対して, 滑らかな正値解の大域的存在を示した. 「11.研究発表」の他に寺田文行(Tournal of Seience Education Axpar)Pro-celdings of the leep TC3 Regional conferenee on microcomputer's in Secondary Education.), 中島勝也(計算機内部数表現に関する研究), 広瀬健(Tournal of enformation Rioces-sing), 和田淳蔵(Tokyo J.Math)の研究業績がある.
なお有馬哲, 橋本喜一朗, 郡敏昭は別に論文を執筆中である.
非線形放物型方程式系と関連する楕円型方程式系の研究

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今年度の研究成果は、"cross-diffusion"と呼ばれる拡散項をもつLotka-Volterra型モデルに対する定常解集合の研究と、退化型拡散項(p-Laplacian)をもつ放物型方程式の解のダイナミックスの研究の二つに分けられる。1.数理生態学における"biodiffusion"のなかには"cross-diffusion"と呼ばれる重要な非線形拡散がある。同一の領域で生存競争している2種以上の生物の固体密度を未知関数として定式化すると、"cross-diffusion"の効果により、拡散係数が固体密度にも依存するような準線形拡散方程式系となる。このようなモデルは1979年に提起され、数値実験では分岐やパターンの形成などの興味深い現象が見られるにもかかわらず、理論的な解析は十分ではない。我々の研究グループは数年前から正値定常解集合の解明に取り組み、正値解が存在するための十分条件や必要条件を見いだしている。今年度は解の多重性に関して非線形拡散がいかなる影響を及ぼすかを調べ、写像度の理論と分岐理論を組み合わせて、
非線形楕円型方程式とその周辺に関する研究

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計画調書の研究目的にかかげた目標に関連した主たる成果は以下の通りである。1.非線形項が境界で特異性を有する半線形楕円型方程式 -Δu(χ)=Κ(χ)u^β(χ)/(1-|χ|)^α χ∈B={χ∈R^N;|χ|<1}に対して、変分的手法により以下の結果を得た。(1)β+1【greater than or equ非自明古典解(C^2(B)∩C^1(B^^-)に属する解)は存在しない。(2)0<α<min(β+1,(β+1)/2+1),α<2^*=(N+1)(N-2) ならば、非自明古典解が存在する。(3)0<β【less than or equal】1,β+1【less than or equal】α<(β+1)/2+1 ならば、Holder連続な非自明解が一意的に存在する。これらの成果は、従来の結果を大幅に改良したもので、その全貌がほぼ解明されたと言える。しかしながら、1<β,(β+1)/2+1【less than or equal】α<β+1 の場合の2.非有界領域における弱解に対するPohozaev型の不等式が、星状領域の外部領域及び柱状領域に対して確立され、準線形楕円型方程式の弱解の非存在に応用された。この結果、解の存在・非存在に関して、星状領域の内部と外部との双対性が明らかにされ、こ
THE MATHEMATICAL ANALYSIS TO NON-LINEAR PHENOMENA THROUGH NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1) The p-Laplace operators is well known as the non-linear modification of the usual Laplacian. These operators or their perturbed operators arise in the model equations for the elastic membrane, nonlinear diffusion phenomena and so on. Moreover, the limit state of solutions at p infinity is of great interest from the mathematical or technological view points. The eigenvalue problem of p-Laplacian has been studied by many authors. Since this problem can be dealt with as a variational problem, many results has been known. However, its limit problem at p infinity had been known because it cannot be described in a variatinal problem. We formulate such problem using the notion of the viscosity solution and obtain some results for the limit eigenvalues and the associate eigenfunctions.2) In the ecological model, a reaction-diffusion equation has the nonlinear diffusion with the p-Laplace operator when the diffusion depends on the population pressure nonlinearly. Yamada has studied such equation and obtain the unique and global existence of a solution and sonic results on the set of stationary solutions. He also study the 3 species cooperative competition-diffusion systems with linear di
Study of nonlinear prabolic systems and related elliptic systems

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(1) Analysis of reaction diffusion systems with cross-diffusion terms : We have discussed reaction diffusion systems with cross-diffusion and reaction of Lotka-Volterra type. These systems appear in mathematical biology. Mathematically, it is very important to derive sufficient conditions for the existence of time-global solutions and get information on the structure of positive stationary solutions (biologically, coexistence states). As to the non-stationary problem a global existence result has been obtained in one and two space-dimensions. For the stationary problem with zero Dirichlet boundary condition, we have studied uniqueness and non-uniqueness of positive stationary solutions as well as sufficient conditions for their existence. It is proved that our system admits multiple existence of postive solutions. Moreover, numerical simulations exhibit complicate structure of positive stationary solutions such as bifurcation of symmetric solutions from semitrivial solutions and, additionally, bifurcation of asymmetric solutions from symmetric ones.(2) Analysis of quasilinear parabolic equations with p-Laplacian and logistic terms : Although the nonlinearity and degeneracy of p-Lap
Resarchs on Solutions of Nonlinear Elliptic Equations and Numcrical Analysis

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Head investigator S.Yotsutani proposed a very accurate numerical computation method to solve Poisson's equation by using a charge simulation method with H.Morishita, N.Kobayashi, H.Takaichi and K.Amano. Recently, He has succeeded to shorten the time of the calculation considerably and improve the accuracy with H.Morishita and K.Anjano, This method is in the conformity with the paerallel computing. Thus it is possible to know the detailed shape of solutions of nonlinear elliptic systems.On the other hand, he developed the mathematical method of investigate the shape of radially symmetric solutions with Y.Kabeya and E.Yanagida. Recently, he has found the systematic change of variables to transform the differential equations arising from the elliptic equations to canonical form with E.Yanagida. Thus, relations between various equations studied one by one independently become very clear, and the understandings encourage the deep understanding of the propertites of each solution.The reserach results of investigators are asfollows. T.lkeda invesitigate the instabiliz ation of the standing pulse solutions of bistable reaction-diffusion systems. Y.Morita showed stabilization of vorticies i
Nonlinear Evolution Equations and Elliptic Equations

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Elliptic Equations (1) Concerning the equation (E) - △u = |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω we obtained the following results.Let Ω = RィイD1NィエD1＼BィイD2R1ィエD2, BィイD2RィエD2 = {x ∈ IRィイD1NィエD1 ; |x|【less than or equal】 R }, 2ィイD1*ィエD1<q< +∞ (2ィイD1*ィエD1 is the critical exponent for Sobolev's embedding HィイD31(/)0ィエD3 (Ω) ⊂ LィイD1qィエD1 (Ω) ), then (E) admits a radially symmetric solution in HィイD11ィエD1 (Ω) ∩ LィイD1qィエD1 (Ω). This fact has been conjectured from the duality between bounded domains and exterior domains.(II) Consider the equation : (E)ィイD2λィエD2 -△u = λu + |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω (1) Let Ω = ΩィイD2dィエD2 × λRィイD1N-dィエD1, (ΩィイD2dィエD2 is a bounded domain in IRィイD1dィエD1), q = 2ィイD1*ィエD1, d【greater than or equal】 1, N 【greater than or equal】 4, then for all λ ∈ (0, λィイD21ィエD2), λィイD21ィエD2 = infィイD2v∈HィイD31(/)0ィエD3 (Ω)ィエD2‖∇ィイD2uィエD2‖LィイD42ィエD4ィイD12ィエD1/‖u‖LィイD42ィエD4ィイD22ィエD2 > 0, (E)ィ
Analysis of nonlinear diffusion equations and related phase transition problems

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In our project we have mainly discussed the stationary and non-stationary problems for the following reaction diffusion systems with quasilinear diffusion terms:(E) u_l = Δ[(1 + αv + γu)u] + uf (u, v), v_l = Δ[(1 + βv + δv)v] + vg (u, v).This is a well-known system which models the habitat segregation phenomenon between two species. In (E) u, v denote the population densities and f, g represent the interaction between u and v such as Lotka-Volterra competition type or prey-predator type.(1) Non-stationary problem. When the system has a cross-diffusion effect, the existence result of global solutions was restricted to the two dimensional case. We have proved that, if α, γ > 0 and β = δ = 0, then (E) admits a unique global solution without any restrictions on the space dimension and the amplitude of initial data. Our strategy is to decouple the system and study reaction-diffusion equations separately. We combine parabolic fundamental estimates with energy estimates of solutions of parabolic equation with self-diffusion. This method is also valid for the case δ > 0; so that the global existence is shown when the space dimension is less than six.(2) Stationary problem. From
Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

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(1)"L^∞-energy method" is invented. This assures the high differentiablity of solutions of quasilinear parabolic equations. By this method, the existence of W^<1. ∞>-solutions for a general doubly nonlinear parabolic equations and the open problem : "porous medium equations admit C^∞-solutions?" is solved affirmatively. Recent studies suggest that this gives a quite powerful tool for various problems.(2)"The theory of nonmonotone perturbations for subdifferentials " is extended to Banach space setting. By this theory, we can treat the existence and regularity of solutions for degenerate parabolic equations in a more natural way than Galerkin' s method and open problems, left unsolved in the usual way, were solved.(3)A Concentration Compactness (CC) theory with partial symmetry is given. The usual CC theory is known to be useful to analyze the problem with lack of compactness. On the other hand, the high symmetry such as the radial symmetry often recovers the compactness. It is studied how the partial symmetry not enough to recover compactness is reflected to CC theory. By this theory, the existence of nontrivial solutions is proved for some quasilinear elliptic equations in i
Research of System of Nonlinear Diffusion Equations and Related Elliptic Differential Equations

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In this project, we have studied the structure of solutions for the following two types of equations : (a) reaction diffusion systems with nonlinear diffusion in mathematical biology and (b) semilinear diffusion equations describing phase transition phenomenaThe first problem in mathematical biology is given by a system of differential equations with quasilinear diffusion of the formu_t=Δ[φ(u,v)u]+au(1-u-v), v_t=Δ[ψ(u,v)v]+bv(1+du-v),under homogeneous Dirichlet boundary conditions. Here u and v denote population densities of prey and predator species, respectively. It is well known that the corresponding stationary problem has a positive steady-state under a suitable condition. Our main interest is to derive useful information on profile and stability of each positive steady-state. In case φ(u,v)=1 and 4,φ(u,v=1+β u, we have shown that the stationary problem has at least three positive solutions if β is sufficiently large and some other conditions are imposed. Moreover, stability or instability of each positive solution is also investigated.The second problem is given by u_t=ε^2u_<xx>+u(1-u)(u-a(x)) with homogeneous Neumann boundary condition, where 0<a(x)<1. When ε is su
Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations

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(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is
Research on the structure of solutions for nonlinear systems of reaction-diffusion equations
Analysis of Reaction-Diffusion Systems and Related Nonlinear Problems

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This research project is concerned with the mathematical formulation of non-uniformity of species in mathematical ecology such as the segregation of two competing species and the spreading of invasive species. These phenomena are described by reaction-diffusion equations with population densities as unknown functions. We have obtained satisfactory results on the structure of positive steady-states for two-species models with nonlinear diffusion and the mechanism of spreading and vanishing for free boundary problems in invasion models

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Misc

Free boundary problems for reaction-diffusion equations arising in ecology

Yoshio Yamada

2022.07
A fee boundary problem for reaction diffusion equation with positive bistable nonlinearity

Maho Endo, Yuki Kaneko, Yoshio Yamada

RIMS Kokyuroku 2121 29 - 40 2019.07

Rapid communication, short report, research note, etc. (bulletin of university, research institution)
Spreading, vanishing and singularity for radially symmetric solutions of a Stefan-type free bounary problem (Developments of the theory of evolution equations as the applications to the analysis of nonlinear phenomena)

Yuki Kaneko, Yoshio Yamada

RIMS Kokyuroku 1997 86 - 95 2016.07

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
On Logistic Equations with Diffusion and Nonlocal Terms (Progress in Qualitative Theory of Ordinary Differential Equation)

Yoshio Yamada

RIMS Kokyuroku 1901 121 - 134 2014.06

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
On a population model witha free boundary and related elliptic problems (Progress in Qualitative Theory of Ordinary Differential Equations)

Yuki Kaneko, Yoshio Yamada

RIMS Kokyuroku 1901 69 - 78 2014.06

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
On mathematical model with cross-diffusion in ecology (Mathematical Sciences of Nonlinear Diffusion), (in Japanese)

Yoshio Yamada

RIMS Kokyuroku 1810 39 - 58 2012.10

Rapid communication, short report, research note, etc. (scientific journal)
A free boundary problem modeling the invasion of species (Analysis on no-equilibria and nonlinear phenomena: from the evolution equations point of view)

Yuki Kaneko, Yoshio Yamada

RIMS Kokyuroku 1792 108 - 117 2012.05

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
On logistic diffusion equations with nonlocal effects

Yoshio Yamada

Proceedings of Seminar on Partial Differential Equations in Osaka 2012 --in honor of Professor Hiroki Tanabe'2 80th birthday 189 - 198 2012

Research paper, summary (national, other academic conference)
Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion (Evolution Equations and Asymptotic Analysis of Solutions)

Yoshio Yamada

RIMS Kokyuroku 1358 24 - 33 2004.02

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
On one-phase Stefan problems for sublinear heat equations

Toyohiko Aiki, HItoshi Imai, Naoyuki Ishimura, Yoshio Yamada

Proceedings of the 3rd Asian Mathematical Conference, 2000, World Sci. Publ. 6 - 11 2002

Research paper, summary (international conference)
Multiple Coexistence for Lotka-Volterra Competition Model with Diffusion (Nonlinear Diffusive Systems: Dynamics and Asymptotics)

Yoshio Yamada

RIMS Kokyuroku 1178 167 - 180 2000.12

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
Asymptotic behavior of solutions to one-phase Stefan problems for sublinear heat equations (Nonlinear Diffusive Systems: Dynamics and Asymptotics)

Toyohiko Aiki, Hitoshi Imai, Naoyuki Ishimura, Yoshio Yamada

RIMS Kokyuroku 1178 36 - 47 2000.12

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
Uniqueness of solutions for a parabolic equation with sublinear term (Nonlinear Evolution Equations and Their Applications), (in Japanese)

Yoshio Yamada

RIMS Kokyuroku 1135 129 - 139 2000.04

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
Analysis of nonlinear phenomenon: bifurcation and travelling wave

Hideaki Oshiro, Shingo Takeuchi, Yoshio Yamada

Riko Journal 1 ( 4 ) 1997

Article, review, commentary, editorial, etc. (bulletin of university, research institution)
Chemical interfacial model: Asymptotic behavior of solutions (Evolution Equation and Its Applications to Nonlinear Problems), (in Japanese)

Masato iida, Yoshio Yamada, Shoji Yotsutani

RIMS Kokyuroku 755 171 - 184 1991.06

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
Evolution equations and their applications to partial differential equations

Yoshio Yamada

1 - 19 1990.04

Lecture material (seminar, tutorial, course, lecture, etc.)
On global solutions for semilinear parabolic equations of higher-order (Evolution Equation and Its Applications), (in Japanese)

Hiroki Hoshino, Yoshio Yamada

RIMS Kokyuroku 698 14 - 35 1989.06

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
$L^p-L^q$ 評価とその応用

山田義雄

第7回発展方程式若手セミナー報告集 20 - 37 1986.04
On a Free Boundary Problem in Ecology

Masayasu Mimura, Yoshio Yamada, Shoji Yotsutani

North-Holland Mathematics Studies 128 ( C ) 107 - 125 1985.01 [International journal]

Research paper, summary (international conference)

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This chapter discusses a free boundary problem for semilinear parabolic equations, which describes the habitat segregation phenomenon in population ecology. The chapter illustrates the global existence, uniqueness, regularity, and asymptotic behavior of solutions for the problem. The stability or instability of each stationary solution is completely determined using the comparison principle. © 1985, Elsevier Inc. All rights reserved.

DOI
Study of free boundary problems arising in ecology

Yoshio Yamada

145 - 153 1984.06
遅れのある半線型拡散方程式の安定性と周期解の分岐について

山田義雄

Mathematics of Oscillatory Phenomena '82 at Hiroshima 15 - 21 1983.03

Rapid communication, short report, research note, etc. (bulletin of university, research institution)
ある種の自由境界条件をもつ放物型方程式

山田義雄

第4回発展方程式若手セミナー報告集 77 - 85 1983.02
On some semilinear Volterra diffusion equations arising in ecology

Yoshio Yamada

Equadiff 5 (Bratislava, 1981), Teubner-Texte Math. 47 370 - 373 1982

Research paper, summary (international conference)
On semilinear Volterra diffusion equations (Mathematical Topics in Biology; '80 December), (in Japanese)

Yoshio Yamada

RIMS Kokyuroku 420 77 - 92 1981.03

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

CiNii
積分項をもつ半線型拡散方程式について

山田義雄

第2回発展方程式若手セミナー報告集 56 - 64 1980.11
On quasilinear wave equations (Applied Analysis for Partial Differential Equations in Applied Sciences), (in Japanese)

Yoshio Yamada

RIMS Kokyuroku 386 53 - 69 1980.05

Rapid communication, short report, research note, etc. (bulletin of university, research institution)
準線型波動方程式と非線型発展方程式の関連

山田義雄

第1回発展方程式若手セミナー報告集 63 - 68 1980.01
非円筒領域における非線型放物型方程式について

山田義雄

非線型発展方程式論，セミナリー・ノート 82 - 106 1977.02

Rapid communication, short report, research note, etc. (bulletin of university, research institution)
On evolution equations governed by subdifferentials (Evolutional System and Free Boundary Problems), (in Japanese)

Yoshio Yamada

RIMS Kokyuroku 264 176 - 192 1976.02

Rapid communication, short report, research note, etc. (bulletin of university, research institution)

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Overseas Activities

数理科学の諸分野に現れる反応拡散方程式系の解析

2011.09

-

2012.03

オーストラリアニューイングランド大学

Internal Special Research Projects

数理生態学に現れる自由境界問題の研究

2015 兼子　裕大, 松澤　寛

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  We have studied a free boundary problem for a certain class of reaction-diffusion equations. Such a problem models the invasion or migration of a biological species which moves toward a new habitat.  The problem has two unknown functions: one is the population density of the species and the other is (a part of ) the boundary of its habitat. The population density is governed by a reaction-diffusion equation and the moving boundary is controlled by Stefan condition.  When a reaction term has two stable and positive equilibrium states, some numerical simulations exhibit different large-time behaviors from known ones.  We have succeeded in getting various theoretical results such as the classification of asymptotic behaviors of solutions into four patterns and the derivation of speeds of spreading free boundaries as time goes to infinity. These result help us to understand the invasion model from the mathematical view-point.
積分項を伴う反応拡散方程式の研究

2013

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　　数理生態学に現れる重要な微分方程式の一つにロジスティック型反応拡散方程式がある．これは生物種の個体数密度 u の変化を記述する方程式で(1) u_t=dΔu+u(a-f(u))の形に記述されるものである．このタイプの方程式の解について興味深いのは、時間的あるいは空間的な非一様性を伴う時空パターンが観測されることである．このようなパターンは生態学的には、棲み分け現象や周期的な個体数変動を意味し、その出現メカニズムを理解することは数学的にも生態学的にも面白いテーマであり、近年活発に研究されている．ただし、従来の研究は(1)のような局所項のみで記述される方程式に限定されていた．しかし、実際の生物の移動・拡散においては、視覚、聴覚の効果が重要な役割を果たすことも多いし、なかには生物種が出す化学物質が影響をもたらすことも多い．このような状況を考慮すると、定式化にあたっては非局所的な相互作用が重要になり、(1) の方程式は(2) u_t=dΔu+u(a-f(u)-k*g(u)), ただし k*g(u)(x)=∫k(x,y)g(u(y))dyの形の、積分項を伴う反応拡散方程式となる．　本研究においては、積分核 k や、相互作用を表す f,g に適当な条件を課し、(2)に境界条件および正値関数の初期条件を設定して考える．この初期値境界値問題に対し、必ず時間大域解が唯一つ存在することを示すことができる．次のテーマは、そのような大域解の時間無限大での漸近挙動を調べることである．一般に、時間無限大での解の漸近挙動には定常問題が密接に関連する．研究成果の一つは、定常問題の正値解を構成する方法として、2通りの方法を開発したことである．一つは分岐理論に基づく基本的な方法で、もう一つは非線形固有値問題ともいうべき、非常に初等的な方法である．また、定常解が安定であるかどうかを判定することは、非定常問題の解の漸近挙動に関わる重要なテーマであるが、有効な一般論がないため、未解決である．本研究では g(u)=bu^2 かつ k が正値核の場合には、正値定常解は安定であることの証明に成功した．これらの結果は2013年5月同済大学（上海）および2013年11月京都大学数理解析研究所での研究集会において講演発表している．
非線形拡散方程式系および関連する界面問題の解析

2002

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今年度の研究テーマは、次のような準線形拡散項を含む反応拡散方程式系　　　　　∂u/∂t=Δ[(1+αv+γu)u]+uf(u,v), ∂v/∂t=Δ[(1+βu+δv)v]+vg(u,v)　　　　に対する非定常問題の解析である。このシステムは同一領域で生存競争する2種の生物について、棲み分け現象を記述する数理モデルとして、1979年にShigesadaらのグループによって提案されたものである。未知関数 u,v は個体数密度を表わし、非線形拡散項は通常の拡散に加え、個体数密度にも拡散が依存することを意味している。反応項 f,g は u,v 間の相互作用を表し、Lotka‐Volterra型の競合モデルを扱う。このシステムについて、ゼロNeumann境界条件を課すと、初期値境界値問題の時間大域的な解の存在について、従来知られている結果は空間次元が２以下のケースに限られていた。また、空間次元についての制約をはずそうとすると、反応項に関する仮定が必要であった。本年度は R.Lu, P.S.Choi氏らとの共同研究によって、α,γ > 0 の場合、もう一方の方程式の拡散項が線形（β=δ=0)ならば、空間次元や初期データの大きさと無関係に時間大域解が一意的に存在することを示すことができた。うまくいった理由は、システムを準線形放物型方程式と半線形放物型方程式に分解し、それぞれの方程式についての基本解評価とself-diffusion 項をフルに活用して解 u,v のアプリオリ評価が得られた点にある。この方法は δ > 0 のときにも適用することができ、空間次元が5以下の場合に大域解の一意的存在を示すことができた。　以上の結果は[1] Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion, to appear in Discrete and Continuous Dynamical Systems.[2] Y. S. Choi, R. Lui and Yoshio Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, to appear in Discrete and Continuous Dynamical Systems.で発表される予定である。
反応拡散方程式系および関連する界面問題の研究

2000

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　本年度の研究成果は主として次のような２つの未知関数u，vに関する反応拡散方程式系　　　　　　　∂u/∂t=μΔu+f(u，v)， ∂v/∂t=νΔv+g(u，v) in Ω×(0，∞)　 Ω: 境界∂Ωで囲まれた領域の正値定常解集合の構造に関するものである。この方程式系は数理生態学分野では、生存競争をしている２種類の生物の個体数密度u，vの変化を記述し、ラプラシアンΔは拡散効果を表す。反応項f，gはu，v間の相互作用を記述し、２種類の生物が競合関係にあるかまたはprey-predator関係にあるかにより関数関係は異なってくる。また、正値定常解は数学的に重要な項であるのみならず、生態学的にも共存解として大きな意味がある。競合モデルについてf，gがLotka-Volterra型の関数として与えられているときは、非常に多くの研究者によって研究されており、例えば正値定常解が存在するための十分条件は広く知られている。しかし、これに反して、解の一意性･非一意性に関する研究は困難を伴い、Dancerらによる研究などいくつかの研究はあるものの十分に満足できるものではない。著者は、数年前から正値定常解の多重性に関する研究に取り組み、いかなる条件下で解が複数個存在しうるか調べている。その結果、拡散係数μ，νをパラメータ空間の点とみなした場合、これらが非常に小さい場合、およびu，v間の相互作用に大きな差があるときにはμ，νが一定の範囲にあるときには正値定常解が２個以上存在することが理論的に明らかになってきた。しかも、空間次元が1のときには、数値解析的にも複数個の解を見出すことができ、解のプロフィールについてもある程度明らかにすることができた。これらの結果は、イタリア・シチリアで開催された「第３回非線形解析学者の国際会議」や日本で開催された「第１回東アジア非線形偏微分方程式シンポジウム」で講演発表した。　なお、秋以降はさらに反応項をより一般化し、f，gがLotka-Volterra 型の以外の関数で与えられるときに定常解集合の構造についてどんな結果が得られるか調べている。たとえば、fがFitzHugh-Nagumo型の3次関数で与えられるとき、正値定常解が存在するための十分条件や、定常解の作る集合の構造も大きく変化することがわかる。このような結果を導くために、現在は分岐理論やコンパクト写像に対する写像度の理論を用いているが、限界もあるため数値実験による解析も視野に入れている。
反応拡散方程式系および関連する楕円型方程式系の解集合の研究

1999

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　本年度の研究成果は Lotka-Volterra 型反応拡散方程式系の正値定常解の研究と sublinear 項を伴う半線形放物型方程式に対する比較定理の研究の２つに大別される。(I) Lotka-Volterra 型の競合モデルに対する反応拡散方程式系の正値定常解は、数理生態学分野においては共存解として大きな意味がある。本研究においては同次ディリクレ境界条件下で共存解が存在するための十分条件を調べるとともに、その一意性・非一意性や安定性について理論的および数値解析的両観点から調べた。とりわけ線形拡散のケースで相互作用の強さを表す係数がある一定の条件をみたすとき、共存解が複数個現れる可能性のあることが明らかになった。より詳しく述べると、２つの拡散係数をパラメータ空間内の点とみなすと、必ず共存解が２個以上現れる範囲のあることを示すことができた。これは、正値定常解の多重性に関する既存の結果に新しい視点を与えるものとなっている。また、空間次元が１の場合に限定すると、相互作用の係数が非常に大きい場合には解の多重性について非常に詳しい情報が得られることも判明してきた。さらに数値解析によって正値定常解を構成するために、従来はいわゆる shooting method を用いてきたが、微妙な解析において精度が低かった。これを改善するために新しい方法として Newton 法を応用したスキームを試みている。(II) sublinear 型の反応項を含む半線形放物型方程式については一般には解の一意性が成立しなくなる。そこで滑らかな関数 f と指数 0＜q＜1 に対して同次ディリクレ条件のもとで∂u/∂u＝Δu＋uｑ＋f(u)の形の方程式に対する非負値解を考え、優解と劣解による比較定理を証明することに成功した。この結果、対応する定常問題の正値解の安定性･不安定性に関し有用な情報が得られる。例えば、空間次元が１のときには上記方程式の正値定常解は相平面の方法ですべて構成することができ、各解の形状も明らかになる。比較定理を適用することにより、それぞれの定常解の安定性、より詳しくは漸近安定性や不安定性を調べることができる。
非線形拡散方程式系および関連する楕円型方程式系の解集合の研究

1998

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　上記研究課題のもと集中的に扱った方程式は非線形拡散項をもつ反応拡散方程式系のうちUt＝△[(1＋αu＋βv) u]＋u (a－bu－cv)、Vt＝△[(1＋γu＋δv)v]＋v (p－qu－rv)の形のものである。これは競合する２種類の生物の固体数密度u、vの時間変化を表現したロトカ・ボルテラ型方程式に、固体数密度に依存する非線形拡散項を加えたものである。生物学的には、棲み分け現象を記述するモデルとして提起されたものであるが、数学的にも初期値境界値を与えたときの大域的可解性や定常解の構成とその形状や安定性の解析が重要な問題となる。得られた成果は(1)非定常問題の大域的可解性と(2)定常解集合の構造の二つに大別される。まず、(1)については空間次元が１または２のときに限定されるが、Yagiによって得られた大域解存在のための十分条件を拡張することに成功した。今後の課題は空間次元が３以上のケースでの解析である。(2)については、同次ディリクレ条件のもとで正値解(共存解とも呼ばれる)がいかなる状況で存在するか、またその個数はどうなるか、に焦点を絞った。定常問題は準線形楕円型方程式系の形になるが、適当な変数変換によりコンパクト写像に対する不動点を求める問題に帰着される。したがって、正値解はDancerによって開発された正錘上の写像度の理論により求められる。この結果は分岐理論の立場からも解釈することができ、共存解のつくる分岐枝がどのようにのびていくかもわかる。さらに、局所的分岐理論と写像度理論を組み合わせて２個以上の共存解が存在するための条件をわかりやすい形で導くことにも成功した。これらの理論的結果を数値実験により検証すると、共存解の集合は非線形拡散の効果により非常に複雑な様相を呈していることが示唆される。定常解の形状や安定性なども含めて今後も解析を要する課題が多い。
非線形放物型方程式系および関連する楕円形方程式系の解集合の研究

1997

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上記課題のもと、［1］反応項をもつ退化放物型方程式の定常解集合の構造と安定性の解析、［2］協力系と呼ばれる反応拡散方程式系の共存解に関する研究を行った。［1］通常の線形拡散項と反応項からなる放物型方程式の解の基本的性質や定常問題の解集合の構造については、Chafee-Infante問題としてきわめて詳細に解析されている。それでは、線形拡散項を退化型非線形拡散項のｐ-ラプラシアンで置き換えたとき、解集合の構造や解のプロフィールにいかなる影響がもたらされるだろうか？空間１次元の場合に限定されるが、線形拡散のケースと比較して興味ある相違がみられた。ひとつは、拡散項と反応項の次数の違いに応じて定常解の集合の分岐構造が大きく異なることである。もうひとつは、退化性により定常解の中にはフラットな形（フラット・ハット）を持つものが現れることである。このため解集合は離散的ではなくなり、比較定理による定常解の安定性の証明もある部分では困難になる。更にフラット・ハットの時間的な変化を知ることも今後の重要な課題となる。［2］協力系と呼ばれる半線形拡散方程式系について、３成分の未知関数から成る場合を調べた。このような問題は数理生態学や化学反応などの分野にあらわれる。とくに、興味があるのは、正値定常解（共存解）であり、いかなる条件下で共存解が現れるか？その個数や安定性はどうか？などの問題が解析の対象となる。線形化方程式に対する強最大値原理を利用して、私達のグループは共存解が存在するための必要十分条件を求め、同時に共存解の一意性と大域的安定性を示すことができた。研究成果の発表1997年７月Coexistence states for some population models with nonlinear cross-diffusion, Forma, Vol.12, 153-166.1997年９月Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian, Technical Report No. 97-11, Waseda Univ.
反応拡散方程式系および関連する楕円型方程式系の解集合の研究

1995

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当研究では反応拡散方程式系のなかでも，とくに相互作用が低階の非拡散項のみならず，高階の拡散項にも非線形拡散として影響を与えるようなモデルを中心に有界な大域解の存在や関連する定常問題の解集合の構造を調べた。数理生態学における "biodiffusion" のなかには "cross-diffusion" と呼ばれる重要な非線形拡散がある。同一の領域で生存競争している2種以上の生物の固体密度を未知関数として定式化すると，"cross-diffusion"の効果により，拡散係数が固体密度にも依存するような準線形拡散方程式系となる。このようなモデルは1979年に提起され，数値実験では分岐やパターンの形成などの興味深い現象が見られるにもかかわらず，理論的なメカニズムの解明は十分ではない。とりわけ，正の定常解は共存解として重要であり，いかなる条件で存在するか，安定であるか，一意的に決まるかなどその解明が待たれる。筆者の研究グループでは，一昨年度，正の関数が作る錐集合上の写像度の理論を適用することにより，正の定常解が存在するための十分条件を発見することに成功した。昨年度の研究においては，分岐理論を組み合わせることにより，定常解はいかなるときに安定となるか，またいかなる状況で2種類以上の共存状態が起こり得るか，より深い理解をめざした。"prey" と"predator" の生存競争モデルにたいしては，"cross-diffusion" の影響による拡散が大きくなればなるほど，適当なパラメータ空間における共存領域が小さくなる，などの点で非線形拡散が共存状態に対して負の作用をすることが示された。また，分岐の方向を調べることにより，複数個の共存解もあり得ることが知られてきている。これらの結果は，"Positive steady-states for prey-predator models with cross-diffusion"（Adv.Differential Equationsに掲載予定），"Coexistence states for some population models with nonlinearcross-diffusion"（Formaに掲載予定）らの論文にまとめられている。

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