Authorship：Principal investigator 

Grant amount：\3640000 （ Direct Cost: \2800000 、 Indirect Cost：\840000 ）

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Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）

Reaction-diffusion systems are mathematical models describing the dynamics of the population densities of particles which appear, disappear and interact each other. Numerical simulations suggest that various dynamic patterns of the population densities appear in reaction-diffusion systems. However, this suggestion has not been proved completely, and thus the theories on reaction-diffusion systems have not been well developed yet.
Our objectives is to clarify how the following two dynamic patterns are generated in reaction-diffusion systems: the pattern with a corner layer; the stairs-like pattern every step of which moves forward with a different speed from each other. In the present investigation we prepare some basic theories which will be useful for the construction and analysis of asymptotic solutions which approximate to those two patterns.

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Authorship：Principal investigator 

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

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Authorship：Principal investigator 

In two-dimensional excitable media, spirals may be formed spontaneously due to the influence of obstacles. For the mathematical understanding of the mechanism of spontaneous spiral formation, we introduce a new free boundary problem that is derived from the modified FitzHugh-Nagumo equation as a singular limit. We prove the existence of traveling spots of this system, which play important role on the spontaneous spiral formation by obstacles. We also study the influence of several obstacles. As more general setting, we consider the inhomogeneous excitable media. We pointed out that the unidirectional failure of propagation is a key of the spiral reentry.

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

In reaction-diffusion systems various shapes of solutions were observed by numerical simulations, however most of them have not rigorously been verified yet. Through this research much information, that will help us to construct asymptotic solutions approximating solutions with `corner layer' and solutions which describe `multi-stage invasion' in population dynamics, have been obtained as follows. (1)Some united viewpoints over several reaction-diffusion systems have been introduced, in order to describe the shapes and the structure of the solutions in their singular limits. The viewpoints will help us to decide whether corner layers do appear or not in some singular limits. (2)The global structure of `single waves' in the Fisher-KPP equation have been shown in collective known facts concerning their existence and stability. Asymptotic solutions which describe multi-stage invasion in cooperation-diffusion systems with many species will be constructed of these single waves.

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

We have deepened our original method and extended its range in order to solve various problems for differential equations. We obtain all the candidate solutions and derive transcendental equations equivalent to problems to be solved.
Especially, as for a cross-diffusion equation,we obtained the stability of stationary solutions in multidimensional case, which is inspired by 1 dimensional case results obtained by our method.
As for linearized eigen-value problems for reaction diffusion equations, we have obtained all exact values of eigen-values and representation formula of eigen-fuctions which have been thought impossible.

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Authorship：Principal investigator 

To construct the quasi-solutions of non-equilibrium growth models, we considered the spontaneous emergence of spirals in an excitable medium. Ueyama et al and his co-workers exhibited the spiral formation influenced by the non-uniform environment using photosensitive BZ reactions and simulations. Ninomiya and his co-workers studied the wave front interaction model as the simplied model of the excitable system and constructed the rotating spots and spirals. These studies enabled us to construct the traveling spots of the singular limit problem of FitzHugh-Nagumo type systems. It revealed the mechanism of spontaneous emergence of spirals in two-dimensional excitable medium.

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

We have been developed the original method for differential equations. We obtain all exact solutions and derive transcendental equations equivalent to the global bifurcation structure of the given differential equations. We have developed this method and applied it various problems. Especially, as for a cross-diffusion equation in 1 dimensional case, we have discovered various devises to analyze transcendental equations, and obtained the method to show the global bifurcation structure. Furthermore, we have started to investigate the stability of stationary solutions including multidimensional case. We have also reveal the blowup phenomena of curvature for plane elastic closed curves and the global structure of all curves.

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Grant amount：\3200000 （ Direct Cost: \2900000 、 Indirect Cost：\300000 ）

The reaction-diffusion equation is widely accepted as a model to describe pattern formations and propagations of some spatial pattern. In this research we study the equation in order to provide a new existence theory for entire solutions and a characterization of the spatial pattern of the solutions. In consequence we mathematically prove the existence and stability of new type of solutions to some class of reaction-diffusion equations.

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

Some free boundary problems describing the spatial segregation of competitive species can be regarded as certain singular limits of some reaction-diffusion systems. The mathematical grounds of this fact have been obtained. It is also shown that a mathematical model describing multi-phase invasion of cooperative species can be approximated by stacked fronts of several Fisher's waves which are mathematical models describing the single invasion of a species.

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

(1) We have shown that, for the Cauchy=Kowalevskaya theorem on a system, it is necessary that the system is similar to Kowalevskian accepting poles on symbols. On the sufficiency, we have succeeded to prove it on some models, but we need refine on the normal form in order to treat the general case. (2) On the systems with coefficients depending only on the time variable, it is necessary and sufficient for the strong hyperbolicity that the system is uniformly diagonalizable as a hyperbolic Fuchsian. (3) The study under the new definition of the p-parabolic system is not satisfactory.

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Grant amount：\47450000 （ Direct Cost: \36500000 、 Indirect Cost：\10950000 ）

We carried out the investigation about the structure of solutions of nonlinear parabolic and elliptic equations. Our main results are as follows : Next, we studied the existence and uniqueness of solutions with moving singularities for a nonlinear parabolic partial differential equation. We also showed that there exists a solution with a moving singularity that changes its type suddenly., and made clear the asymptotic behavior of singular solutions that converges to a singular steady state. We also studied a chemotaxis system, and made clear the structure of self-similar solutions that blows up by concentrating to a point in finite time.
For a reaction-diffusion system, which is called a Gierer-Meinhardt system, we studied the mathematical structure of pattern formation, and also made clear the behavior of time-dependent solutions.

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Grant amount：\4030000 （ Direct Cost: \3400000 、 Indirect Cost：\630000 ）

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

We explain principal results.
Let ω be an integrable holomorphic one form defined in a neighborhood ∪ of the disk D^<2n>(1)⊂C^n, n【greater than or equal】2. Assume that the holomorphic foliation F(ω) of codimension one defined by ω is transverse to the boundary S^<2n-1>(1) of D^<2n>(1). By Mobius transformation, we can suppose that the only one singular point of ω inside D^<2n>(1) is the origin 0.
Theorem([6]) If F(ω) has a leaf L such that L has the following properties (i)〜(iii), then n=2.
(i) 0∈L^^-, (ii) L is closed in ∪＼Sing(ω), (iii) L is transverse to each shpere S^<2n-1>(r), 0<r【less than or equal】1.
Let ω be a holomorphic one form defined in a neighborhood ∪ of D^<2n>(1)⊂C^n, n【greater than or equal】3 such that Sing(ω)_∩S^<2n-1>(1)=φ. Let ξ be a holomorphic vector field defined in ∪.
Theorem([7]) If ω(ξ)=0 and ξ is transverse to S^<2n-1>(1), then ω is not integrable.
Theorem([8]) Let X be a polynomial vector field on C^n, n【greater than or equal】2 with isolated singularities. If the holomorphic foliation F(X) defined by solutions of X on CP (n) has singularities of hyperbolic type, the following conditions are equivalent.
(i) F(X) has n separatrices on C^n and is transverse to a sequence of spheres S^<2n-1>(p_j, R_j)⊂C^n where <lim>___<j→∞> R_j=+∞.
(ii) X is linear (of Poincare hyperbolic type) in some affine chart on C^n.

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Grant amount：\3200000 （ Direct Cost: \3200000 ）

3重結節点のみからなるネットワークf構造を持つ界面に対し,各弧が平均曲率流に従い,3重結節点ではヤングの法則を満たし,境界とは直交するように接している場合について,界面がどのようなダイナミクスに従うかについて研究を進めた.特に,考える領域の境界の曲率がダイナミクスに及ぼす影響について考察した.本年度は特に,3重結節点が複数ある場合について,定常解の存在条件を明らかにし,その情報から安定性を決定する方法について研究を進め,以下のような成果を上げた.
まず,定常界面の安定性に関する理論的考察を元に,3重結節点が1個の場合について得られている存在条件を帰納法を用いて一般化することにより,複数個の3重結節点を持つ界面の存在条件を明らかにした.さらには,条件を境界からの寄与と内部構造からの寄与の項に分解し,境界の曲率の符号と不安定性次元の関係を明らかにした.
次に,与えられた領域において,負の長さを許した場合の定常状態が存在するための条件と安定性との関わりについて調べた.これは古典的なFermat-Steiner問題と関係する興味深い変分問題であるが,境界条件の違いから自由度の高い難しい問題となる.これまでの研究により,凸領域において複数個の3重結節点を持つ界面に対して定常状態の存在が示されているが,ここでは非凸な領域についても研究を進め,変曲点で退化しない場合には定常解は退化せず,従って双曲的な性質を持つことを明らかにした.
その他,ネットワーク上の領域における線形固有値問題の主固有値の最小化問題,熱方程式の解の臨界点の位置に関する研究を行い,その基本的な性質について詳細に調べた.

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

For the understanding of various nonlinear phenomena observed in nonlinear diffusive systems, a key is to analyze the mechanism for the appearance of singularities such as blow-up, concentration, and transition layers. In this research, we aim to develop new methods for the analysis of stable pattern formation, pattern dynamics in higher dimensional space, and long-time behavior of scalar reaction-diffusion equations.
First, we studied the stability of stationary solutions in shadow systems, and showed that the convexity of spatial structure plays an important role under the assumption that nonlinear term has skew-gradient structure.
Second, we studied the behavior of solutions of Fujita-type equations, and made clear the relation between the convergence rate of solutions and decay rate of initial data.
Third, we studied a minimization problem for the principal eigenvalue of elliptic problem with indefinite weight. We showed the minimizer must be of bang-bang type, and obtained a minimizer in the case of one-dimensional space. We also considered the case of cylindrical domain, and found that a sort of symmetry-breaking must occur.

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Grant amount：\8400000 （ Direct Cost: \8400000 ）

1.We studied a one-dimensional Ginzburg-Landau equation in a ring, which is a mathematical model in a superconducting wire. When the wire is uniform, we revealed the global bifurcation structure for the two physical parameters and determined which solutions are minimizer of the energy functional. We also studied the configuration of the phase of solutions to the Ginzburg-Landau model in the wire with non-uniform thickness.
2.We studied how the solution structure of a nonlinear equation is affected by the geometry of a domain. This approach would be developed to the Ginzburg-Ladau equation.
3.An asymptotic behavior of the time evolutionary Ginzburg-Landau equations was studied. Some spectral result for the linearized operator of the equations was also obtained
4.A variational method to the transition layer problem in reaction-diffusion equations was developed. This approach would be applied to a model of the superconductivity.
5.Numerical computations for a BEC model and several Ginzburg-Landau models were achieved. We also discovered new pattern-dynamics arising in such nonlinear dissipative systems. In particular we proved the existence of solutions related to dynamics of front waves to reaction-diffusion equations.

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Grant amount：\3700000 （ Direct Cost: \3700000 ）

反応拡散系から現れる樹状形状を研究するため,まず,結晶成長に現れる進行波解について調べた.そのため,結晶成長のモデルの一つであるアレン・カーン方程式を扱った.空間2次元における進行波解の存在およびその漸近安定性について示し,東京工業大学・谷口助教授と共著論文にまとめた.研究目的の一つである異方性を含むモデルに関しては,異方性と外力を含む曲率流モデルのV字型進行波解の存在を示した.また,異方性が強くなると,進行波解は,だんだんと折れ線に近づき,クリスタライン運動に収束することも証明した.
常微分方程式系の解は,すべて有界となるような系であるにもかかわらず,拡散項を付けた反応拡散系には,有限時間で爆発するような解が存在することを「拡散誘導爆発」という.拡散という空間均一化の効果を加えることにより,空間非一様性がどんどん大きくなり,有限時間で爆発してしまう現象であり,パターン形成との関係が深い.この現象は,一体どのような非線形性に対して起きるのか調べるために,常微分方程式系における非線形項と線形項の関係を調べた.ミネソタ大学H.Weinberger氏との共同研究により,非線形項がp次斉次式の場合に線形項が大域的挙動にどのような影響を与えるのかを調べ,論文にまとめた.斉次項をもつ常微分方程式系でも,線形項が様々な影響を与えることがわかった.
拡散だけでなく境界条件も爆発に影響を与える.M.Fila氏,J.L.Vazquez氏との共同研究により斉次ノイマン境界条件では爆発する解があるが,斉次ディリクレ境界条件では爆発しないような系の構成に成功した.

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Grant amount：\7500000 （ Direct Cost: \7500000 ）

Masaharu Nagayama (one of investigators of the present research project) has devised a computer code that can analyze bifurcation structures in a neighborhood of double bifurcation points. This code deals with bifurcation phenomena of pulse solutions to mono-stable reaction-diffusion systems, and is equipped with the following two functions : (1)It can find a critical point and construct its bifurcation branch, (2)It can extend existing bifurcation branches. In order to devise the code, we consider the reaction-diffusion system on a finite interval (0,L) subject to the periodic boundary condition where L is a fixed large positive number. From the phase condition we obtain the equation that determines the propagating velocity of traveling pulse, and by the Keller method we express the dependency on a parameter p included in the equation systems. The problem formularized as in the above is numerically solved by the Newton method in the computer code. We note that a solution is a set of {solutions to reaction-diffusion systems, c, p}. When a traveling pulse bifurcates from a standing pulse, there appear two zero-eigenvalues, one of which is a trivial one trivial one corresponding to parallel translation. Our code applies to not only this case but also the cases where two crucial zero-eigenvalues exist except the trivial one.
The head investigator have dealt with standing and traveling combustion pulses of a mathematical model for self-propagating high-temperature syntheses including both the cooling effect and raw material supply system. Employing a piece-wise constant function for the reaction term, we have studied the existence of pulse solutions in a mathematically rigorous way, and also the collision dynamics of combustion pulses on a circle domain.

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

First, Lou-Ni-Yotsutani [DCDS 2004] investigated a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions and revealed the structure of all solutions.
This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method, which are the combination of the modern method of PDE and classical analysis and algebra.
There are a lot of interesting problems for which our method is applicable.
A problem of the Oseen's spiral flow is one of them, for which we obtained the complete bifurcation diagram in Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003].
Matsumoto-Murai-Yotsutani [Pisa, 2005] gave the complete answer for a problem to determine curves with the least energy under the given length.
Second, Kosugi-Morita-Yotsutani [CPAA 2005, J.Math.Phy. 2005] have revealed the complete Global bifurcation branches one dimensional Ginzburg-Landau equations with periodic boundary conditions.

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Grant amount：\1700000 （ Direct Cost: \1700000 ）

We have three themes. The first is to establish the necessary and sufficient condition for the Cauchy-Kowalevskaya theorem for systems of partial differential equations including the generalization to the, Nagumo type. On the original Cauchy-Kowalevskaya theorem, we obtained clearer proof. On the theorem of Nagumo type, we obtained a proof on the necessity and also a proof on the sufficiency in a special case. The second is the characterization of the strong hyperbolicity on systems. On the example which is pointwisely diagonalizable but not strongly hyperbolic by Petrovsky, we have already known that it changes to a strongly hyperbolic system by any hyperbolic perturbation. We showed that this phenomenon occurs generally for the systems with time-dependent coefficients. To show this, we apply the solvability of the Cauchy problem for the systems of Fuchs type. We also succeeded the generalization of the structure of the solvability. The third is the solvability of the Cauchy problem for p-parabolic systems to the future. Unfortunately, we obtained an idea to solve this problem, but finally we cannot achieve it as the complete form. In these researches, the comparison between the calculation on the non-commutative ring of the meromorphic formal symbols and that on the holomorphic pseude-differential operators has played an essential role.
At first, the Kac problem is not the theme of this project. We obtained a good knowledge on the non-commutative groups through the research on the determinatnt theory on non-commutative ring and it brings a viewpoint on the framework of the existence of the counterexamples on the Kac problem by Sunada. As a result, we obtained concrete example of the domains which change from convex to nonconvex smoothly and for which the Kac problem is affirmative by proving mathematically Watanabe's conjecture by the numerical try. This is the first offer of a nonconvex domain for which the Kac problem is affirmative.

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

First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida.
Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.
Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida.
As related problems, we are investigating a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.-M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.

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Grant amount：\11300000 （ Direct Cost: \11300000 ）

A helical wave is observed in self-propagating high-temperature syntheses (SHS), for instance. One can create a high-quality uniform product by the SHS when a combustion wave keeps its profile and propagates at a constant velocity. When experimental conditions are changed, however, the planar traveling wave may lose its stability and give place some non-uniform ones. Actually, a planar pulsating wave appears through the Hopf bifurcation of planar traveling wave. Moreover, we observe a wave that propagates in the form of spiral encircling the cylindrical sample with several reaction spots. This wave is called a helical wave since it has been shown by our 3D numerical simulation that the isothermal surface of the wave has some wings and it helically rotates down as time passes on. Similar helical waves are observed also in propagation fronts of polymerizations in laboratory and they are obtained also by numerical simulation of some autocatalytic reactions as well as the SHS.
We have been studied the existing condition of helical wave and the transition process of wave patterns from traveling mode to pulsating mode and/or helical mode, and we have obtained the following results:
1. A stable helical wave can bifurcate directly from a planar traveling wave.
2. Even if a traveling wave is stable in R, the corresponding planar traveling wave can be unstable in the band domain as well as in the cylindrical domain, and a helical wave takes the place of planar traveling wave.
3. There are no stable helical wave when the band width L is small or the radius R of cylindrical domain is small.
4. Helical waves with different numbers of reaction spots can coexist stably.

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

前年度までに実施された熱平衡解からの分岐現象の数理解析的研究(水平方向の周期の比は1:√<3>,レイリー数が分岐パラメータ)と対流パターンに対応する分岐曲線のニュートン法による追跡によってつぎのことが明らかになった.
(1)臨界レイリー数においては余次元2の分岐が起こり,ロール型の対流パターンと長方形型の対流パターンが分岐すること.
(2)分岐したロール型の対流パターンは,追跡した範囲内においては,安定であること.
(3)分岐直後の長方形型の対流パターンも安定であるが,あるレイリー数における2次分岐を経て不安定化すること.分岐直後の正六角形状の対流パターンは不安定であるが,長方形型の対流パターンから2次分岐した安定解の枝と結合した後は安定になること.
上記の研究成果は,レイリー数が臨界値より大きい場合には安定な対流パターンが多重に存在することも示している.これを受けて,平成13年度には,典型的なレイリー数に対するパターン選択問題を取り上げ,発展方程式系の直接数値シミュレーションによって選択されるパターンを観察するという立場で研究を堆進した.その結果,つぎのようなことが判明した.
(i)臨界値よりやや大きくレイリー数を選ぶと,長方形型パターンが観測されるが、プラントル数Prが小さいとき(例えば,Pr=1)には,長方形型の対流パターンもしばらくの間は持続するものの,いずれはロール型のパターンに変形することが判明した.すなわち,プラントル数が小さいときの,長方形型のパターンが安定に存在する範囲は非常に狭いことが推測される.
(ii)プラントル数Prが大きいとき(例えば,Pr=10)には,長方形型の対流パターンが安定に存在することを臨界値よりやや大きなレイリー数については確認した.レイリー数をしだいに大きくすれば対流パターンも変形されてゆくが,上記の(3)で表現されているような正六角形状のパターンへの接続までには至らなかった.

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Grant amount：\2200000 （ Direct Cost: \2200000 ）

本研究では,拡散の役割を調べるために,反応拡散系における拡散と非線形性の関係を重点的に調べている.特に爆発問題と拡散の役割を重視して研究している.昨年度に引き続き次の3つのテーマを中心に研究を行ってきた.
1.拡散誘導爆発解の爆発指数
2.空間2次元における反応拡散系の極限問題
3.拡散と非線形拡散の関係
テーマ1は,線形項を導入することによって起きる線形誘導爆発問題の爆発指数および拡散誘導爆発指数を数値計算によってそれぞれ求め,興味深い結果を得た.しかし,数学的な証明を付けることにはまだ成功していない.
テーマ2については,2つの側面からアプローチを行った.1つは,界面方程式の運動を記述するのに基本的であると思われる進行波解についての研究,もう1つは,別の種類に極限問題を導出するための予備的な研究である.前者については進行波解の厳密解を求め,すべての進行波解を決定することに成功した.これは,2000年に論文として出版された.また,現在,その進行波解の大域的漸近安定性を調べ,論文として投稿中である.後者については,パルス型界面による相転移問題を考えるため,パルス進行波解の厳密解の構成を行った.
テーマ3については,反応拡散系と非線形拡散を含む方程式系との関係を調べている.これによって,反応拡散系によって非線形拡散を含む方程式系が近似できることが分かった.現在,これについては,論文を作成中である.

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Grant amount：\3700000 （ Direct Cost: \3700000 ）

We studied the stability and dynamics of vortices of the Ginzburg-Landau equation which arises as a model describing a macroscopic superconducting phenomenon. We first showed the existence and stability of a single vortex solution in a disk with a variable coefficient. Next, to investigate the motion law of vortices, we derive an explicit form of a singular limit equation as the parameter goes to infinity. By virtue of this form we obtained some dynamical properties of vortices.
We also studied some dynamical system problems and solution structures of elliptic equation to obtain several new results.

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Grant amount：\2200000 （ Direct Cost: \2200000 ）

1. 森田、二宮両研究分担者の研究(裏面の第3、第4の論文)により、従来にも増して拡散方程式における拡散項の役割の解明が重要なことが認識されるようになってきた。そこで、その基礎になる線形拡散方程式系(parabolic system)の基礎理論の整備が急務となり、このテーマを優先して研究した。従来のPetrowskyによるp-parabolic systemの定義は相似変換に不変でないので、p-parabolic systemの定義から考え直さなければならなかった。幸い、我々の微分作用素を成分に持つ行列の重み付きの行列式の理論によりp-parabolic systemの相似変換に不変な新しい自然な定義を確立した。さらに、我々のformal symbolsのクラスにおける擬Jordan標準形の理論により新しい定義の弱い意味での妥当性と、変換行列が滑らかであるという付加的仮定のもとにではあるが、新しい定義によるクラスでCauchy問題のH無限適切性が成り立つことを示すことに成功した(裏面の第2論文)。定義の妥当性を示すために、多くの典型的具体例を構成したが、その多くは今回の科研費で購入した2台のコンピュータにより四ツ谷研究分担者の指揮のもとにアルバイトに依託してさせた数式処理による計算実験により構成できた。
2. 系に対する南雲型のCauchy-Kowalevskayaの定理のための必要十分条件については、研究代表者が「予想」を公表していて、最も簡単な場合には空間次元が1ならば予想の正しいことを証明していた。今年度は、従来、意外と困難であるといわれている多次元化に取り組み、成功した。こちらも、アイデアの多くが、研究代表者の学部卒業研究における数式処理による実験に由来する(大学間交流筑波研究集会で口頭発表、論文作成中、基礎理論は裏面の第1論文)。
3. 双曲系に関する研究においては、特性根の多重度が一定で高々2の場合に、主要部のシンボルの固有ベクトルの陪特性帯に沿っての挙動を明らかにしたが、その結果を解空間の構造の解明に結び付ける作業は未完成で、部分的解明に留まっている。(愛媛大学における研究会で口頭発表)
以上の結果は学術論文にまとめるだけでなく我々の手法のすべてを解説した報告集として公表する。

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Grant amount：\13500000 （ Direct Cost: \13500000 ）

Concerning the above research project, we obtained the following results.
(1) We carried out the study on skew-gradient systems, which are a generalized activator-inhibitor systems. When a steady state is characterized as a min-max point of an energy functional, we showed that the steady state is stable regardless of parameter values. We also make clear the relation with the so-called Turing instability.
(2) We carried out numerical simulations on a model equation which describes a spatial pattern formation observed In a colony of some bacteria. It is shown that such spatial patterns appear as a history of the process.
(3) Two-species competition system is a main problem in the theory of mathematical ecology. We gave a mathematically rigorous verification for the fact that the system reduces to a Stephen -type free boundary value problem in a singular limit where the competition rate is infinitely large.
(4) The Gierer-Meinhardt system is a mathematical model for the morphogenesis in Mathematical biology. We studied stability of spiky stationary patterns, and gave some criteria for the stability and Instability.

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

本研究では,拡散の役割と非線形性の関係を調べるために,拡散のもつ性質のうち,通常予想されてこなかった意外な側面の発見を行い,それによって拡散のもつ性質を外延しようと試みてきた.現在のところ,拡散の性質が際立った状態で現れる問題に注目して研究を行っている.特に,研究代表者等によって近年発見された拡散誘導現象
反応拡散系に見られる拡散誘導爆発
の仕組みを研究している.爆発問題は,特異性のある問題なので,拡散のもつある性質が,拡大されて導き出されることが予想されるからである.
また,拡散誘導爆発は,拡散の効果によって爆発しない常微分方程式系が爆発することがあることを結論づける興味深い結果である.この結果は単に拡散の役割が空間一様化だけでなく,非線形性との兼ね合いで爆発にまで影響を与えることを述べただけにとどまらず,モデル方程式を建てる際にも拡散のこの意味での役割も考慮に入れる必要があることを示唆している.ここで作られた方程式系は人工的なモデルであったが,数理生物学や化学反応などに現れるモデルにおいても,拡散誘導爆発が見られることが数値計算によって明らかになってきた.
拡散誘導爆発が起きる状況を把握するために,その自己相似解の果たす役割と方程式系の退化の関係を調べている.

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

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

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Grant amount：\1000000 （ Direct Cost: \1000000 ）

反応拡散系の解構造における拡散の役割について調べてきた.拡散は一般に空間一様化に働くように考えられてきた.事実拡散係数が大きいときは非線形方程式でもその類推はある程度正しいことが知られている.しかし,拡散不安定性のように必ずしも空間一様化と考えてはいけないことも知られてきた.これは拡散項を付けることによって,一様な解が不安定化し,解は時間とともに非一様化していくことがあることを示している.拡散不安定性の発見は後のパターン形成の問題に大きな影響を与えたように,拡散の新しい役割を調べることは意義深いことと思われる.
本研究では特に爆発問題と拡散の役割について考察している.常微分方程式系のすべての解は有界であるにも関わらず(実際にはすべての解は原点に収束する),それに拡散項をつけた反応拡散方程式系の解で有限時間に爆発するような方程式系を作った(溝口・柳田氏との共同研究).このような現象を拡散誘導現象と呼ぶことにする.本来,解の爆発は線形問題では生じず,非線形問題特有の現象である.従って爆発の有無という点では拡散の効果はあまり大きいもののように考えられてこなかった.また拡散が空間一様化に働くとすれば常微分方程式系の解に近づき,有界に留まることが期待されるが,上述の結果によって,拡散を省いただけの常微分方程式系からだけでは解の大域的存在を決定できないことが結論づけられる.

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Grant amount：\4500000 （ Direct Cost: \4500000 ）

研究実施計画に基づいて研究を推進し以下で述べる成果を得た。
1)相互作用のある拡散系は、統計物理や数理生物の多くの興味深いモデルを含み、その確率解析は無限次元拡散過程の研究にも重要な意味をもつ。その観点から従来、定常分布が多様に存在する状況下でのエルゴード的挙動を研究してきたが、今回は定常分布が自明な場合または存在しない場合のエルゴード的挙動の研究に取り組み、有限系からの近似におけるエルゴード的挙動に関する新しい結果を得た。(Cox-Greven-志賀)
また、相互作用のある拡散系と一般化された逆正弦法則の問題との関連も明らかになりつつあり、論文として準備中である。(志賀)
2)無限次元線型マルコフ系の新しいクラスを導入し、対応する標本リアプノフ指数の定義可能性の証明およびカップリング径数に関する漸近挙動を調べ、径数が小さい領域では有限系、無限系ともに同一のオーダーをもつことを証明した。(古尾谷-志賀) このアイディアはランダム環境中のランダムウォークの生存確率の漸近解析に適用可能であり、その結果は論文として掲載予定である。(志賀)
3)ランダムウォークおよびブラウン運動について種々の観点から研究を進め、2次元ランダムウォークのポテンシャル核の精密な漸近解析(深井-内山)、さらに時空的観点からのウィーナーテストの問題を解決した。(深井-内山) また、ブラウン運動が導く道空間上のある種の保測変換と数々のブラウン汎関数の同分布性との関連を解明した。(高岡)
4)線型拡散方程式に対する混合問題の非負解の一意性が成り立つための領域の形に関する必要十分条件を与えた。(村田)また、常微分方程式系とそれに拡散項を付けた非線形拡散方程式を解の爆発問題の観点から研究し、常微分方程式系のあらゆる解は原点に収束するのに,その常微分方程式系に拡散を付けた方程式の解は爆発するという興味深い例を見つけた。(二宮)

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Grant amount：\2500000 （ Direct Cost: \2500000 ）

交付申請書に記載した研究実施計画に基づいて研究を推進し以下で述べる成果を得た。
1)相互作用のある拡散系は、無限次元拡散モデルの重要なクラスであり、統計物理や数理生物の多くの興味深いモデルを含んでいる。その拡散系でとくに拡散係数が一般の関数の場合には、定常分布の完全な記述を与えるという定常分布問題は重要かつ未解決である。基礎の空間が立方格子の場合で相互作用が均質的かつ拡散係数が有界のときには、この問題はすでに志賀が解決したが、拡散係数が非有界の場合には一般に前者とは異なる現象が現われることを新たに指摘し、各成分が非負かつ相互作用が不偏ならば定常分布はすべて空間的に均質的であることを証明した。
2)無限次元モデルのエルゴード的挙動を、近似する有限次元モデルから観測する問題に取り組んだ。相互作用が推移的ならば、有限次元モデルから適当な時空スケーリングにより無限系の定常状態のパラメータの揺動を観測できることを相当程度に確立できた。さらに相互作用が再帰的の場合にも2次元では集団平均過程のスケーリング極限の存在を証明した。(T.Cox, A.Grevenとの共同研究として発表予定。)
3)無限次元線型マルコフ系の新しいクラスを導入し、対応する標本リアプノフ指数の定義可能性を証明した。さらに標本リアプノフ指数のカップリング径数に関する漸近挙動を調べ、これについては有限系、無限系ともに同一のオーダーをもつことを証明した。(この結果は古尾谷祐との共同論文として現在まとめている。)
4)多粒子のマルコフ力学系に対する流体力学極限の問題はすでに多くの研究がある。それに対し内山は今回、ある古典力学に従う多粒子系の流体力学極限を調べマクロな運動を支配する非線形拡散方程式の導入に成功した。
5)線型拡散方程式の正値解の一意性問題や競争的非線型拡散方程式などでも村田、二宮により重要な成果を挙げることが出来た。

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