非線形楕円型方程式 (系) に対する特異摂動問題を中心に研究を実施した. 非線形シュレディンガー方程式に対する特異摂動問題に関しては, 局所的な変分法によるアプローチを研究代表者田中は J. Byeon 氏と共に開発し, ポテンシャル関数の極大点, 鞍点に凝集する解の構成に成功した. この構成法は非常に広いクラスの非線形項に対して適用可能であり, 従来の Lyapunov-Schmidt
法による極限方程式の解の一意性, 非退化性を要求する存在結果を大きく拡張するものである. なお, 本年度の研究において 1 点に与えられた数のピークが凝集する multi-peak 解の存在の構成にも成功している. このようなmulti-peak 解の存在は非退化条件なしには証明されていなかったものである.
常微分方程式の Lagrange 系に対する特異摂動問題に関しては, 高振動解の adiabatic invariant を用いたプロファイルの決定および与えられた admissible なプロファイルをもつ解の構成を P. Felmer 氏, S. Martinez 氏らと共に行い成功した.
また 2 次の相互作用をもつ非線形シュレディンガー方程式系について研究代表者は分担者小澤および林氏と共に研究に取り組み, 初期値問題の局所および大域可解性, さらには定在波解の存在を様々な設定の下で行った. また研究分担者小薗は Lax-Milgram 定理の一般化およびその楕円型方程式系への応用を, また連携研究者足達, 佐藤は準線形楕円型方程式, 非線形シュレディンガー方程式系の解の漸近挙動の研究等を行い, 塩路は非線形楕円型方程式の球対称解の研究を行い, 既存の結果をほぼすべて含む, 球対称解の一意性定理を導いた.

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

It is very important to pursue mathematical study for Partial Differential Equations which describe various phenomena arising in natural sciences such as Physics. However, since the behavior of solutions of wave equation, a typical example of hyperbolic equation, and heat equation, a typical example of parabolic equation, is quite different to each other, the method of study and the group of researchers are both separated. Based on the fact that wave equation with strong dissipation acquires some parabolicity which was detected by research representative, we studied some abstract hyperbolic equation with strong dissipation and clarified the mechanism of acquiring the parabolicity and revealed the hierarchical structure underlying between hyperbolicity and parabolicity

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.

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.

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.

We tried to construct a new mathematical theory for clarifying the mechanisms of nonlinear and complex phenomena in the real world and to make use of it to handle some concrete problems arising in the material and life science. In fact, our new mathematical theory provided an effective approach for some open questions. This is one of the most important achievements in this research, and the so-called "Life Mathematics" has been evolved in the same framework

研究目的にかかげた目標に関する次の幾つかの興味ある成果が得られた。
(1)我々の先行研究によって、p-Laplacianを主要項に持つ準線形放物型方程式
u_t=△_p u+uに対する初期値境界値問題に対して、全ての解軌道を引き付ける「大域アトラクター」が、L^2で構成され、さらにそれが無限次元を持つ事実が知られていたが、これはかなり特殊な状況であり、非線形楕円型方程式に関するLyusternik-Scnirelman理論からも、その無限次元性は導出できるという難点があった。
uをαu-b(x)|u|^q uとしても、大域アトラクターの存在とその無限次元性が導かれる
ことが示された。これは、より一般的な非線形項f(x, u)に対しても、同様な結果が成立することを示唆する、重要な発見である。
(2)多孔質媒質中を流れる流体(溶媒)の速度及び温度と流体中の溶質の濃度の振舞いを記述する、2または3次元有界領域におけるBrin kman-Forchheimer方程式の時間大域解の存在と一意性が、H^1に属する初期値に対して、示された。
これによって、この方程式に対する、大域アトラクターの構成の出発点がクリアーされたことになる。
また、よく知られているように、3次元空間におけるナビエ・ストークス方程式の一意的時間大域解の存在問題が未解決大問題である事実と比較すると、非常に興味深い知見を与えている。

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 the strong force condition of Gordon. We remark that our condition is given as a property of the energy surface S={(q, p);H(q, p) =E} not on the Hamiltonian H(q, p).

(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 shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.
iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long.

(1)L^p空間では不可能であるが、L^∞空間ではじめて可能になるエネルギー評価の新たな手法である「L^∞-energy method」が開発され、様々な非線形方物型方程式に極めて有効であることが明らかにされつつある。この手法が、走化性粘菌の行動を記述する非線形放物型方程式系にも有効であることがわかり、従来の研究より大幅に弱い条件のもとで、解の存在、一意性が得られることが示された。他の非線形放物型方程式系への応用が期待される。
(2)系を記述する種々のパラメーターの値が同じであっても、系の状態は必ずしも同じにはならず、系の状態が、パラメーターの値のみならず、その過去の履歴に依存して決定される現象を、ヒステレシス効果と呼ぶ。ヒステレシス項を有するfood-prey-predator model方程式の解の存在、正値性、有界性、一意性を保証する従来の条件が大幅に緩められた。
ヒステレシス効果を表す項は、可能なすべてのヒステレシスループ(これは、未知関数に依存する)から作られる領域の指示関数(indicator function)の劣微分作用素で与えられるため、極めて強い非線形性を有している。このため、従来の研究では、ヒステレシス項を特徴付ける関数の2回微分までがすべて有界であるという強い条件が必要であったが、「L^∞-energy method」を応用することにより、これを2回連続的微分可能性条件にまで弱めることができた。さらに、解の一意性に対する証明が改良され、空間次元が4以下の場合まで示すことができるようになった。

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 phenomena
The first problem in mathematical biology is given by a system of differential equations with quasilinear diffusion of the form
u_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 sufficiently small, it is known that this problem admits various kinds of steady-state solutions. In particular, we are interested in steady state with transition layers and spikes. Here transition layer for a solution means a part of u(x) where u(x) drastically changes from 0 to 1 or 1 to 0 in a very short interval. Such oscillating solutions have been studied by Ai-Chen-Hastings and our group, independently. It has been proved that any transition layer appears only in a neighborhood of x such that a(x)=1/2 and that any spike appears only in a neighborhood of x such that a(x) takes its local maximum or minimum. We have also established more information on profiles of multi-transition layers and multi-spikes, their location and the relationship between profile and stability of steady-state solution with transition layers.

We study the existence and multiplicity of solutions of nonlinear differential equations via variational methods. In particular, we study singular perturbation problems.
1.We study the existence and multiplicity of solutions of nonlinear scalar field equations : -Δu+V(x)u=f(u) in R^N. Usually in such a problem global conditions on nonlinearity f(u)(ex.global Ambrosetti-Rabinowitz condition) are required to ensure the existence of solutions. In this study we tried to obtain an existence result without such global assumptions and we find that it is possible if we require sufficiently fast decay of the potential V(x).
2.We also study singular perturbation problem : -Δu+λ^2a(x)u=|u|^<p-1>u in R^N, where a(x)【greater than or equal】0. As a limit problem as λ→∞, a Dirichlet boundary value problem -Δu=|u|^<p-1>u, u|_<∂Ω>=0 in Ω≡{x ∈R^N;a(x)=0} appears. We assume Ω consists of several bounded connected components Ω_1,【triple bond】, Ω_κ and for given solutions u_i(x) of the Dirichlet problem in Ω_i, we try to find a solution u_λ(x) in R^N whose limit is u_i(x) in Ω_i (connecting problem). We succeed to find a solution joining Mountain Pass solutions without non-degeneracy conditions. Also we show that there are infinitely many sign-changing solutions that are connectable with Mountain Pass solutions.
3.For 1-dimensional Allen-Cahn equations and Schrodinger equaitons, we study the characterization of a family of solutions in the setting of singular perturbation. More precisely, we consider a family of solutios with increasing number of layers or spikes. We give a characterization of such a family using "limit enery function" or "envelop function". Conversely for addmissible patterns we construct corresponding families of solutions via variational methods.

The subjects of this project are the following :(1)Modelings of phase transition phenomena with their mathematical theory(2)Formulations of the related optimal control problems and numerical simulations(3)Return of the research results to the school educationConcerning (1), on the basis of the fundamental laws of the thermo-mechanics, we investigated the structure (phase change, component separation, damage evolution, ordering of atoms) of materials from the view-point of the theory of dymanical systems. Especially, during the last two years (2003-2004) of the project, we treated a class of models for irreversible phase change phenomena, taking account of the process of damage, and evolved its mathematical theory. In our research the subdifferential theory, which have been accumulated for these 15 years, worked very well and we had a big progress more than we expected. Concerning (2), proposing some formulations of optimal control problems in which control parameters are described by a class of hysteresis functions, we finished almost the theoretical part of the problems and tried partially their numerical tests. It was pointed out that there are still some questions which should be improved. Many papers treating subjects (1)and (2)were published and we organized two international conferences during this research project.As far as (3)is concerned, many practices were reported in the meetings "Mathematical analysis of phase transitions and their related mathematics education" which we organized every year. We paid our attention to time-dependent phenomena noticed easily in our daily lives, verifying very carefully their value-added educational aspects as teaching materials of mathematics or sciences for the secondary school. We expect that this research enables to provide new teaching materials by which students can perceive usefulness of mathematics

(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 infinite cylindrical domains.
(4)The classical "Principle of Symmetric Criticality (PSC)" by R.Palais assures that under suitable conditions, critical points in the subspace with the symmetry give real critical points in the whole space, but is restricted to the system with variational structures. PSC is extended to a more general theory which covers the elliptic systems without full symmetry or evolution equations including time evolution terms.
(5)A new degree theory is established. It can teat mutivuled operators including subdifferential operators and cover nonlinear PDE with various multivaluedness nature.
(6)The theory of nonmonotone perturbations for subdifferentials is ameliorated to cover the initial-boundary value problems and time periodic problems for magneto-micropolar fluid equations.

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 the view-point of mathematical biology, it is very important to study positive stationary solutions and to know their number. We have tried to get some conditions for the multiplicity of such positive solutions. In particular, the multiple existence is established if interactions are very large in case of competition model with linear diffusion or if one of cross-diffusion is very large in case of prey-predator model.

We study the existence problems for nonlinear differential equations via variational methods. We mainly dealt with nonlinear elliptic problems and Hamiltonian systems.
1. We study the existence and multiplicity of positive solutions of nonlinear scalar field equations in unbounded domains. In particular, we are concerned with equations which depend on the space variable x and we investigate the effects of the inhomogeneity (dependence on the space variable x) on the set of solutions of the scalar field equation. We find a very delicate dependence ― very small inhomogeneity induces a big change in the set of soluitons ― and we find an example of nonlinear scalar field equation which has 4 positive solutions after very small but not zero pert perturbation.
2. We also consider the singular perturbation problems for nonlinear elliptic problems. We get 2 results : (a) for 1-dimensional setting we introduce a new finite dimensional reduction and we succeed to prove the existence of solutions with a cluster of interior or boundary layers for inhomogeneous Allen-Cahn type equations. We also succeed to prove the existence of solutions with a cluster of spikes for nonlinear Schrodinger equations. (b) We give a mountain pass characterization of positive solutions for a wide class of nonlinear elliptic equations. As an application, we show the existence of a spike solution for a wide class of nonlinear elliptic equations including asymptotically linear equations.
3. For Hamiltonian systems we deal with singular Hamiltonian systems with 2-body type singularities. In case the potential V(q) has more than 3 strong force type singularities we find a family of very complex solutions which are related with symbolic dynamical systems. We also deal with the case the set S of singularity is not a point and it has a positive volume. We consider the case where V(q) 〜 - 1/ dist (q, S)^α and we find the existence of non-collision solutions for all positive α > 0, that is, even for weak force case 0 < α < 2.

In this project, various linear or nonlinear systems such as phase transition models And reaction-diffusion equations from the view-point of the linear or non-linear Operator theory, and a lot of results, which are of high level especially from an interdisciplinary point of view, has been established. For instance,
(1) A stability theory for a class of nonlinear systems of parabolic PDEs, which includes a mathematical model for solid-liquid phase transition in the mesoscopic length scale, was evolved. In this theory, a new concept "the local in space stability" was introduced. This is quite reasonable from the physical point of view ; in fact, within this concept the growth or disapprearance of phases are able to be theoretically explained.
(2) It is a very convenient mathematical approach to describe various nonlinear time dependent processes as (autonomous or nonautonomous) dynamical systems, when one wants to know the asymptotic behavior of the processes as time goes to infinity. There is a concept of global attractors of dynamical systems for this purpose. We established a general method for the construction of global attractors of dynamical processes of a class wide enough.

1. It is shown that a way of phenomenological description of the mosaic state in a superconductor under an applied magnetic field leads to consider the minimizing problem of the Gibbs free energy under the constraints of complete expulsion of magnetic field from the parts of superconductor.
2. The initial-boundary value problem for the time-dependent Ginzburg-Landau-Maxwell equations is considered. The global existence and uniqueness theorems of L_2 weak or strong solutions are established via Fadeo-Galerkin's method. As to the parabolic version, the local existence of L_3 solutions is obtained by the semigroup approach for both bounded and exterior problems.
3. Numerical experiments of solutions to the parabolic version of the time dependent Ginzburg-Landan-Maxwell equations are obtained by the finite difference methods.

(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-Laplacian brings about the difficulty, it also gives remarkable nonlinear phenomenon. We have obtained satisfactory understanding on the structure of stationary solutions in higher space dimension as well as one dimension. In particular, we also have studied profiles of stationary solutions and proved interesting results on flat hats which stem from degenerate diffusion. Furthermore, we could show interesting information on the temporal and spatial change of non-stationary solutions.

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)ィイD2λィエD2 has a nontrivial solution, which gives a generalization of the well-known result of Brezis-Nirenberg to unbounded cylinders. (2) Let Ω = ΩィイD2dィエD2 x RィイD1N-dィエD1 and let ΩィイD2dィエD2 be a d-dimensional annulus. ・ If q 【greater than or equal】 NィイD2dィエD2 = 2 (N -d+1)/(N-d+1-2) , then (E)ィイD2λィエD2 admits no nontrivial weak solution.
・ If q < NィイD2dィエD2, then (E)ィイD2λィエD2 admits a nontrivial weak solution.
These results reveal the fact that the d-dimensional symmetry reduces the effective dimension by (d-1).
(III) Consider (E)ィイD21ィエD2 -Δu + u = a(x) |u|ィイD1q-2ィエD1u + f(x) x ∈ IRィイD1NィエD1, 2 < q < 2ィイD1*ィエD1 o < a(x), |a(x) - 1| 【less than or equal】 CeィイD1λ|x|ィエD1, λ > 0 It is shown that if ‖f‖ィイD2H-1(RィイD1NィエD1)ィエD2 is sufficiently small, then (E)ィイD21ィエD2 has at least two positive solutions. Furthermore, we found that for the case where f = 0 and q < 2ィイD1*ィエD1 is close enugh to 2ィイD1*ィエD1,the multiplicity of positive solutions depends upon the topological property (su as category) of the set {x ∈Ω ; u(x) = maxィイD2x∈ΩィエD2 }.The analysis of this phenomenon will be an interesting subject to study in future.
Parabolic Equations (I) It has been well known that weak solutions of porous medium equations enjoy the Holder continuity. However, the existence of smooth (local) solutions has been left as an open problem for long time. Otani-Sugiyama gave an affirmative answer to this open problem, by developing the LィイD1∞ィエD1-energy method, which was introduce by themselves to show the local existence of WィイD11,∞ィエD1-solutions for more general doubly nonlinear parabolic equations. This is the most fascinating result among our results obtained in this reseach project.
(II) It was left as an unsolved problem to determine the asymptotic behabiour of solutions of (P) uィイD2tィエD2, -Δu = |u|ィイD12ィイD1*ィエD1-2ィエD1u x∈Ω, u(x) = 0 x∈∂Ω. To this problem, the following partial answer was obtained. 「Let Ω = {x ∈ RィイD1nィエD1 : |x|< 1 } and the solution u (x.t) be positive, radially symmetric and monotone decreasing with respect to r = |x|. Then u blows up in a finite time or becomes a global solution and satisfies the following property : 「There exists a sequence {tィイD2nィエD2 } such that |∇u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1δィエD1(0) (u - x), |u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1σィエD1(0) (u - x). 」 This result give some information about the problem above to some extent. However, since strong technical condtions are assumed. We need further in vestigation to solve this problem in a natural setting.

The main objective of this project is to treat various nonlinear phenomena with phase transitions from interdisciplinary points of view. Our research is concerned with their modellings and theoretical analysis by using tools in nonlinear functional analysis for instance the subdifferential operaors theory of convex functions.
During the term of this project we found that many solid-liquid or solid-solid phase transition models are able to be handled within the perturbation theory of nonlinear evolution equations generated by subdifferentials in Hilbert spaces. Moreover, we pro- posed a class big enough of dynamical processes, including nonlinear phenomena in our consideration as typical examples, and evolved the stability theory for it ; in fact, we suc- ceeded in the construction of global attractors. In our set-up, one of the most important characteristics is that the domain of dynamical process depends upon time.
Also, as co-products of our results obtained in this project, an important question, which had been remained open for 30 years, about nonlinear elliptic operators was solved. This is a big contribution in this field, too.

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、正値定常解が2個以上存在する状況を新たに発見した。
2.p-Laplacianを含む拡散方程式にたいしてChafee-Infanteタイプの非線形項を付け加え、解の挙動、定常解集合の構造、安定性を研究した。空間次元1のケースに限定されるが、定常解集合の構造を完全に解明することができた。とくにp-Laplacianの退化性のため、定常解集合の構造は非退化のときと全く異なり、非可算集合となる。さらに、解のプロフィール、解の分岐構造、解の個数、安定性について今まで知られていなかった情報が得られた。今後は、空間次元の高いときの解集合の構造も調べたい。

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、この分野における重要な知見が得られた。
3.Pohzaev型の(不)等式に依らない、正値解の非存在の為の新たな手法の端緒が開かれた。これは、領域は平行移動不変性と正値解の一意性の議論を組み合わせた議論によるもので、正値解の一意性がよく調べられている、固有値問題、sub-linear(sub-principal)caseに対して有力な道具を提供するものである。この手法のより一般的な場合への拡張が期待される。
その他、これに関連する周辺の成果も多数得られている。

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, which converges at most only one iteration for a tridiagonal system, and n iterations for a block tridiagonal system.
The generalized convergence theorems to the improved SOR method with orderings are also considered, and we study necessary and sufficient conditions for a matrix to be a generalized diagonally dominant.
Using the notation 'basic LUL factorization' of matrices, we give some techniques to obtain special multiple relaxation parameters such that the spectral radius of the iterative matrix is zero for the Hessenberg matrices and a class of matrices.

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境界が時間とともに変動する、非柱状領域における初期値境界値問題、周期問題の強解の存在と一意性が、柱状領域における熱伝導と粘性流に対してそれぞれ知られている結果を含む極めて一般的な枠組みで解決された。しかしながら、このように多くの成果があげられた一方、最終目標であったシュレディンガー混合型方程式系の多くを含む統一的理論を構築するという課題については、いくつかの有力な手がかりは得られたものの、達成するには至らなかった。今後の課題としたい。

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。周期解が必ずしも軌道安定になるとは限らないこともわかり、将来さらに詳しい解析を進める必要がある。また、[1]の方程式系をディリクレ境界条件の下で解析すると、解の漸近挙動を調べる際にノイマンのときにはなかった難しさがあらわれる。定常問題の解集合の構造が単純ではない点である。対応する楕円型方程式系の正値解を見つけることが重要であり、不動点定理や写像度の理論を使って解の存在を示すことができた。ただし、解集合の情報は完全ではなく、今後も研究を続けなければならない。[2]は二種類の生物が競合している状況をモデルとしていて、二つの方程式の間で拡散効果が相互作用をしている。このような数学的難点を克服して、かなり自然な条件のもとで有界な大域界を校正することができた。今後は、この解の時間無限大での漸近挙動を調べる予定である。

2次元共形場理論の出発点は,複素解析的にouカーstateとin stateを記述することにある。4次元共形場理論においても複素解析的手法を適用し,粒子場と反粒子場とが,各々複素平面のはりあわせとして関係していることを示し,共形場理論への出発点を与えることができた。
S^4上のディラック作用素と赤道S^3上のハミルトニアンを複素ベクトル場を成分とする行列で具体的に表示することにより,S^4上の調和スピノ-ルの特徴づけを与え,またS^3上のハミルトニアンの固有値および完全固有スピノ-ル系を求めた。一方S^3へのSU(2,D)の左及び右からの作用より得られる最高ウエイメト表現に附随した球函数を2次元複素座標により表示した。これは初等的な結果であるが新しい。この球函数の族が上記固有スピノ-ルを系統的に与えることがわかる。この固有函数系に自然に附随してS^3上の無限次元グラスマン多様体が構成される。このグラスマン多様体の各元はS^4の北半球,南半球のスピノ-ルに境界系件を与えていると考えられる(witten's idea)が,このtransmission問題を考え解訳した。とくにディラック作堂素の指数定理の直接計算による証明が得られた。さらに進んでフェルミオン・フォック空間を導入した。ヴィラソロ代数の4次元の類似を探することが今後の問題となる。(以上 郡)
量子群の研究に関しいは,A_< nー1>型のヘッケ代数により量子群Vg(gl(n+1))の表現の指標を訳定する研究が行なわれた(上野)
この他,函数解析の基本的定理に関して,Whitteyーschwartzによるdistribntionの特徴づけの定理の精密化が得られた(垣田)

今年度における研究目標はかなり達成できた。特に頂点(回転の中心となる点)をもつ回転面に関しては予想より非常に良い結果を得た。まず基本的性質として、頂点をもつ回転面上の極の作る集合は、頂点を中心とするある半径の閉球をなすことがわかった。次に回転面上の極が2個以上あるための同値条件を、関数L(t)だけの簡単な性質だけで表すことができた。ただし、L(t)は、頂点を中心とする半径tの円の長さを表す。マンゴルトの回転面と呼ばれる回転面を定義し、この回転面上の最小軌跡を完全に決定することに成功した。いくつかの興味深い例を作ることにも成功した。非連結な最小軌跡をもつ回転面、ある点でガウス曲率が正であるが、共役点を持たない回転面の例、極を沢山もつマンゴルトの回転面の例等を見つけた。またマンゴルトの回転面上において、極の作る閉球の半径は、L(t)だけで表せるある幾何学的な等式を満たすことも証明した。最後に主定理を述べるのに必要な定義をしておく。Mを回転面、PをMの頂点とする。各q(Mの点でPでない点)に対して、μ_qをPからでて、qを通る測地線、またt_qをqからでてpを通る測地線とする。ただし、各測地線のパラメ-タ-はその孤長にとるものとする。dをM上のリ-マンの距離関数とするとき、次の定理を証明した。
主定理、Mをマンゴルトの回転面とする。各xに対して、xの最小軌跡C_xは空集合であるか、C_x=u_x〔d(p,x)∞〕である。ただし、xはt_xに沿うxの第一共役点を表す。

対称臨界性原理とは、「Banach空間X上で定義された汎関数Jに対し、ある群Gの作用に関して不変な部分空間X_G上でのJの臨界点が、X全体でのJの臨界点を与える」という原理である。この原理は、Jの汎関数(フレッシェ)微分を、劣微分作用素を含むかなり一般的な多価作用素Aに置き換えても成立することが、本研究により示されている。さらに、作用素Aが、必ずしも変分構造を有していない場合に対して拡張することが可能(AがG-共変であれば十分)であり、時間発展を伴う発展方程式に対して有用であろうことが期待されていた。(1)非線形放物型方程式、波動方程式、シュレンディンガー方程式等への応用考えるとき、時間に関する微分作用素d/dtがどのような空間でG-共変となるかを調べることは重要であるが、今回L2(0,T;X),X=L2(Ω),L2(Ω)xL2(Ω),H10(Ω)xL2(Ω),などでそのG-共変性(G=O(N))が確かめられた。これらの空間は、上記の諸方程式を抽象発展方程式に帰着させるときに現れる基本的な空間であり、これは、今後の発展方程式への応用研究において重要な知見である。(2)放物型方程式の時間大域解の漸近挙動を解析する際、コンパクト性は強力なな道具を提供する。一般の非有界領域では欠如しているソボレフの埋蔵定理にかかわるコンパクト性が、回転対称性を有する関数からなる部分空間においては、恢復するという事実に基づき、ある種の回転対称性を有する非有界領域におけp-Laplace作用素と爆発項を含む非線形放物型方程式の対称大域解のW1,p-有界性が確かめられた

On the theme of researching the theory of viscosity solutions of differential equations and its applications, we investigated viscosity solutions of boundary value problems, weak KAM theory, regularity of viscosity solutions, optimizations problems, several kinds of asymptotic problems in differential equations, curvature flows and motions of phase boundaries, mass transportation problems, problems in engineering and economics. Based on the investigations done before, we have succeeded to obtain many, new observations on each of subjects listed above. Our contributions to research on Aubry sets in weak KAM theory and its application to asymptotic problems are significant

　非有界領域における臨界ソボレフ指数を越える非線形性を有する非線形楕円形方程式に対して、以下の興味深い結果がえられた。方程式　（Ｅ）　－△ｕ＝｜ｕ｜q-2ｕ　ｘ∈Ω、　ｕ（ｘ）＝ｘ∈∂Ωに対して、次の定理を得た。(1)「Ω＝ＲN＼ＢR、・ＢR＝｛ｘ∈ＲＮ；｜ｘ｜≦Ｒ｝、２*＜ｑ＜∞（２*は、ソボレフ型埋蔵Ｈ10（Ω）⊂Ｌq（Ω）の臨界指数）とするとき、（Ｅ）はＨ1O（Ω）⊂Ｌq（Ω）に属する（球対称）非自明解をもつ（大谷・橋本哲）・」１＜ｑ≦２*の場合には、非自明解が存在しないことが既に知られており、（大谷・橋本貴）、有界領域に対する既知の結果（１＜ｑ＜２*のき存在、２*≦ｑ＜∞のとき非存在＞との双対性から、解の存在が予想されていたが、この長年の未解決問題が肯定的に解決された意義は大きい。　また今まで、非有界領域における臨界ソボレフ指数を越える非線形性を有する非線形楕円形方程式に対する変分的解法が皆無であったが、この定理を証明するために、「適当な変換によって、外部領域における問題を、円環領域における境界に特異性を有する楕円形方程式に変換し、これを変分的な方法で解くことに帰着する」という全く新しい技法が開発されたことは極めて意義深い。(2) Ω＝Ωｄ×ＲN-d、Ωdをｄ-次元円環領域とする。非自明解の存在に関して、今まで解明されていなかった場合２*＜ｑ≦Ｎd=２(N-d+1)/N-d-1が解明された：（ⅰ）２*＜ｑ≦Ｎdの場合は、非自明解が存在し、（ⅱ）ｑ＝Ｎdの場合は、解が存在しない。即ち、既存の結果とあわせると「（ⅰ）１＜ｑ≦Ｎdのとき存在（ⅱ）Ｎd≦ｑのとき非存在」となり、この問題に対する完全な解答が得られたことになる。この事実から、「領域のｄ次元対称性は、実効的次元を（ｄ－１）次元だけ減ずる効果をもたらす」ことが結論づけられた。

臨界ソボレフ指数の非線形性を有する、楕円型方程式に関していくつかの興味深い結果が得られた。方程式（Ｅ）　-Δu=λu+│u│q-2u x∈Ω, u(x)=0 x∈∂Ωに対して、次の定理が成り立つ。（１）Ω=Ωd&#215;　RN-d（RNの非有界柱状領域）、q=2*（2*は、ソボレフ型埋蔵H10 (Ω)⊂Lq(Ω)の臨界指数）、d≧1,N≧4とするとき、任意のλ∈(0,λ1)、λ1=infν∈H10(Ω)∥∇u∥2L2/∥u∥2L20に対し、(E)は非自明解をもつ（大谷・石渡）。（この結果は、有界領域に対してよく知られているBrezis-Nirenbergの結果の非有界柱状領域への拡張を与えている。また、λ≦0の場合には、(E)は非自明界を持たないことが知られている（大谷・橋本）。）（２）Ω=Ωd&#215;　RN-d,Ωdをd-次元円環領域とするとき、q>Nd=2(N-d+1)/(N-d+1-2)ならば、(E)は非自明界（適当なクラスに属する弱解）を持たない（大谷・橋本）。（2＜q≦2*に対しては、(E)は非自明解をゆるすことが知られており（大谷・石渡）、2*＜q≦Ndの場合を解決することが、今後の課題となろう。）　次に、当面の目標であった、劣微分作用素に対しての写像度を定義することに成功した（大谷・小林）。これは、従来の理論の欠点（例えば「deg(f,G,p)≠0　であってもf(x)=pの解がGに存在するとは限らない」など）を取り除いたものであり、Leray-Schauder型の写像度の理論に沿った拡張になっている。この意味で、従来の理論を大きく越える枠組みを提供するものであり、非線形偏微分方程式への応用が期待される。この為には、摂動理論の確立が今後の最重要課題となろう。研究成果の発表1997　Periodic problems for heat convection equations in noncylindrical domains, Funkcialaj Ekvacioj 40 19-391997　Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains, Houston J. Math. 23, 267-2901997　Nonexistence of positive solutions for some quasilinear elliptic equations in striplike domains, Discrete and Continuous Dynamical Systems 3, 565-578