Updated on 2023/02/02

TANAKA, Kazunaga

##### Scopus Paper Info
###### Paper Count: 0  Citation Count: 0  h-index: 11

Citation count denotes the number of citations in papers published for a particular year.

Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Professor
Homepage URL

### Concurrent Post

• Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

### Research Institute

• 2022
-
2024

Waseda Research Institute for Science and Engineering   Concurrent Researcher

### Education

•
-
1986

Waseda University   Graduate School, Division of Science and Engineering   Mathematics

•
-
1986

Waseda University   Graduate School, Division of Science and Engineering   Mathematics

•
-
1982

Waseda University   Faculty of Science and Engineering   Mathematics

### Degree

• Waseda University   Doctor of Science

• 早稲田大学   理学修士

• Waseda Univeristy   Master of Science

### Research Experience

• 1999
-

Waseda University, Professor

• 1994
-
1998

Waseda University, Associate Professor

• 1992
-
1994

Nagoya University, Associate Professor

• 1992
-
1994

Nagoya University, Associate Professor

• 1990
-
1992

Nagoya University, Lecturer

• 1986
-
1990

Nagoya University, Research Assistant

### Professional Memberships

•

Mathematical Society of Japan

### Research Areas

• Mathematical analysis

### Research Interests

• variational problems, Hamiltonian systems, nonlinear elliptic problems

### Papers

• Symmetric ground states for doubly nonlocal equations with mass constraint

Silvia Cingolani, Marco Gallo, Kazunaga Tanaka

Symmetry   13 ( 7 )  2021.07

View Summary

We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem.

5
Citation
(Scopus)
• Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

S. Cingolani, M. Gallo, K. Tanaka

Nonlinearity   34 ( 6 ) 4017 - 4056  2021.06

View Summary

We study existence of solutions for the fractional problemwhere N 2, s ∈ (0, 1), m > 0, μ is an unknown Lagrange multiplier and satisfies Berestycki-Lions type conditions. Using a Lagrangian formulation of the problem (P m ), we prove the existence of a weak solution with prescribed mass when g has L 2 subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev's mountain in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in Hirata and Tanaka (2019 Adv. Nonlinear Stud. 19 263-90); Ikoma and Tanaka (2019 Adv. Differ. Equ. 24 609-46). A multiplicity result of infinitely many normalized solutions is also obtained if g is odd.

9
Citation
(Scopus)
• Semi-classical states for logarithmic Schrödinger equations

Norihisa Ikoma, Kazunaga Tanaka, Zhi Qiang Wang, Chengxiang Zhang

Nonlinearity   34 ( 4 ) 1900 - 1942  2021.04

View Summary

In this paper, we investigate semi-classical bound states for logarithmic Schrödinger type equations with a potential function which has a finite number of singularities of at most logarithmic strength. We construct localized solutions concentrating at a logarithmic type singular point of the potential, and we also characterize the asymptotic limiting profile of the localized solutions. To accomplish these, we develop new penalization techniques for treating the difficulties associated with the non-smoothness of the variational formulation and the singularity of the potentials.

3
Citation
(Scopus)
• Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass

Silvia Cingolani, Kazunaga Tanaka

Springer INdAM Series   47   23 - 41  2021

View Summary

We study existence of radially symmetric solutions for the nonlocal problem: where (formula presetned) a unknown Lagrange multiplier. Using a Lagrange formulation of the problem (1 ), we develop new deformation arguments under a version of the Palais-Smale condition introduced in the recent papers (Hirata and Tanaka, Adv Nonlinear Stud 19:263–290, 2019; Ikoma and Tanaka, Adv Differ Equ 24:609–646, 2019) and we prove the existence of a ground state solution for the nonlinear Choquard equation with prescribed mass, when F satisfies Berestycki-Lions type conditions.

3
Citation
(Scopus)
• A Note On Deformation Argument For L2 Normalized Solutions Of Nonlinear Schrödinger Equations And Systems

Norihisa Ikoma, Kazunaga Tanaka

Advances in Differential Equations   24 ( 11-12 ) 609 - 646  2019.11

View Summary

Abstract. We study the existence of L2 normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions, we develop new deformation arguments for the constrained functional on (Formula Presented). As applications, we give other proofs to the results of [5, 8, 22]. As to the results of [5, 22], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [8], via our deformation result, we can show the existence of vector solution without using constraint related to the Pohozaev identity.

• Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches

Jun Hirata, Kazunaga Tanaka

Advanced Nonlinear Studies   19 ( 2 ) 263 - 290  2019.05

View Summary

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝ N (N ≥ 2): (Equation Presented) where g(ξ) ∈ C(ℝ, ℝ), m > 0 is a given constant and μ ∈ ℝ is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗)m. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313.345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347.375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in ℝ N : Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253.276], it enables us to apply minimax argument for L 2 constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem (Equation Presented).

20
Citation
(Scopus)
• Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well

Silvia Cingolani, Kazunaga Tanaka

Revista Matematica Iberoamericana   35 ( 6 ) 1885 - 1924  2019

View Summary

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −ε2Δv + V (x) v = ε1α (Iα ∗ F(v))f(v) in RN , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|N-α is the Riesz potential, F ∈ C1(R, R), F´(s) = f(s) and ε > 0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as ε → 0, to a local minima of V (x) under general conditions on F(s). Our result is new also for f(s) = |s|p-2s and applicable for p ∈ (NN+α, NN+-α2 ). Especially, we can give the existence result for locally sublinear case p ∈ (NN+α , 2), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least cupl(K)+1 solutions concentrating around K as ε → 0, where K ⊂ Ω is the set of minima of V (x) in a bounded potential well Ω, that is, m0 ≡ infxεΩ V (x) < infxε∂Ω V (x) and K = {x ∈ Ω; V (x) = m0}.

14
Citation
(Scopus)
• Multi-bump solutions for logarithmic Schrödinger equations

Kazunaga Tanaka, Chengxiang Zhang

Calculus of Variations and Partial Differential Equations   56 ( 2 )  2017.04  [Refereed]

View Summary

We study spatially periodic logarithmic Schrodinger equations:
{ - Delta u + V(x)u = Q(x)u log u(2), u &gt; 0 in R-N, (LS)
u is an element of H-1 (R-N),
where N &gt;= 1 and V(x), Q(x) are spatially 1-periodic functions of class C-1. We take an approach using spatially 2L-periodic problems (L &gt;&gt; 1) and we show the existence of infinitely many multi-bump solutions of (LS) which are distinct under Z(N)-action.

39
Citation
(Scopus)
• Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

Silvia Cingolani, Louis Jeanjean, Kazunaga Tanaka

Journal of Fixed Point Theory and Applications   19 ( 1 ) 37 - 66  2017.03

View Summary

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations LA,Vħu=f(|u|2)uinRNwhere N≥ 3 , LA,Vħ is the Schrödinger operator with a magnetic field having source in a C1 vector potential A and a scalar continuous (electric) potential V defined by LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ RN such that (Formula presented.). For ħ> 0 small we prove the existence of at least cupl (K) + 1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→ 0.

7
Citation
(Scopus)
• Remarks on the clark theorem

Guosheng Jiang, Kazunaga Tanaka, Chengxiang Zhang

Journal of Nonlinear and Convex Analysis   18 ( 8 ) 1421 - 1434  2017

View Summary

The Clark theorem is important in critical point theory. For a class of even functionals it ensures the existence of infinitely many negative critical values converging to 0 and it has important applications to sublinear elliptic problems. We study the convergence of the corresponding critical points and we give a characterization of accumulation points of critical points together with examples, in which critical points with negative critical values converges to nonzero critical point. Our results improve the abstract results in Kajikiya [4] and Liu-Wang [7].

• Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well

Silvia Cingolani, Louis Jeanjean, Kazunaga Tanaka

Calculus of Variations and Partial Differential Equations   53 ( 1-2 ) 413 - 439  2015.05

View Summary

We consider singularly perturbed nonlinear Schrödinger equations (Formula Presented) where (Formula Presented) and f is a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domain (Formula Presented) such that (Formula Presented) and we set (Formula Presented). For (Formula Presented) small we prove the existence of at least (Formula Presented) solutions to (0.1) concentrating, as (Formula Presented) around K. We remark that, under our assumptions of f, the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.

15
Citation
(Scopus)
• Continuum of solutions for an elliptic problem with critical growth in the gradient

David Arcoya, Colette De Coster, Louis Jeanjean, Kazunaga Tanaka

Journal of Functional Analysis   268 ( 8 ) 2298 - 2335  2015.04

View Summary

We consider the boundary value problem(Pλ)u∈H01(Ω)∩L∞(Ω):-δu=λc(x)u+μ(x)|∇u|2+h(x), where Ω⊂RN, N≥3 is a bounded domain with smooth boundary. It is assumed that c{greater-than but not equal to}0, c, h belong to Lp(Ω) for some p>N/2 and that μ∈L∞(Ω). We explicitly describe a condition which guarantees the existence of a unique solution of (Pλ) when λ<0 and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of (P0). It crosses the axis λ=0 if (P0) has a solution, otherwise it bifurcates from infinity at the left of the axis λ=0. Assuming that (P0) has a solution and strengthening our assumptions to μ(x)≥μ1>0 and h{greater-than but not equal to}0, we show that the continuum bifurcates from infinity on the right of the axis λ=0 and this implies, in particular, the existence of two solutions for any λ>0 sufficiently small.

27
Citation
(Scopus)
• Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions

David Arcoya, Colette De Coster, Louis Jeanjean, Kazunaga Tanaka

Journal of Mathematical Analysis and Applications   420 ( 1 ) 772 - 780  2014.12  [Refereed]

View Summary

In this note we present some uniqueness and comparison results for a class of problem of the form
-Lu = H(x, u, del u) + h(x), u is an element of H-0(1)(Omega) boolean AND L-infinity(Omega), (0.1)
where Omega subset of R-N, N &gt;= 2 is a bounded domain, L is a general elliptic second order linear operator with bounded coefficients and H is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp. (C) 2014 Elsevier Inc. All rights reserved.

22
Citation
(Scopus)
• Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains

Jaeyoung Byeon, Kazunaga Tanaka

Calculus of Variations and Partial Differential Equations   50 ( 1-2 ) 365 - 397  2014.05  [Refereed]

View Summary

In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem:
-Delta u = u(p) in Omega(t) u=0 on partial derivative Omega(t).
Here 1 &lt; p &lt; N+2/N-2 when N &gt;= 3, 1 &lt; p &lt; infinity when N = 2 and Omega(t) and is a tubular domain which expands as t -&gt; infinity. See (1.6) below for a precise definition of expanding tubular domain. When the section D of Omega(t) is a ball, the existence of multi-bump positive solutions is shown by Dancer and Yan (Commun Partial Differ Equ, 27(1-2), 23-55, 2002) and by Ackermann et al. (Milan J Math, 79(1), 221-232, 2011) under the assumption of a non-degeneracy of a solution of a limit problem. In this paper we introduce a new local variational method which enables us to show the existence of multi-bump positive solutions without the non-degeneracy condition for the limit problem. In particular, we can show the existence for all N &gt;= 2 without the non-degeneracy condition. Moreover we can deal with more general domains, for example, a domain whose section is an annulus, for which least energy solutions of the limit problem are really degenerate.

11
Citation
(Scopus)
• On a system of nonlinear Schrodinger equations with quadratic interaction

Nakao Hayashi, Tohru Ozawa, Kazunaga Tanaka

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   30 ( 4 ) 661 - 690  2013.07  [Refereed]

View Summary

We study a system of nonlinear Schrodinger equations with quadratic interaction in space dimension n &lt;= 6. The Cauchy problem is studied in L-2, in H-1, and in the weighted L-2 space &lt; x &gt; L--1(2) = F(H-1) under mass resonance condition, where &lt; x &gt; = (1 + vertical bar x vertical bar(2))(1/2) and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations. (c) 2012 Elsevier Masson SAS. All rights reserved.

52
Citation
(Scopus)
• High energy rotation type solutions of the forced pendulum equation

Patricio Felmer, André De Laire, Salomé Martínez, Kazunaga Tanaka

Nonlinearity   26 ( 5 ) 1473 - 1499  2013.05

View Summary

In this article we study the existence and asymptotic profiles of high-energy rotation type solutions of the singularly perturbed forced pendulum equation ε2u'ε+sin uε= ε2ε in (-L, L). We prove that the asymptotic profile of these solutions is described in terms of an energy function which satisfy an integro-differential equation. Also we show that given a suitable energy function E satisfying the integro-differential equation, a family of solutions of the pendulum equation above exists having E as the asymptotic profile, when ε → 0. © 2013 IOP Publishing Ltd & London Mathematical Society.

1
Citation
(Scopus)
• Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

Jaeyoung Byeon, Kazunaga Tanaka

Journal of the European Mathematical Society   15 ( 5 ) 1859 - 1899  2013

View Summary

We consider a singularly perturbed elliptic equation ε 2Δu - V (x)u + f(u) = 0, u(x) > 0 on ℝN, lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ε ℝN: The singularly perturbed problem has corresponding limiting problems ΔU - cU + f(U) = 0, U(x) > 0 on ℝN, lim |x|→∞U(x) = 0, c > 0: Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f ε C1 In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods. © European Mathematical Society 2013.

35
Citation
(Scopus)
• Nonlinear elliptic equations in strip-like domains dedicated to Professor Antonio Ambrosetti

Jaeyoung Byeon, Kazunaga Tanaka

Advanced Nonlinear Studies   12 ( 4 ) 749 - 765  2012

View Summary

We study the existence of a positive solution of a nonlinear elliptic equation Mathematical equation represented where k ≥ 2 and D is a bounded domain in Rℓ, ℓ ≥ 1. We give almost necessary and sufficient condition on f (ℰ) for the existence of a positive solution, which is inspired by the works of Berestycki-Lions [1] and Berestycki- Gallouët-Kavian [3].

• A local mountain pass type result for a system of nonlinear Schrödinger equations

Norihisa Ikoma, Kazunaga Tanaka

Calculus of Variations and Partial Differential Equations   40 ( 3 ) 449 - 480  2011

View Summary

We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ1, μ2, β > 0 and V1(x), V2(x): RN → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ε RN the least energy level m(P) for non-trivial vector solutions of the rescaled "limit" problem: We assume that there exists an open bounded set Λ ⊂ RN satisfying We show that (*) possesses a family of non-trivial vector positive solutions which concentrates-after extracting a subsequence e{open}n → 0-to a point P0 ε Λ with m(P0) = infPεΛm(P). Moreover (v1e{open}(x), v2e{open}(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling. © 2010 Springer-Verlag.

50
Citation
(Scopus)
• Nonlinear scalar field equations in ℝN: Mountain pass and symmetric mountain pass approaches

Jun Hirata, Norihisa Ikoma, Kazunaga Tanaka

Topological Methods in Nonlinear Analysis   35 ( 2 ) 253 - 276  2010

View Summary

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝN : -Δu = g(u) in ℝN', u ∈ H1 (ℝN). We give an extension of the existence results due to H. Berestycki, T. GaIloüe't and O. Kavian [2]. We take a mountain pass approach in H1(ℝ N) and introduce a new method generating a Palais-Smale sequence with an additional property related to Pohozaev identity. ©2010 Juliusz Schauder Center for Nonlinear Studies.

• Sign-Changing multi-bump solutions for nonlinear schrödinger equations with steep potential wells

Yohei Sato, Kazunaga Tanaka

Transactions of the American Mathematical Society   361 ( 12 ) 6205 - 6253  2009.12  [Refereed]

View Summary

We study the nonlinear Schrodinger equations:
(P(lambda)) -Delta u+(lambda(2)a(x)+1)u = vertical bar u vertical bar(p-1)u, u is an element of H(1)(R(N)),
where p &gt; 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) &gt;= 0,0 &lt; lim inf(vertical bar x vertical bar -&gt;infinity) a(x) &lt;= lim sup(vertical bar x vertical bar -&gt;infinity) a(x) &lt; infinity and a(-1)(0) consists of 2 connected bounded smooth components Omega(1) and Omega(2).
We study the existence of solutions (u(lambda)) of (P(lambda)) which converge to 0 in R(N) \ (Omega(1) boolean OR Omega(2)) and to a prescribed pair (v(1)(x), v(2)(x)) of solutions of the limit problem:
-Delta v(i) + v(i) = vertical bar v(i)vertical bar(p-1)v(i) in Omega(i)
(i = 1, 2) as lambda -&gt; infinity.

25
Citation
(Scopus)
• Uniqueness of radially symmetric positive solutions for - Δ u + u = up in an annulus

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka

Journal of Differential Equations   245 ( 5 ) 1198 - 1209  2008.09

View Summary

In this article we prove that the semi-linear elliptic partial differential equation- Δ u + u = up in Ω,u > 0 in Ω, u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus {x ∈ RN | a < | x | < b}, with N ≥ 2, 0 < a < b ≤ ∞. We also show the positive solution is non-degenerate. © 2008 Elsevier Inc. All rights reserved.

21
Citation
(Scopus)
• Highly oscillatory behavior of the activator in the Gierer and Meinhardt system

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka

Mathematische Annalen   340 ( 4 ) 749 - 773  2008.04

View Summary

In this article we construct a new type of solutions for the Gierer and Meinhardt system -ε2uxx + u &=&u 2/v in (0,L),- vxx+ v &=& u2 in (0, L) with boundary conditions u x (0) = u x (L) = 0 and v x (0) = v x (L) = 0. As ε approaches zero, we construct a family of positive solution (u ε , v ε ) such that the activator u ε oscillates c 0/ε times, with c 0 in an appropriate range, while the inhibitor remains close to a limiting profile, which is a strictly decreasing function. © 2007 Springer-Verlag.

2
Citation
(Scopus)
• The fixed energy problem for a class of nonconvex singular Hamiltonian systems

C. Carminati, É Séré, K. Tanaka

Journal of Differential Equations   230 ( 1 ) 362 - 377  2006.11

View Summary

We consider a noncompact hypersurface H in R2 N which is the energy level of a singular Hamiltonian of "strong force" type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type. © 2006 Elsevier Inc. All rights reserved.

11
Citation
(Scopus)
• High frequency solutions for the singularly-perturbed one-dimensional nonlinear Schrödinger equation

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka

Archive for Rational Mechanics and Analysis   182 ( 2 ) 333 - 366  2006.10

View Summary

This article is devoted to the nonlinear Schrödinger equation [InlineMediaObject not available: see fulltext.] when the parameter ε approaches zero. All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We prove that for every envelope, there exists a family of solutions reaching that asymptotic behavior, in the case of bounded intervals. We use a combination of the Nehari finite dimensional reduction together with degree theory. Our main contribution is to compute the degree of each cluster, which is a key piece of information in order to glue them.

11
Citation
(Scopus)
• On the number of positive solutions of singularly perturbed ID nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka

Journal of the European Mathematical Society   8 ( 2 ) 253 - 268  2006  [Refereed]

View Summary

We study singularly perturbed 1D nonlinear Schrodinger equations (1.1). When V ( x) has multiple critical points, ( 1.1) has a wide variety of positive solutions for small epsilon and the number of positive solutions increases to infinity as epsilon --&gt; 0. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V (x). Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

5
Citation
(Scopus)
• Multiple stable patterns for some reaction-diffusion equation in disrupted environments

Takanori Ide, Kazuhiro Kurata, Kazunaga Tanaka

Discrete and Continuous Dynamical Systems   14 ( 1 ) 93 - 116  2006.01  [Refereed]

View Summary

We study the existence of multiple positive stable solutions for
-epsilon(2)Delta u(x) = u(x)(2) (b(x) - u(x)) in Omega, partial derivative u/partial derivative n (x) = 0 on partial derivative Omega.
Here epsilon &gt; 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

• Multiple stable patterns for some reaction-diffusion equation in disrupted environments

T Ide, K Kurata, K Tanaka

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   14 ( 1 ) 93 - 116  2006.01

View Summary

We study the existence of multiple positive stable solutions for
-epsilon(2)Delta u(x) = u(x)(2) (b(x) - u(x)) in Omega, partial derivative u/partial derivative n (x) = 0 on partial derivative Omega.
Here epsilon &gt; 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

• A remark on periodic solutions of singular Hamiltonian systems

Shinji Adachi, Kazunaga Tanaka, Masahito Terui

Nonlinear Differential Equations and Applications   12 ( 3 ) 265 - 274  2005.12  [Refereed]

View Summary

In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force. In particular for potential V (t, q) similar to - 1/dist(q, D)(alpha), where D is a compact C-3-surface in R-N, we prove the existence of a non- collision periodic solution.

2
Citation
(Scopus)
• A remark on periodic solutions of singular Hamiltonian systems

S Adachi, K Tanaka, M Terui

NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS   12 ( 3 ) 265 - 274  2005.12  [Refereed]

View Summary

In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force. In particular for potential V (t, q) similar to - 1/dist(q, D)(alpha), where D is a compact C-3-surface in R-N, we prove the existence of a non- collision periodic solution.

2
Citation
(Scopus)
• A positive solution for a nonlinear schrödinger equation on ℝN

Louis Jeanjean, Kazunaga Tanaka

Indiana University Mathematics Journal   54 ( 2 ) 443 - 464  2005  [Refereed]

View Summary

We prove the existence of a positive solution for a class of equations of the form
-Delta u + V(x)u = f (u), u is an element of H-1 (R-N).
On the nonlinearity f, only conditions around 0 and at infinity are required.

137
Citation
(Scopus)
• Multi-clustered high-energy solutions for a phase transition problem

PL Felmer, S Martinez, K Tanaka

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS   135   731 - 765  2005

View Summary

We study the balanced Allen-Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions mu(epsilon) (x) with an increasing number of layers as epsilon -&gt; 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

• Multi-clustered high-energy solutions for a phase transition problem

Patricio L. Felmer, Salomé Martínez, Kazunaga Tanaka

Royal Society of Edinburgh - Proceedings A   135 ( 4 ) 731 - 765  2005  [Refereed]

View Summary

We study the balanced Allen-Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions mu(epsilon) (x) with an increasing number of layers as epsilon -&gt; 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

9
Citation
(Scopus)
• Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

L Jeanjean, K Tanaka

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   21 ( 3 ) 287 - 318  2004.11

View Summary

We consider a class A equations of the form
-epsilon(2) Deltau + V(x)u = f(u), u is an element of H-1(R-N).
By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as epsilon --&gt; 0. We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function xi --&gt; f(xi)/xi. We deal with asymptotically linear as well as superlinear nonlinearities.

• Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

Louis Jeanjean, Kazunaga Tanaka

Calculus of Variations and Partial Differential Equations   21 ( 3 ) 287 - 318  2004.11  [Refereed]

View Summary

We consider a class A equations of the form
-epsilon(2) Deltau + V(x)u = f(u), u is an element of H-1(R-N).
By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as epsilon --&gt; 0. We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function xi --&gt; f(xi)/xi. We deal with asymptotically linear as well as superlinear nonlinearities.

154
Citation
(Scopus)
• A note on a mountain pass characterization of least energy solutions

Louis Jeanjean, Kazunaga Tanaka

Advanced Nonlinear Studies   3 ( 4 ) 445 - 455  2003.11

View Summary

We consider the equation
-u" = g(u), u(x) is an element of H-1 (R) (0.1)
Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on R-N.

32
Citation
(Scopus)
• A note on a mountain pass characterization of least energy solutions

L Jeanjean, K Tanaka

ADVANCED NONLINEAR STUDIES   3 ( 4 ) 445 - 455  2003.11  [Refereed]

View Summary

We consider the equation
-u" = g(u), u(x) is an element of H-1 (R) (0.1)
Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on R-N.

• Multiplicity of positive solutions of a nonlinear Schrodinger equation

YH Ding, K Tanaka

MANUSCRIPTA MATHEMATICA   112 ( 1 ) 109 - 135  2003.09  [Refereed]

View Summary

We consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation:
-Deltau + (lambdaa(x) + b(x))u = u(p), u &gt; 0 in R-N, (P-lambda)
where p is an element of (1, N+2/N-2) if N greater than or equal to 3 and p is an element of (1, infinity) if N = 1, 2, and a (x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Omega := int a(-1) (0) consisting of k components Omega(1),..., Omega(k) and the first eigenvalues of -Delta + b(x) on Omega(j) under Dirichlet boundary condition are positive for all j = 1, 2,..., k. Under these conditions we show that (P-lambda) has at least 2(k) - 1 positive solutions for large lambda. More precisely we show that for any given non-empty subset J subset of {1, 2,... k}, (P-lambda) has a positive solutions u(lambda)(x) for large lambda. In addition for any sequence lambda(n) --&gt; infinity we can extract a subsequence lambda(ni) along which ulambda(ni) converges strongly in H-1 (R-N). Moreover the limit function u(x) = lim(i--&gt;infinity) ulambda(ni) satisfies
(i) For j is an element of J the restriction u\Omega(j) of u(x) to Omega(j) is a least energy solution of -Deltav + b(x)v = v(p) in Omega(j) an v = 0 on partial derivativeOmega(j).
(ii) u(x) = 0 for x is an element of R-N \ (boolean ORjis an element ofJ Omega(j)).

88
Citation
(Scopus)
• Multiplicity of positive solutions of a nonlinear Schrodinger equation

YH Ding, K Tanaka

MANUSCRIPTA MATHEMATICA   112 ( 1 ) 109 - 135  2003.09  [Refereed]

View Summary

We consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation:
-Deltau + (lambdaa(x) + b(x))u = u(p), u &gt; 0 in R-N, (P-lambda)
where p is an element of (1, N+2/N-2) if N greater than or equal to 3 and p is an element of (1, infinity) if N = 1, 2, and a (x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Omega := int a(-1) (0) consisting of k components Omega(1),..., Omega(k) and the first eigenvalues of -Delta + b(x) on Omega(j) under Dirichlet boundary condition are positive for all j = 1, 2,..., k. Under these conditions we show that (P-lambda) has at least 2(k) - 1 positive solutions for large lambda. More precisely we show that for any given non-empty subset J subset of {1, 2,... k}, (P-lambda) has a positive solutions u(lambda)(x) for large lambda. In addition for any sequence lambda(n) --&gt; infinity we can extract a subsequence lambda(ni) along which ulambda(ni) converges strongly in H-1 (R-N). Moreover the limit function u(x) = lim(i--&gt;infinity) ulambda(ni) satisfies
(i) For j is an element of J the restriction u\Omega(j) of u(x) to Omega(j) is a least energy solution of -Deltav + b(x)v = v(p) in Omega(j) an v = 0 on partial derivativeOmega(j).
(ii) u(x) = 0 for x is an element of R-N \ (boolean ORjis an element ofJ Omega(j)).

88
Citation
(Scopus)
• A remark on least energy solutions in RN

Louis Jeanjean, Kazunaga Tanaka

Proceedings of the American Mathematical Society   131 ( 8 ) 2399 - 2408  2003.08  [Refereed]

View Summary

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R-N:
-Deltau = g(u), u is an element of H-1(R-N),
where N greater than or equal to 2. Without the assumption of the monotonicity of t bar right arrow g(t)/t, we show that the mountain pass value gives the least energy level.

248
Citation
(Scopus)
• Clustering layers and boundary layers in spatially inhomogeneous phase transition problems

K Nakashima, K Tanaka

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   20 ( 1 ) 107 - 143  2003  [Refereed]

• Clustering layers and boundary layers in spatially inhomogeneous phase transition problems

Kimie Nakashima, Kazunaga Tanaka

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire   20 ( 1 ) 107 - 143  2003

View Summary

The existence of solutions with multiple transition layers for the spatially inhomogeneous phase transition problem is discussed. The location and multiplicity of the transition layers, especially the clustering layers and the boundary layers were studied. The existence of complicated dynamics, described in terms of symbolic sequence of integers was proved using Conley index theory.

43
Citation
(Scopus)
• A remark on least energy solutions in R-N

L Jeanjean, K Tanaka

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   131 ( 8 ) 2399 - 2408  2003  [Refereed]

View Summary

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R-N:
-Deltau = g(u), u is an element of H-1(R-N),
where N greater than or equal to 2. Without the assumption of the monotonicity of t bar right arrow g(t)/t, we show that the mountain pass value gives the least energy level.

• An elementary construction of complex patterns in nonlinear schrödinger equations

Manuel Del Pino, Patrido Felmer, Kazunaga Tanaka

Nonlinearity   15 ( 5 ) 1653 - 1671  2002.09

View Summary

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

21
Citation
(Scopus)
• A positive solution for an asymptotically linear elliptic problem on ℝn autonomous at infinity

Louis Jeanjean, Kazunaga Tanaka

ESAIM - Control, Optimisation and Calculus of Variations   7 ( 24 ) 597 - 614  2002.08  [Refereed]

View Summary

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on R-N. The main difficulties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the "Problem at infinity" is autonomous, in contrast to just periodic, can be used in order to regain compactness.

103
Citation
(Scopus)
• Existence of positive solutions for a class of nonhomogeneous elliptic equations in RN

Shinji Adachi, Kazunaga Tanaka

Nonlinear Analysis, Theory, Methods and Applications   48 ( 5 ) 685 - 705  2002.02

View Summary

The existence of positive solutions for a class of nonhomogeneous elliptic equations in RN was analyzed. The mountain pass minimax value was strictly less than the first level of the break down of (PS)c-condition for I(u). Through the Mountain Pass Theorem, a critical point was obtained which was a positive solution of the nonhomogeneous elliptic equation.

21
Citation
(Scopus)
• Existence of positive solutions for a class of nonhomogeneous elliptic equations in R-N

S Adachi, K Tanaka

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   48 ( 5 ) 685 - 705  2002.02  [Refereed]

• A positive solution for an asymptotically linear elliptic problem on R-N autonomous at infinity

L Jeanjean, K Tanaka

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS   7 ( 24 ) 597 - 614  2002  [Refereed]

View Summary

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on R-N. The main difficulties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the "Problem at infinity" is autonomous, in contrast to just periodic, can be used in order to regain compactness.

103
Citation
(Scopus)
• Multiple positive solutions for nonhomogeneous elliptic equations

S Adachi, K Tanaka

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   47 ( 6 ) 3783 - 3793  2001.08  [Refereed]

View Summary

We study the existence of multiple positive solutions for nonhomogeneous superlinear elliptic problems in R-N:
{-Deltau + u = g(x,u) + f(x) in R-N,
{u &gt; 0 in R-N,
{u is an element of H-1 (R-N),
where g(x,u) is a superlinear function such that g (x,0) equivalent to 0 and f is an element of H-1(R-N), f greater than or equal to 0, f not equivalent to 0. A typical example of g(x,u) is g(x,u) = a(x)u(P), where 1 &lt; p &lt; (N+2)/(N-2) if N greater than or equal to 3, 1 &lt; p &lt; infinity if N = 1, 2. We show the existence of multiple positive solutions under suitable conditions on g(x, u) and f(x).

• Multiple positive solutions for nonhomogeneous elliptic equations

S Adachi, K Tanaka

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   47 ( 6 ) 3783 - 3793  2001.08  [Refereed]

View Summary

We study the existence of multiple positive solutions for nonhomogeneous superlinear elliptic problems in R-N:
{-Deltau + u = g(x,u) + f(x) in R-N,
{u &gt; 0 in R-N,
{u is an element of H-1 (R-N),
where g(x,u) is a superlinear function such that g (x,0) equivalent to 0 and f is an element of H-1(R-N), f greater than or equal to 0, f not equivalent to 0. A typical example of g(x,u) is g(x,u) = a(x)u(P), where 1 &lt; p &lt; (N+2)/(N-2) if N greater than or equal to 3, 1 &lt; p &lt; infinity if N = 1, 2. We show the existence of multiple positive solutions under suitable conditions on g(x, u) and f(x).

• Four positive solutions for the semilinear elliptic equation: -Delta mu+mu=alpha(chi)mu(p)+f(chi) in R-N

S Adachi, K Tanaka

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   11 ( 1 ) 63 - 95  2000.08  [Refereed]

View Summary

We consider the existence of positive solutions of the following semilinear elliptic problem in R-N:
[GRAPHICS]
where 1 &lt; p &lt; N+2/N-2 (N greater than or equal to 3), 1 &lt; p &lt; infinity (N = 1, 2), a(x) is an element of C(R-N), f(x) is an element of H-1(R-N) and f(x) greater than or equal to 0. under the conditions;
1 degrees a(x) is an element of (0,1] for all x is an element of R-N,
2 degrees a(x) --&gt; 1 as \x\ --&gt; infinity,
3 degrees there exist delta &gt; 0 and C &gt; 0 such that
a(x) - 1 greater than or equal to -Ce-(2+delta)\x\ for all x is an element of R-N,
4 degrees a(x) not equal 1,
we show that (*) has at least four positive solutions for sufficiently small \\f\$$H-1(RN)) but f not equal 0. • Trudinger type inequalities in RN and their best exponents Shinji Adachi, Kazunaga Tanaka Proceedings of the American Mathematical Society 128 ( 7 ) 2051 - 2057 2000 [Refereed] View Summary We study Trudinger type inequalities in R-N and their best exponents alpha(N). We show for alpha is an element of (0, alpha(N)), alpha(N) = N omega(N-1)((N-1)) (omega(N-1) is the surface area of the unit sphere in R-N), there exists a constant C-alpha&gt;0 such that [GRAPHICS] for all u is an element of W-1,W-N (R-N) \ {0}. Here Phi(N) (xi) is defined by [GRAPHICS] is also shown that (*) with alpha greater than or equal to alpha(N) is false, which is different from the usual Trudinger's inequalities in bounded domains. 237 Citation (Scopus) • Four positive solutions for the semilinear elliptic equation: - Δu + u = a(x)up + f(x) in ℝN Shinji Adachi, Kazunaga Tanaka Calculus of Variations and Partial Differential Equations 11 ( 1 ) 63 - 95 2000 View Summary We consider the existence of positive solutions of the following semilinear elliptic problem in ℝN: (Formula Presented) where 1 &lt p &lt N + 2/N - 2 (N ≥ 3), 1 &lt p &lt ∞ (N = 1, 2), a(x) ∈ C(ℝN), f(x) ∈ H-1 (ℝN) and f(x) ≥ 0. Under the conditions: 1° a(x) ∈ (0, 1) for all x ∈ ℝN, 2° a(x) → 1 as |x| → ∞, 3° there exist δ &gt 0 and C &gt 0 such that a(x) - 1 ≥ -Ce-(2+δ)|x| for all x ∈ ℝN, 4° a(x) ≢ 1, we show that (*) has at least four positive solutions for sufficiently small ||f||H-1(ℝN) but f ≢ 0. 92 Citation (Scopus) • Scattering solutions for planar singular hamiltonian systems via minimization P. Felmer, K. Tanaka Adv. Diff. Eqn. 5 1519 - 1544 2000 • Hyperbolic-like solutions for singular Hamiltonian systems Patricio Felmer, Kazunaga Tanaka Nonlinear Differential Equations and Applications 7 ( 1 ) 43 - 65 2000 [Refereed] View Summary We study the existence of unbounded solutions of singular Hamiltonian systems: q + del V(q) = 0, were V(q) similar to - 1/\q\alpha is a potential with a singularity. For a class of singular potentials with a strong force alpha &gt; 2, we show the existence of at least one hyperbolic-like solutions. More precisely, for given H &gt; 0 and theta(+), theta(-) is an element of SN-1, we find a solution q(t) of (*) satisfying [GRAPHICS] 6 Citation (Scopus) • Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds, K. Tanaka Ann. Inst. H. Poincare Anal. Non Lineaire. 17 1 - 33 2000 • Trudinger type inequalities in R^N and their best exponents S. Adachi, K. Tanaka Proc. Amer. Math. Soc. 128 2051 - 2057 2000 • Scattering solutions for planar singular hamiltonian systems via minimization P. Felmer, K. Tanaka Adv. Diff. Eqn. 5 1519 - 1544 2000 • Hyperbolic-like solutions for singular Hamiltonian systems P Felmer, K Tanaka NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 7 ( 1 ) 43 - 65 2000 [Refereed] View Summary We study the existence of unbounded solutions of singular Hamiltonian systems: q + del V(q) = 0, were V(q) similar to - 1/\q\alpha is a potential with a singularity. For a class of singular potentials with a strong force alpha &gt; 2, we show the existence of at least one hyperbolic-like solutions. More precisely, for given H &gt; 0 and theta(+), theta(-) is an element of SN-1, we find a solution q(t) of (*) satisfying [GRAPHICS] • Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds K Tanaka ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 17 ( 1 ) 1 - 33 2000.01 [Refereed] View Summary We study the existence of periodic solutions of singular Hamiltonian systems as well as closed geodesics on non-compact Riemannian manifolds via variational methods. For Hamiltonian systems, we show the existence of a periodic solution of prescribed-energy problem: (q) double over dot + del V(q) = 0, 1/2\(q) over dot\(2) + V(q) = 0 under the conditions: (i) V(q) &lt; 0 for all q is an element of R-N \ {0}; (ii) V(q) similar to -l/\q\(2) as \q\ similar to 0 and \q\ similar to infinity. For closed geodesics, we show the existence of a non-constant closed geodesic on (R x SN-1, g) under the condition: g((s,x)) similar to ds(2) + h(0) as s similar to +/- infinity, where h(0) is the standard metric on SN-1. (C) 2000 Editions scientifiques et medicales Elsevier SAS. • Uniqueness of positive radial solutions of semilinear elliptic equations in R-N and Sere's non-degeneracy condition Y Kabeya, K Tanaka COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 24 ( 3-4 ) 563 - 598 1999 [Refereed] • Uniqueness of positive radial solutions of semilinear elliptic equations in R-N and Sere's non-degeneracy condition Y Kabeya, K Tanaka COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 24 ( 3-4 ) 563 - 598 1999 [Refereed] • On Keplerian N-body type problems, A. Ambrosetti, K. Tanaka in Nonlinear Analysis and continuum mechanics: Paper for the 65-thrbirthday of James Serrin (G. Buttazzo, G.P. Galdi, E. Lanconelli, P.rPucci ed.) Springer 15 - 25 1998 • On Keplerian N-body type problems, A. Ambrosetti, K. Tanaka in Nonlinear Analysis and continuum mechanics: Paper for the 65-th birthday of James Serrin (G. Buttazzo, G.P. Galdi, E. Lanconelli, P.Pucci ed.) Springer 15 - 25 1998 • Multiple Positive solutions for some nonlinear elliptic systems K. Tanaka Topological mathods in nonlinear Analysis 10 15 - 45 1997 • Multiple Positive solutions for some nonlinear elliptic systems K. Tanaka Topological mathods in nonlinear Analysis 10 15 - 45 1997 • Periodic solutions of first order singular Hamiltonian systems K Tanaka NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 26 ( 4 ) 691 - 706 1996.02 • Periodic solutions of first order singnlar Hamiltonian systems K. Tanaka Nonlinear Analysis,Theory,Methods &amp; Applications 26 791 - 706 1996 • A REMARK ON POSITIVE RADIAL SOLUTIONS OF THE ELLIPTIC EQUATION DELTA-U+K(VERTICAL-BAR-X-VERTICAL-BAR)U((N+2)/(N-2))=0 IN R(N) Y SASAHARA, K TANAKA PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 123 ( 2 ) 527 - 531 1995.02 View Summary We consider the following semilinear elliptic equation involving critical Sobolev exponents: -Delta u = K(\x$$u((n+2)/(n-2)) in R(n),
u(x) --&gt; 0 as \x\ --&gt; infinity,
where n greater than or equal to 3, K(r) is an element of C([0, infinity), R). We grove the existence of a positive radial solution with asymptotic behavior C/\x$$n-2) at \x\ = infinity under the conditions (i) K(r) &gt; 0 for all r &gt; 0, (ii) K(0) = K(infinity), and (iii) there exist C, delta &gt; 0 such that K(r) greater than or equal to K(0) - Cr-delta for small r &gt; 0 and K(r) greater than or equal to K(0) - Cr--delta for large r &gt; 0. • A REMARK ON POSITIVE RADIAL SOLUTIONS OF THE ELLIPTIC EQUATION DELTA-U+K(VERTICAL-BAR-X-VERTICAL-BAR)U((N+2)/(N-2))=0 IN R(N) Y SASAHARA, K TANAKA PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 123 ( 2 ) 527 - 531 1995.02 [Refereed] View Summary We consider the following semilinear elliptic equation involving critical Sobolev exponents: -Delta u = K(\x$$u((n+2)/(n-2)) in R(n),
u(x) --&gt; 0 as \x\ --&gt; infinity,
where n greater than or equal to 3, K(r) is an element of C([0, infinity), R). We grove the existence of a positive radial solution with asymptotic behavior C/\x\(n-2) at \x\ = infinity under the conditions (i) K(r) &gt; 0 for all r &gt; 0, (ii) K(0) = K(infinity), and (iii) there exist C, delta &gt; 0 such that K(r) greater than or equal to K(0) - Cr-delta for small r &gt; 0 and K(r) greater than or equal to K(0) - Cr--delta for large r &gt; 0.

• Homoclinic orbits on non-compact Riemannian manifolds for second order Hamiltonian systems

F. Gianonni, L. Jeanjean

Red.Sem.Mat.Univ.Padova   93   153 - 176  1995

• Homoclinic orbits on non-compact Riemannian manifolds for second order Hamiltonian systems

F. Gianonni, L. Jeanjean

Red.Sem.Mat.Univ.Padova   93   153 - 176  1995

• A NOTE ON GENERALIZED SOLUTIONS OF SINGULAR HAMILTONIAN-SYSTEMS

K TANAKA

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   122 ( 1 ) 275 - 284  1994.09  [Refereed]

View Summary

We consider T-periodic solutions of singular Hamiltonian systems with weak force q + delV(q, t) = 0, where V(q, t) approximately - 1/\q\alpha near q = 0 with alpha is-an-element-of (0, 2) . In particular, we study some properties of generalized T-periodic solutions, which were introduced by Bahri and Rabinowitz.

• A prescribed-energy problem for a conservative singular Hamiltonian system

Kazunaga Tanaka

Archive for Rational Mechanics and Analysis   128 ( 2 ) 127 - 164  1994.06

15
Citation
(Scopus)
• A note on the existence of multiple homoclinic orbits for a perturbed radial potential

K. Tanaka

Nonlinear Differential Equations and Applications   1   149 - 162  1994

• A note on generalized solutions of singular hamiltonian systems

Kazunaga Tanaka

Proceedings of the American Mathematical Society   122 ( 1 ) 275 - 284  1994

View Summary

We consider T-periodic solutions of singular Hamiltonian systems with weak force In particular, we study some properties of generalized T-periodic solutions, which were introduced by Bahri and Rabinowitz. © 1994 American Mathematical Society.

19
Citation
(Scopus)
• PERIODIC-SOLUTIONS WITH PRESCRIBED ENERGY FOR SOME KEPLERIAN N-BODY PROBLEMS

A AMBROSETTI, K TANAKA, E VITILLARO

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   11 ( 6 ) 613 - 632  1994  [Refereed]

View Summary

We prove the existence of periodic solutions with prescribed energy for a class of N-body type problems with Keplerian like interaction.

• A PRESCRIBED-ENERGY PROBLEM FOR A CONSERVATIVE SINGULAR HAMILTONIAN SYSTEM

K TANAKA

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   128 ( 2 ) 127 - 164  1994  [Refereed]

• A note on the existence of multiple homoclinic orbits for a perturbed radial potential

K. Tanaka

Nonlinear Differential Equations and Applications   1   149 - 162  1994

• PERIODIC-SOLUTIONS WITH PRESCRIBED ENERGY FOR SOME KEPLERIAN N-BODY PROBLEMS

A AMBROSETTI, K TANAKA, E VITILLARO

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   11 ( 6 ) 613 - 632  1994  [Refereed]

View Summary

We prove the existence of periodic solutions with prescribed energy for a class of N-body type problems with Keplerian like interaction.

• A PRESCRIBED ENERGY PROBLEM FOR A SINGULAR HAMILTONIAN SYSTEM WITH A WEAK FORCE

K TANAKA

JOURNAL OF FUNCTIONAL ANALYSIS   113 ( 2 ) 351 - 390  1993.05  [Refereed]

• NON-COLLISION SOLUTIONS FOR A 2ND-ORDER SINGULAR HAMILTONIAN SYSTEM WITH WEAK FORCE

K TANAKA

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   10 ( 2 ) 215 - 238  1993  [Refereed]

View Summary

Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems:
[GRAPHICS] We assume V(q, t) &lt; 0 for all q, t and V(q, t), V(q)(q, t) --&gt; 0 as Absolute value of q --&gt; infinity. Moreover we assume V(q, t) is of a form: V(q, t) = - 1/\q\alpha + U(q,t) where 0 &lt; alpha &lt; 2 and U(q, t) is-an-element-of C2 ((R(N)\{0}) x R, R) is a T-periodic function in t such that \q\alpha U(q, t), \q\alpha+1 U(q)(q, t), \q\alpha+2 U(qq)(q, t), \q\alpha U(t)(q, t) --&gt; O as Absolute value of q --&gt; 0.
For alpha is-an-element-of (1, 2], we prove the existence of a non-collision solution of (HS). For alpha is-an-element-of (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.

• Non-collision solutions for a second order singular Hamiltonian system with weak force

K. Tanaka

Annales de Institut Henri Poincare : Analyse non lineaire   10   215 - 238  1993

• A prescribed energy problem for a singular Hamiltonian system with a weak force

K. Tanaka

Journal of Functional Analysis   113   351 - 390  1993

• Multiple periodic solutions of a superlinear forced wave equation

Kazunaga Tanaka

Annali di Matematica Pura ed Applicata   162 ( 1 ) 43 - 76  1992.12

View Summary

We study the existence of forced vibrations of nonlinear wave equation: {Mathematical expression} where g(ξ)∈C(R, R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t). © 1992 Fondazione Annali di Matimatica Pura ed Applicata.

7
Citation
(Scopus)
• MULTIPLE PERIODIC-SOLUTIONS OF A SUPERLINEAR FORCED WAVE-EQUATION

K TANAKA

ANNALI DI MATEMATICA PURA ED APPLICATA   162   43 - 76  1992

View Summary

We study the existence of forced vibrations of nonlinear wave equation: [GRAPHICS} where g(xi) is-an-element-of C(R, R) is a function with superlinear growth and f(x, t) is a function which is 2pi-periodic in t. Under the suitable growth condition on g(xi), we prove the existence of infinitely many solution of (*) for any given f(x, t).

• HOMOCLINIC ORBITS IN A 1ST ORDER SUPERQUADRATIC HAMILTONIAN SYSTEM - CONVERGENCE OF SUBHARMONIC ORBITS

K TANAKA

JOURNAL OF DIFFERENTIAL EQUATIONS   94 ( 2 ) 315 - 339  1991.12  [Refereed]

• SOME RESULTS ON CONNECTING ORBITS FOR A CLASS OF HAMILTONIAN-SYSTEMS

PH RABINOWITZ, K TANAKA

MATHEMATISCHE ZEITSCHRIFT   206 ( 3 ) 473 - 499  1991  [Refereed]

• SOME RESULTS ON CONNECTING ORBITS FOR A CLASS OF HAMILTONIAN-SYSTEMS

PH RABINOWITZ, K TANAKA

MATHEMATISCHE ZEITSCHRIFT   206 ( 3 ) 473 - 499  1991

• Homoclinic orbits in a first order superquadratic hamiltonian system: Convergence of subharmonic orbits

Kazunaga Tanaka

Journal of Differential Equations   94 ( 2 ) 315 - 339  1991

View Summary

We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JHz(t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z∞(t) and subharmonic solutions (zT(t))Tε{lunate}N (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z∞(t) in Cloc1(R,R2N) as T → ∞. © 1991.

87
Citation
(Scopus)
• Existence of infinitely many solutions for some superlinear elliptic equations,

R. Kajikiya, K. Tanaka

Journal of Mathematical Analysis and Applications 共著   149   313 - 321  1990

• HOMOCLINIC ORBITS FOR A SINGULAR 2ND-ORDER HAMILTONIAN SYSTEM

K TANAKA

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   7 ( 5 ) 427 - 438  1990

• Existence of infinitely many solutions for some superlinear elliptic equations,

R. Kajikiya, K. Tanaka

Journal of Mathematical Analysis and Applications 共著   149   313 - 321  1990

• Homoclinic orbits for a singular second order hamiltonian system

K. Tanaka

Annales de i'institut henri Poincare Analyse non Iineaire   7   427 - 438  1990

• MORSE INDEXES AT CRITICAL-POINTS RELATED TO THE SYMMETRIC MOUNTAIN PASS THEOREM AND APPLICATIONS

K TANAKA

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   14 ( 1 ) 99 - 128  1989  [Refereed]

• Morse indices at critical points related to the symmetric mountain pass theorem and applications

K. Tanaka

Communications in partial differential equations   14   99 - 128  1989

• INFINITELY MANY PERIODIC-SOLUTIONS FOR THE EQUATION .2. UTT-UXX+/-/U/P-1U=F(X,T)

K TANAKA

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   307 ( 2 ) 615 - 645  1988.06  [Refereed]

• Infinitely many periodic solutions for the equation utt-uxx ± |u|P-1=f(x, t), II

K. Tanaka

Transaction of the American Mathematical Society   307   615 - 645  1988

• Forced Vibrations for a Superlinear Vibrating String Equation

Kazunaga Tanaka

North-Holland Mathematics Studies   148 ( C ) 247 - 266  1987

View Summary

Many mathematicians are concerned with free vibrations—that is, they assume f(x, t) = 0 and find non trivial solutions. This chapter discusses the nonlinear vibrating string equation. © 1987, Publishing Committee of Lecture Notes in Numerical and Applied Analysis

• INFINETELY MANY PERIODIC-SOLUTIONS FOR A SUPERLINEAR FORCED WAVE-EQUATION

K TANAKA

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   11 ( 1 ) 85 - 104  1987.01  [Refereed]

• Infinitely many periodic solutions for a superlinear forced wave equation

Kazunaga Tanaka

Nonlinear Analysis   11 ( 1 ) 85 - 104  1987

9
Citation
(Scopus)
• DENSITY OF THE RANGE OF A WAVE OPERATOR WITH NONMONOTONE SUPERLINEAR NONLINEARITY

K TANAKA

PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES   62 ( 4 ) 129 - 132  1986.04  [Refereed]

• Density of the range of a wave operator with nonmenotone superlinear nonlinearity

K. Tanaka

Proceedings of the Japan Academy   62A   129 - 132  1986

• Forced vibrations for a superlinear vibrating string equation

K. Tanaka

in Recent topics in nonlinear PDE, III (Tokyo 1986), North-Holland     247 - 266  1986

• On the range of wave operators

Kazunaga Tanaka

Tokyo Journal of Mathematics   8 ( 2 ) 377 - 387  1985

3
Citation
(Scopus)
• Infinitely many periodic solutions for the equation : utt-uxx ±|u|S-1u=f(x, t)

K. Tanaka

Communications in partial differential equations   10   1317 - 1345  1985

• On the range of wave operators

K. Tanaka

Tokyo Journal of Mathematics   8   377 - 387  1985

• INFINITELY MANY PERIODIC-SOLUTIONS FOR THE EQUATION - UTT-UXX+/-LULS-1U=F(X,T)

K TANAKA

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   10 ( 11 ) 1317 - 1345  1985  [Refereed]

• On the number of positive solutions of singularly perturbed 1D NLS

P. Felmer, S. Martinez, K. Tanaka

J. Eur. Math. Soc.   to appear

• High frequency chaotic solutions for a slowly varying dynamical system

P. Felmer, S. Martinez, K. Tanaka

Ergodic Theory Dynamical Systems   to appear

• High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation

P. Felmer, S. Martinez, K. Tanaka

Arch. Rat. Mech. Anal   to appear

• On the number of positive solutions of singularly perturbed 1D NLS

P. Felmer, S. Martinez, K. Tanaka

J. Eur. Math. Soc.   出版予定

• High frequency chaotic solutions for a slowly varying dynamical system

P. Felmer, S. Martinez, K. Tanaka

Ergodic Theory Dynamical Systems   出版予定

• High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation

P. Felmer, S. Martinez, K. Tanaka

Arch. Rat. Mech. Anal   出版予定

### Books and Other Publications

• 変分問題入門 : 非線形楕円型方程式とハミルトン系

田中, 和永

岩波書店  2018.06 ISBN: 9784007307706

• Nonlinear Problems 2 (in Japanese)

K. Tanaka

Iwanami  2000

• 非線形問題 2

田中和永

岩波書店  2000

### Research Projects

• 古典場の理論における微分型相互作用の数学解析

Project Year :

2019.04
-
2024.03

View Summary

微分型シュレディンガー方程式をはじめとする微分型相互作用を持つ古典場模型の非線型偏微分方程式は、ピカール逐次近似の枠組において、必然的に微分の損失を伴うため、その回避を巡って方程式に応じた個別の方法論が提案されているが、未だに本質的な理解に至っていない。本研究の目的は、近年の微分型シュレディンガー方程式の初期値問題の時間大域的存在を保障する新しい閾値の変分解析的理解を足掛かりとして、微分型相互作用の大域的構造を(a)漸近解析(b)調和解析(c)変分解析の三つの方法論に基づいて明らかにする事である。微分型シュレディンガー方程式をはじめとする微分型相互作用を持つ古典場模型の非線型偏微分方程式は、ピカール逐次近似の枠組において、必然的に微分の損失を伴うため、その回避を巡って方程式に応じた個別の方法論が提案されているが、未だに本質的な理解に至っていない。本研究の目的は、近年の微分型シュレディンガー方程式の初期値問題の時間大域的存在を保障する新しい閾値の変分解析的理解を足掛かりとして、微分型相互作用の大域的構造を(a)漸近解析、(b)調和解析、(c)変分解析の三つの方法論に基づいて明らかにする事である。令和元年度は、漸近解析班は自己相似解の研究を中心に、調和解析班は非線型ポテンシャルの研究・特性法の函数空間論的定式化を中心に、変分解析班は輪郭分解の基礎理論の研究を中心に研究を進めた。特に、通常の非線型微分シュレディンガー方程式(Derivative Nonlinear Schr"odinger Equation)の自己相似解を世界に先駆けて構成する事が出来た。求めるべき自己相似解が一つ決めた時刻の状態で決定されてしまう事に注目し、その時刻での波動函数を振幅函数と位相函数に分離して夫々が満たすべき微分方程式を導出したところ、位相函数が本質的には振幅函数で決定されてしまうと云う新しい知見を得た。これにより、振幅函数の大域解の構成に問題の全てが帰着する事になった。振幅函数の従う非線型常微分方程式を解析する事によって、大域解を構成する事が出来たので、自己相似解を具体的に表示する事が出来た。研究計画は予定通り順調に進んでいる。現在までの研究で思いがけない着想が幾つか得られており、今後の進展に繋がる事が期待される。令和2年度は、令和元年度に引き続き、漸近解析班は自己相似解の研究を中心に、調和解析班は非線型ポテンシャルの研究・特性法の函数空間論的定式化を中心に、変分解析班は輪郭分解の基礎理論の研究を中心に研究を進める。後半からは、漸近解析班・調和解析班は分散構造の研究、変分解析班はハミルトン構造の研究にも着手する予定である

• Research on Dispersive Equations and Harmonic Analysis

Project Year :

2018.10
-
2023.03

• 変分的手法による非局所非線形楕円型方程式の研究

Project Year :

2017.04
-
2022.03

View Summary

非線形楕円型方程式に対する L2-制限問題等について研究を行った. L2-制限問題は物理的に重要な対象である. ここでは新たな変形理論 (deformation theory) を開発し, ミニマックス法等により L2-制限問題の解の存在を示し, また多重度を得ることに成功した. ここで展開されている変形理論は通常の Palais-Smale 条件の下のものではなく, Pohozaev 関数の効果を加えた Palais-Smale-Pohozaev 条件 (PSP) の下で展開され, 非線形スカラーフィールド方程式をはじめとする種々の問題に適用可能であると共に証明の簡略化を与える. 有名な結果である Berestycki-Lions の結果を制限問題を経由せずに証明を与えることも可能である.さらに拘束条件に対応する関数空間内の超局面 (さらに一般に submanifold) 上で PSP 条件の下での変形理論を展開することにより, 連立シュレディガー方程式系に対する L2 制限問題等を扱うことが可能となった. Bartsch-Soave らの結果もこの枠組みで扱うことができ, さらなる発展が見込まれる.非局所問題, 特に Choquard 方程式に対しても L2-制限問題を考察し, Lagrangian formulation の下で (PSP) 条件の下での変形理論を用いることにより解の存在を, さらにある種の対称性の下で無限個の解の存在を示した. この手法は fractional Laplacian を伴う非線形スカラーフィールド方程式等へも拡張可能であり, 発展が期待される.変形理論に関しては特異摂動下での変形理論を見直し, 新たな勾配流を構成した. 特異摂動下での L2-制限等への応用が期待される.L2-constraint problem に対する新たなアプローチを見いだし, 非局所問題に対する L2-constraint problem に対しても存在定理を見いだしている.これらの結果は fractional Laplacian を伴う問題等への拡張, さらには特異摂動問題への発展が見込まれる.今までの研究を踏まえ, 非局所問題に対する特異摂動問題に取り組む. 特に L2 制限問題を重視する.非局所問題に対する特異摂動問題はその難しさから, アプローチを行うための道具だての準備から始める必要がある. 今までに整備を行った (PSP) 条件の下での変形理論等は非局所方程式に対する特異摂動問題に対しても有効であることが期待される. いわゆる tail minimizing method に対応する flow の構成もなされており, 準備は整いつつある.従来, 特異摂動問題は非線形項がべき関数のように線形よりも高い増大度をもつものが扱われてきた. 本研究では対数的非線形性に代表される線形よりも低い増大度をもつ劣線形性をもつ非線形方程式に対する特異摂動問題も研究テーマに加え取り組む

• Study on null forms in global space-time in the framework of equalities

Project Year :

2016.04
-
2019.03

• Mathematical Analysis of Critical Interactions in Classical Fields

Project Year :

2014.04
-
2019.03

View Summary

Regarding the critical interactions in the theory of classical fields, we have studied equations of nonrelativistic fields and of semirelativistic fields and related inequalities appearing in the analysis of those equations. As for the nonlinear Shr"odinger equations, we proved blow-up of solutions in non-gauge invariant setting.Moreover, we have improved the Br\'ezis-Gallou"et argument by renormalizing higher order energy to cover higher order nonlinearities in the global Cauchy problem for nonrelativistic and semirelativistic equations

• Analysis of the nonlinear elliptic eigenvalue problems and inverse problems

Project Year :

2013.04
-
2017.03

View Summary

In this study, we consider the inverse and direct bifurcation problems of nonlinear eigenvalue problems. For the direct problems, we establish the precise asymptotic formulas for the eigenvalue problems which have biological and physical background. For the inverse bifurcation problems, we consider the typical inverse problem for elliptic equations to understand well the structure of inverse problems. In particular, we concentrate on the study of the global structure of bifurcation curves for some nonlinear ordinary differential equations. We apply these precise asymptotic properties to the typical inverse bifurcation problem and obtained some fundamental and new results in this direction

• Research of Navier-Stokes equations in undounded domains by real analysis and the energy method

Project Year :

2013.04
-
2017.03

View Summary

For the Boussinesq equations, we established the unique existence of the solutions, and obtained the asymptotic behavior up to the second order.Besides, for the stationary Navier-Stokes euqations on two-dimensional whole plane and exterior domains, we introduced a new assumption on the symmetry of domains, external force and the boundary value, and showed the existence of solutions which decay at infinity.Further, under a weaker assumption on the symmetry, we showed the global asymptotic stability of the stationary solutions under arbitrary perturbations in the L2-space, together with the speed of decay measured by various norms

• Variational study of nonlinear elliptic problems

Project Year :

2013.04
-
2017.03

View Summary

Via variational approached, we study nonlinear elliptic problems. In particular, we develop a new variational approach and we construct concentrating solutions for several singular perturbation problems, where uniqueness and non-degenerary of solutions of limit problems are not known and the Lyapunov-Schmidt reduction method is not applicable.We also introduce new variational approaches based on the scaling properties the problems, which enable us to study ground states and other solutions for several nonlinear elliptic equations and systmes

• Study on variational problems, optimization problems and nonlinear partial differential equations

Project Year :

2013.04
-
2016.03

View Summary

In this research, we studied the structure of solutions of nonlinear ellipticpartial differential equations arising in mathematical models of quantum phenomena and pattern formation problems of the mathematical biology. Especially, we investigated the precise asymptotic behavior of eigenvalues of the Laplacian with the mixed boundary condition on a thin domain and the precise asymptotic behavior of the least energy of a nonlinear variational problem associated with the Bose-Einstein condensation. We also studied the structure of global minimizers for 3-component FitzHugh-Nagumo system and the parameter range of existence and non-existence of non-constant steady patterns for several nonlinear rection-diffusion systems, e.g., the prey-predator model with a cross-diffusion effect

• Study on oscillation and singularity in mass resonance

Project Year :

2013.04
-
2016.03

View Summary

Mass resonance phenomenon of systems of nonlinear Schr"odinger equations is studied. We have found an appropriate Lagrangean for the system of Schr"odinger equations with quadratic interactions, by which we are able to give a clear explanation of the roles of mass resonance phenomenon as well as of complex conjugate in wavefunctions.A variational setting has been introduced for the analysis on standing waves associated with mass resonance. We proved the existence and (essential) uniqueness of ground states.Moreover, examples of interactions were found for the blow-up solutions with arbitrarily small Cauchy data under mass resonance condition

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

Project Year :

2012.04
-
2015.03

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

• Studies on a unified point of view on global theories of nonlinear elliptic equations

Project Year :

2009.04
-
2014.03

View Summary

We studied nonlinear elliptic equations arising in various fields of mathematical physics by means of variational analysis, ordinary differential equations, and viscosity techniques. We studied orbital stability of standing waves, explicit blow-up solutions, and exponential decay of ground states for systems of nonlinear Schr"odinger type equations

• Study on asymmetry and anisotropy in blow-up theory for nonlinear parabolic equations

Project Year :

2011.04
-
2013.03

View Summary

Blow-up phenomenon of positive solutions to the Cauchy problem for nonlinear heat equation of Fujita type is studied. The blow-up phenomenon was first observed and proved by Hiroshi Fujita about half a century ago. There is a large literature on the subject, especially on the formation of blow-up solutions in configuration space up to the blow-up time. We have proved that the blow-up time is characterized by means of the spherical average of the Cauchy data. We removed the assumptions of uniformity and isotropy on the Cauchy data, which were necessary in the former works by Lee and Ni (Trans. Ams, 1992) and Gui and Wang (JDE 1995). We described how the ODE structure controls the blow-up phenomenon

• Study of structures of solutions to variational problems, optimization problems and nonlinear partial differential equations

Project Year :

2010.04
-
2013.03

View Summary

We proved existence of non-constant stationary solutions to nonlinear reaction-diffusion equations arising in some pattern formation problems, for example, an Allen-Chan modelwith non-homogeneous environment and a Chemotaxis model with saturation effect

• Asymptotic analysis and inverse problems of the nonlinear elliptic eigenvalue problems

Project Year :

2009.04
-
2013.03

View Summary

In this study, we investigated the precise asymptotic properties of the eigenfunctions and eigenvalues of nonlinear elliptic equations, and studied the inverse eigenvalue problems associated with thedirect problems. We clarify the global and local structures of the bifurcation curves for the equations with several types of nonlinear terms. As for inverse problems, we mainly studied the inverse bifurcation problems for logistic type equations. By using the theory of ordinary differential equations and the asymptotic expansion formulas for bifurcation curves, we determined the unknown nonlinear terms by the asymptotic behaviors of the bifurcation curves

• Project Year :

2009
-
2012

View Summary

This research is concerned with the Navier-Stokes equations on either the whole plane or two-dimensional exterior domains. It was shown that, if there exists a small stationary external force with strong symmetry, the equation has a small stationary solution decaying rapidly at infinity. It was also shown that, if the stationary solution above is sufficiently small, it is stable under initial perturbation without restriction on the size. The Navier-Stokes equaions in an infinite layer is also studied. It is shown that, if the equation is treated in the Besov spaces, nontrivial solutions with no external forces exist if p is infinite, and that these solutions correspond to the Poiseuille flows.

• Project Year :

2009
-
2012

View Summary

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.

• Project Year :

2009
-
2011

View Summary

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.

• Project Year :

2008
-
2011

View Summary

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.

• Project Year :

2006
-
2008

• Project Year :

2005
-
2008

• Project Year :

2005
-
2007

View Summary

Analysis for the heat convectin problems for the system of Oberbeck-Boussinesq equations in the horizontal strip domain. The existence of bifurcation curve of roll-type solutions is proved for the ten times Rayleigh number of critical Rayleigh number by a computer assisted proof . The second bifurcation point of stationary roll-type solutions is determined by a computer assisted proof. The hexagonal-type and rectangle-type solutions of 3-dimensional problems are also proved for the existence by a computer assisted proof at least for rather small Rayleigh numbers.
The cocoon bifurcation for the Michelson system is analyzed and proved for the existence bf infinitely many bifurcations of heteroclinic orbits to the saddle-node periodic orbit by a topological method and a computer assisted proof.
The driven-cavity problem of 2-dimensional Navier-Stokes equation is solved for rather large Reynolds numbers compared to the existing verified result by a Newton type computer assited proof.

• Project Year :

2005
-
2007

View Summary

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).

• Project Year :

2004
-
2007

View Summary

(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.

• Project Year :

2003
-
2006

View Summary

We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.
1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body
2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.
3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid
4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum.

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(萌芽研究)

Project Year :

2003
-
2005

View Summary

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。

• Project Year :

2003
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2005

View Summary

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.

• Project Year :

2003
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2005

View Summary

1. Kurata studied the existence and qualitative properties of optimal solutions to several optimization problems for nonlinear elliptic boundary value problems arising in mathematical biology and nonlinear heat conduction phenomena. Kurata also proved the existence of multiple stable patterns in population growth model with Allee effect, symmetry breaking phenomena of the least energuy solution to nonlinear Schroedindger equation and asymptotic profile of radial solution with vortex to 2-dimensional nonlinear Schroedinger equation.
2. Okada studied numerial simulation and numerical analysis of nonlinear paratial differential equations. Especially, he constructed boundary spline function by using Newton extrapolation polynomials.
3. Sakai studied Hele-Shaw free boundary problem in the case that initial data has a cusp and found sufficient conditions to specify the typical pheneomena.
4. Isozaki discovered the relationship between the hyperbolic geometry and inverse problem. He also studied the inverse conductivity problem with discontinuous inclusions and found a numerical algorithm to detect discontinuities.
5. Jimbo continued his research on the study of solution structure of the Ginzburg-Landau equation arising in superconductivity under heterogeneous environments. He also studied the spectrum of elliptic operator associated with the Maxwell equation and proved characterization of eigenvalues and proved a perturbation formula by using weak forms.
6. Tanaka studied concentration phenomena of solutions and clustered solutions for nonlinear elliptic singular perturbation problems. Especially, he constructed high frequency solution to nonlinear Schroedinger equations and multi-clustered high energy
Solutions to a phase transition prolem.

• Project Year :

2002
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2004

View Summary

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.

• Project Year :

2002
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2004

View Summary

We studied nonlinear elliptic eigenvalue problems with several parameters. Our main concern was to clarify the asymptotic properties of eigenvalue parameters and associated eigenfunctions by using variational methods and singular perturbation approaches when a parameter was very large.
We studied two-parameter ordinary differential equations with two pure power nonlinear terms. We established several asymptotic formulas for eigenvalues by using several variational approaches. We also studied the two-parameter problems which were related to the simple pendulum problems. We established the precise asymptotic formulas for the solutions with boundary layers when a parameter was very large under the Dirichlet boundary conditions. The formulas obtained here were totally different from those for the associated one-parameter simple pendulum problems.
For problems with one-parameter, we studied the problems which were related to the simple pendulum problems. We first studied ordinary differential equations and established precise asymptotic formulas for the boundary layers when a parameter was large under the Dirichlet boundary conditions. We next extended this result to the nonlinear elliptic eigenvalue problems in a smooth bounded domain. We studied nonlinear elliptic eigenvalue problems with pure power nonlinearity and established the asymptotic formula for the eigenvalues in L-2 framework. We also treated the perturbed simple pendulum problems in a bounded domain. It is known that if parameter is large, then an associated solution is nearly flat inside the domain. We have succeeded to establish the precise asymptotic formulas for the interior behavior of the solutions to understand precisely how flat the solutions were inside a domain when a parameter was very large. The formulas obtained here were exactly represented by using the nonlinear term.

• Project Year :

2001
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2004

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In a joint work with Yoshihiro Shibata, we obtained a sufficient condition on time-independent external forces for the unique existence of a stationary solution in a certain class of the Navier-Stokes equation in exterior domains of dimension n greater than or equal to 3 by using the duality between the Lorentz spaces and real interpolation. Our class is a natural generalization of the so-called physically reasonable solutions, and our suffirient condition gives a unified view for the case with zero velocity at infinity and the case with non-zero velocity at infinity.
Next, in a joint work with Yuko Enomoto and Yoshihiro Shibata, we verified the stability in the weak-Ln space of the stationary solution above for time-evolution under small initial perturbation in the weak-Ln space, and showed that the smallness above can be taken uniformly in the velocity at infinity of the stationary solution.
Furthermore, by using real interpolation for sublinear operators, we generalized these results for time-dependent external forces, and obtained a sufficient condition for the unique existence of the corresponding time-periodic or almost periodic solutions. We also showed the stability of these solutions in weak-Ln spaces under perturbations on the external forces and initial data uniform in the velocity at infinity of the solutions.
On the other hand, as a preparation for generalized the results above for general unbounded domains, we generalized the Lp-theory on the boundary value problem for the Stokes equation in a layer domain, in a joint work with Takayuki Abe for higher-order Sobolev spaces and Besov spaces, and obtained a sufficient condition on the external forces for the unique existence of the solution of the boundary value problem. In particular, we showed that the uniqueness of the solution fails in the case p=infinity, and that the Poiseuille flow can be characterized as the solution with zero as the external forces and boundary values.

• Project Year :

2001
-
2003

View Summary

Keller and Segel derived the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. So this model is called as the Keller-Segel system. Chirdress and Percus conjectured that there exists a threshold number (8π) such that if an initial value is smaller than 8π, the solutions exist globally in time, on the other hand if an initial value is greater than 8π, the chemotactic collapse can occure. Our objective is to solve this conjecture. We study the Keller-Segel system in R^2 because this system is rarely studied in hole space. Especially we treated the self-similar solution and solved almost positively the Chirdress and Percus conjectuer. For this purpose we encountered several problems and solved these. We enumerate these problems : (1)determine decay order of the self-similar solutions (2)derive the Lieuville type theorem (3)reduction to one elliptic equation from an system of two elliptic equations (4)show the radial symmetry of solutions by using the moving plane method (5)determine the global blanch of solutions (6)solve the Chirdress and Percus conjectuer.

• Project Year :

2000
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2003

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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 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.

• Project Year :

2000
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2003

View Summary

1.Stability of the Oseen flow in the n-dimensional exterior domain (n>2).
2.Stability of the Couette flow and the Poiseuille flow in the infinite layer.
3.Rate of convergence of the non-stationary flow to the stationary flow of compressible viscous fluid.
4.Resolvent estimate of solutions to the Stokes equation with Neumann boundary condition.

• Project Year :

2001
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2002

View Summary

1. Kurata studied the following:
(1) breakdown of the monotonicity of the minimizer to a one-dimensional Cahn-Hilliard energy with inhomogeneous weight and the existence of non-topological solution to a nonlinear elliptic equation arising from Chern-Simons-Higgs theory.
(2) optimal location of a obstacle in an optimization problem for the first Dirichlet eigenvalue to Schrodinger operator.
2. Jimbo studied the existence of stable vortex solutions and the non-existence of permanent current in a convex domain to Ginzburg-Landau equation with magnetic effect. Tanaka constructed solutions with complex patterns to inhomogeneous Allen-Cahn equation and nonlinear Schrodinger equation. Murata studied the structure of positive solutions to elliptic equation of skew-product type and classifies the Martin boundary and Martin kernel completely.
3. Mochizuki studied the inverse spectrum problem for Dirac operator and Sturm-Liouville operator by interior datas. Sakai studied the asymptotic behavior of the moving boundary for Hale-Shaw flow when the initial region has an angle in details.

• Project Year :

2000
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2002

View Summary

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.

• Project Year :

2000
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2001

View Summary

1. (1) We studied the nonlinear two-parameter problem -u"(x) + λu(x)^q = μu(x)^p,u(x) > 0,x ∈(0,1),u(0) = u(1) = 0. Here 1 < q < p are constants and λ,μ > 0 are parameters. We established precise asymptotic formulas with exact second term for variational eigencurve μ(λ) as λ→∞. We emphasize that the critical case concerning the decaying rate of the second term is p = (3q - 1)/2 and this kind of criticality is new.
(2) We considered the nonlinear two-parameter problem u"(x) + μu(x)^p = λu(x)^q, u(x) > 0, x ∈ I = (0,1), u(0) = u(1) = 0, where 1 < q < p < 2q + 3 and λ,μ > 0 are parameters. We established the three-term spectral asymptotics for the eigencurve λ = λ(μ) as μ→∞ by using a variational method on the general level set due to Zeidler. The first and second terms of'λ(μ) do not depend on the relationship between p and q. However, the third term depends on the relationship between p and q, and the critical case is p = (3q-1)/2.
2. We considered the nonlinear eigenvalue problem -Δu = λf(u), u > 0 in Ω,u = 0 on ?∂Ω, where Ω=B_R={x ∈ R^N : |x| <R} or A_<a,R> = {x ∈ R^N : a< |x| <R} (N【greater than or equal】2) and λ>0 is a parameter. It is known that under some conditions on f and g, the corresponding solution u_λ develops boundary layers when λ>> 1. We established the asymptotic formulas for the width of the boundary layers with exact second term and the estimate of the third term.
3. We considered several elliptic partial differential equations and parabolic systems related to nonlinear eigenvalue problems and obtained some existence results and qualitative properties of the solutions.

• Project Year :

1999
-
2001

View Summary

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.

• Project Year :

1999
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2000

View Summary

The following is the abstract for the main results obtained under this research project.
1. In the research of Hamiltonian systems, Ito generalized the notion of complete integrability for Hamiltonian systems to that for general vector fields. He proved that the integrability of an analytic vector field is equivalent to the existence of a convergent normalizing transformation near an equlibrium point that are non-resonant and elliptic. It gives an answer to the Poincare center problem.
2. In the research of ergodic theory, Morita studied the zeta function associated with two dimensional scattering billiards problem. He succeeded in extending it meromorphically to a half plane with its real part greater than some negative constant.
3. In the research of bifurcation theory of dynamical systems, Kokubu studied the generalization of Conley index theory to slow-fast systems which are singularly perturbed vector fields. He defined transition matrices when the slow variables are of dimension one, and obtained a general method for proving the existence of periodic or heteroclinic orbits.
4. By using variational method for singular Hamiltonian systems, Tanaka proved the existence of orbits such as (1) scattering type ; (2) periodic orbits under the class of perturbation of type -1/γ^2 ; (3) unbounded and chaotic motions for systems whose potential have two singular points.
5. In the research of symplectic/contact geometry, Ono succeeded in constructing the Floer homology with integer coefficients. Nakai studied 1st order PDE's from the viewpoint of foliation theory and Web geometry. In particular, he defined affine connections for those PDE's with finite type, and used them to study the singularities associated with the foliation defined by their solutions.

• Project Year :

1999
-
2000

View Summary

T.Nagi, Y.Naito and K.Yoshida studied the Keller-Segel system which is the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. Naito and Yoshida with N.Muramoto studied the self-similar solution to the Keller-Segel system. Then the Keller-Segel system is reduced to an elliptic equation with an parameter σ, and obtained solutions of two types, one of which is in a low critical lebel, the other is in a high cirtical lebel. When the parameter is large, there is no self-similar radial solution. This means two solutions are connected by a global blanch. Nagai with T.Senba and T.Suzuki considered the location of blow-up points. These results are announced at each conference or workshop e.g, IMS Workshop on Reaction-Diffusion Systems at HongKong (December 6-10, 1999), The Third World Congress of Nonlinear Analysis at Catania (July 19-26, 2000). H.Usami treated the oscillation problem to the second order ordinary differential equations and the asymptotic properties of the variational eigenvalues. Y.Mizuta completed the results by Koskela concerning the uniqueness property for Sobolev functions.

• Project Year :

1999
-
2000

View Summary

1. Kurata studied the following :
(1) estimates of the second and third derivatives of fundamental solutions to magnetic Schrodinger operators with non-smooth potentials and the Calderon-Zygmund property of certain operators.
(2) estimates of the heat kernel of magnetic Schrodinger operators.
(3) exiestence and further properties of the optimal configuration to several optimization problems for the first Dirichlet eigenvale. Especially, we find a symmetry-breaking pheneomena of the optimal configuration for certain symmetric domains. We also studied the regularity of the free boundary associated with the optimal configuration.
2. Jimbo studied the non-existence of stable non-constant solution to Ginzburg-Landau equation with magnetic effect.
3. Tanaka studied discontinuous phenomena for solutions under the perturbation to nonlinear ellitic equation -Δu+u=u^p.
4. Murata studied the structure of positive solutions to elliptic and parabolic equations of second order on non-compact Riemannian manifolds.
5. Mochizuki studied the blow-up and the asymptotic behavior of solutions to KPP equation and inverse spectrum problem for Sturm-Liouville operator by interior datas.
6. Ishii showed the convergence of geometric approximation method for the Gauss curvature flow.
7. Sakai studied the condition of the existence of a measure which has a fine support and makes the same potential outside the polygon in two-dimensional case.

• Project Year :

1998
-
1999

View Summary

(1) Eigenvalue Problems of Elliptic Equations : Two-parameter eigenvalue problems for semilinear elliptic equations are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Two-parameter Ambrosetti-Prodi problems are also studied. We investigate the relation between parameters and the number of solutions.
(2) Positive Solutions of Elliptic Equations : Semilinear second-order elliptic euations are considered in unbounded domains. We establish multiplicity results for positive solutions and uniqueness theorems for positive solutions.
(3) Positive Solutions of Quasilinear Ordinary Differential Equations : Quasilinear ordinary differential equations whose leading term is one-dimensionai pseudo-Laplacian are considered. We obtain asynrptotic representations of positive solutions. As an application of these results, we show existence of several types of positive solutions of exterior Dirichlet problems for quasilinear elliptic equations.
(4) Mathematical Models Describing Aggregation Phenomena of Molds : We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We clarify the relation between parameters and the number of self-similar solutions.
(5) Nonnegative Nontrivial Solutions of Quasilinear Elliptic Equations and Elliptic Systems : We establish necessary and/or sufficient conditions for quasilinear elliptic equations, as well as quasilinear elliptic systems, to possess nontrivial nonnegative entire solutions. Several Liouville type theorems are also obtained.

• Project Year :

1997
-
1999

View Summary

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.

• Project Year :

1997
-
1998

View Summary

1. Kurata studied the following :
(1) unique continuation theorem and an estimate of zero set of solutions to Schrodinger operators with singular magnetic fields.
(2) finiteness of the lower spectrum of uniformly elliptic operators singular potentials.
(3) Liouville type theorem for Ginzburg-Landau equation and existence and its profile of the least energy solution to nonlinear Schrodinger equation with magnetic effect.
(4) existence of non-topological solution to a nonlinear elliptic equation arising from Chern-Simons-Higgs theory
2. Jimbo studied existence and zero set of stable non-constant solution to Ginzburg-Landau equation.
3. Tanaka studied Hamilton system, uniquness and non-degeneracy of positive solution to a nonlinea elliptic equation, and the construction of multi-bump solutions.
4. Murata studied uniqueness of non-negative solution to parabolic equation.
5. Mochizuki studied global existence and blow-up of solutions to reaction-diffusion systems.
6. Ishii studied dynamics of hypersurfaces and homogenization of Hamilton-Jacobi equation.
7. Sakai studied Hale-Shaw flow in the case that initial domain has a corner.

• Project Year :

1997
-
1998

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(1) T.Kori investigated the theory of the index of a gauge coupled Dirac operator with Grassmannian boundary condition, especially he gave a direct method of calculations not using the Atiyah-Patodi-Singer theory for those problems on the four dimensional hemisphere.
(2) T.Kori.proved a formula about the chiral anomaly of gauge coupled Dirac operators. Here he proved that the index of a gauge coupled Diracoperator on S^4 is equal to the index of the geometric Dirac operator on the hemisphere with a Grassmannian boundary condition comming from the vector potential, that is, the effect by the gauge is absorbed in the boundary condition. This result will be published in the Proceeding of the conference on Geometric Aspects of Partial Differential Equations as one volume of AMS Contemporary Mathematics series.
(3) The problem of extension of spinors from the boundary to the interior or the exterior as zero mode spinors is solved. By this an analogy of the Laurent expansion theorem for zero mode spinors is obtained. Thus the concepts of meromorphic spinors and their residues are introduced. He proved the residue theorem on a domain in S^4. Many theorems that are counterparts of what are known in complex function theory are expected to hold in our framework of spinor analysis. This will be our next project.

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(奨励研究(A))

Project Year :

1997
-
1998

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変分的手法により非線型微分方程式の解の存在問題の研究を行った.特に本年度は(1)R^Nにおけるnonlinear scalar field equation,(ii)ハミルトン系の非有界軌道の存在等を主に扱った.(i)R^Nにおけるnonhlinear scalar field equationに関しては,まず軸対称な空間依存性をもつ方程式-Δu+V(|x|)u=u^pの正値解の一意性を考察し,Kwongによる一意性の結果の非常に簡略化された証明を得ることができた.またその一意性の応用として周期ポテンシャルをもつ非線型楕円型方程式-Δu+V(x)u=u^pのあるクラスに対してmulti-bump solutionが存在することを,特に無限個の正値解が存在することを示すことができた.
(ii)ハミルトン系に関しては,2体問題型のポテンシャルV(q)〜-1/(|q|^α)(α>0)に対して無限から来て無限に飛びさる軌道の変分的な構成を考え,与えられたH>0をtotal energyとしてもち,さらに与えられた入射角,出射角をもつ軌道の存在を空間次元Nに関する制限なしでstrong force条件(α>2)の下で示した.ここで空間次元が2のときは回転数を有効に利用することができ比較的容易に証明はなされるが,N 3の場合は異なりR^N＼{0}上のループ空間のtopologyに関する考察が必要不可欠となることに注意して頂きたい.なおH=0のときは古典力学における放物軌道に対応し,非常に興味ある問題であるが,その存在は今後の課題としたい.
(iii)上記の(i),(ii)以外にもMoser-Trudinger型の不等式の最良指数についても研究を行い,Ogawa,Ozawaにより導入されたスケール不変なMoser-Trudinger型の不等式は有界領域の場合と異なり最良指数を達成しないことを示した.

• Project Year :

1997
-
1998

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The following is the abstract for the main results obtained under this research project.
1. In the research of integrable systems, Miyaoka proved that all isoparametric hypersurfaces in the sphere with six principal curvatures are homogeneous. For the proof, she used the isospectrality of the family of the shape operators on the focal set of isoparametric hypersurfaces. This is a remarkable result to solve the conjecture by Yau, and shows close connection between the theory of hypersurfaces and that of integrable systems.
2. In the research of ergodic theory, using transfer operators method, Morita obtained Fredholm determinant representation for the Selberg zeta function Z(s) of closed geodesics on hyperbolic Riemann surface with finite area. He investigated the spectral properties of the transfer operators, and then obtained some analytic information of Z(s).
3. By using variational methods, Tanaka studied unbounded solutions of singular Hamiltonian systems of the two-body type. Namely he proved the existence of a hyperbolic-like solutions for a class of singular potentials with the strong force alpha > 2. Also, he studied the prescribed energy problem for Hamiltonian systems of the same form with alpha = 2, and showed variationally the existence of periodic solutions on the zero energy surface. This led to an existence theorem of closed geodesics on noncompact Riemannian manifold.
4. In the research of symplectic geometry, Ono solved the Arnold's conjecture for general closed symplectic manifold by constructing Floer homology for periodic Hamiltonian and Gromov-Witten invariant. He generalized this approach further and studied constructions of Floer homology for Lagrangian intersection.
5. In the research of complex dynamical systems, Shiga showed some similar properties between limit sets of Kleinian group and Julia sets in theory of complex dynamical systems.

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

Project Year :

1996

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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、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。

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

Project Year :

1996

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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。

• Project Year :

1994
-
1996

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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, 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.

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(奨励研究(A))

Project Year :

1995

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ハミルトン系の周期解,ホモクリニック解および非線型楕円型方程式の解の存在問題を変分的手法により研究し,次の研究実績をあげることができた.
1.特異なハミルトン系に対する周期解の存在問題は,従来2体問題に関連したラグランジュ系に対してのみ考察されていた.本研究においては,より一般的なハミルトン系で特異点をもつものに対して周期解の存在を考え,ミニマックス法と有限次元近似をあわせて用いることにより,特異なハミルトン系のクラスで周期解の存在が保証されるものを得ることができた.近年,ハミルトン系の周期解の存在問題はsymplectic幾何学の視点からも重要であることが認識され,盛んに研究が行われている.特異なハミルトン系に対してはenergy surface{(p,q);H(p,q)=h}はnon-compactとなり,non-compact集合に対してsymplecticな不変量を導入する問題と密接に関連するものと思われる.この関連を研究するため現在prescribed energy problemを初めてとして研究を続行している.また特異なハミルトン系に対する周期解の多重性も重要な問題である.この問題についても現在研究を続行している.
2.ホモクリニック解の存在については,non-compactなリーマン多様体上である種のラグランジュ系を考え,ミニマックス法により,その存在を得た.この結果はR^Nの場合であっても,ホモクリニック解の新しい存在結果を与えていると思われる.
3.非線型楕円型方程式に関してはR^N(N【greater than or equal】3)上でΔu+K(|x|)u^<(N+2)/(N-2)>=0を考察した.この方程式は微分幾何学における山辺の問題と関連した重要な方程式である.ここでは特に球対称解u(|x|)の存在を考察し,変分的手法により,その存在を非常に一般的なK(|x|)に対して示した.従来,球対称解の存在問題はシューチング法で扱われることが多いが,変分的手法を導入することによりより一般的な存在結果を得ることができた

• Project Year :

1991
-
1992

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By using some properties of potential-kernels of logarithmic type, we gave a definitive solution of the following well-known problem. "Does the totality of convolution kernels satisfying the domination principle coincide with the closure of the set of Hunt convolution kernels?" In this connection, we proved that a potential-kernel is spectral synthetic if it satisfies the domination principle. Applying to the theory of potential-kernels, we obtain that with a given potential-kernel satisfying the domination principle, its resolvent formed by nice potential-kernels is associated. Suggested by the sweeping-out process, we worked out an arc-variation to investigate the analytic capacity. In the study of the classical harmonic function theory, it is remarkable to determine domains on which non-zero subharmonic functions are not integrable.
Potential-kernels of logarithmic type possess recurrent semi-group,s which is closely related with the probability theory. In the study of the probability theory, we obtain a criterion of the transiency of Ornstein-Uhlenbeck type processes and results concerning optimal diffusion processes.

• 科学研究費助成事業(名古屋大学)  科学研究費助成事業(一般研究(C))

Project Year :

1991

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極小曲面の安定性を調べるのに重要な第2変分から定まるヤコビ作用素の研究とかかわって以下の結果を得た。
1、3次元リ-マン多様体が局所的に等質であるための条件を曲率テンソルを用いてあらわした。これはI、M、Singerが問題としていたところのものに対する3次元の場合の部分的解等を与えている。(大和一夫)
2、Lie接触構造の中で特に共形多様体の接球束上の構造に対して正規接続を共形接続から具体的に構成した。(佐藤肇)
3、ハミルトン系のホモクリニック軌道あるいは周期軌道の存在を変分を用いて研究した。モ-ス理論及び関数解析的な手法を用いることにより2〜3の存在定理を得た。(田中和永)
4、解析的偏微分方程式のGevrey族空間での可解性を研究したものによって定まる様子を詳しく調べた。特に指数が非正の場合のGevrey族空間での可解性の研究が偏微分方程式において始めて取り扱われた。(三宅正武)
5、領域の形状とそこの調和関数の境界挙動の関係を可積分性の観点から調べた。(鈴木紀明)
6、有限な全曲率を持つ種数Oの3次元コ-クリッド空間の極小曲面のガウス写像からきまるシュレジンガ-作用素のO固有値に対応する固有関数の代数的な構成方法を与えた。これを使って種数Oの場合のヤコビ作用素のindexは一般にはガウス写像の写像度dを使って2dーlとなることがわかった。(江尻典雄)
7、高全次元の場合の極小曲面のindexには単位法束上の解析が興味ある対象であり上の結果達と深い関係が期待される。

• 非線形放物型方程式のアトラクターの研究

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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

• New development of the qualitative theory of nonlinear parabolic and elliptic equations

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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

### Specific Research

• 2016   平田 潤

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非線形楕円型方程式に対する研究を特異摂動問題を中心に研究を行った. その際, 非局所問題への対応を意識し, 局所問題と非局所問題を統一的に扱う理論の構築を目指し, 従来扱うことのできなかった磁場を伴う非線形シュレディンガー方程式および対数的非線形性を持つ非線形シュレディンガー方程式を扱った.特に磁場を伴う非線形シュレディンガー方程式に対しては, 特異摂動問題での設定の下で, ポテンシャルウェル内に凝集する解の多重度を示した. また対数的非線形性を持つ非線性シュレディンガー方程式に対しては空間周期的なポテンシャルを持つものを考え, multi-bump 解の存在を示した.&nbsp;さらに臨界点理論における Clark の定理に対して, その臨界点集合の構造に関する Kajikiya および Liu-Wang の結果を改良することに成功した.

• 2015

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非線形楕円型方程式に対する特異摂動問題の研究を行った. &nbsp;主に数理物理における半古典極限 (semi-classical limit) に関連する状況において凝集解の存在, 多重度を考察した.具体的には, 一般的な非線形項を伴うシュレディンガー方程式のポテンシャル関数 V(x) の極小点に凝集する解の多重度を研究し, V(x) の極小点のなす集合の大きさを測る位相幾何的な量 (cup-length) を用いて凝集解の個数を評価できることを示した.また磁性の効果を加味した magnetic-Schroedinger 方程式に対しても同様の結果が成立することを示した.また柱状領域を摂動した領域での非線形 Dirichlet 問題を考察し, 非常に一般的な状況において不安定な特異摂動解の存在を示した.

• 2003

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R^N における非線型楕円型方程式の研究, 特にその特異摂動問題に関する研究において成果が得られた.まず, 代表者は L. Jeanjean 氏との共同研究により R^N における非線型 Schrodinger方程式に対する特異摂動問題に関して進歩がみられた. 特に, 今まで扱うことができなかった広いクラスの非線型項 --- 漸近的に線型の増大度をもつ非線型項をも許容するクラス --- に対しても epsilon -> 0 のときポテンシャル V(x) の極小点に集中し,スパイクを形成する解の族の構成に成功した. ここにおいて, 昨年までの研究で得られたleast energy solution の Mountain Pass Theorem による特徴付けが非常に重要な役割を果たしている.また Y. Ding との共同研究においては非線型 Schrodinger 方程式に対して異なるタイプの特異摂動問題を考察し, 従来知られていたものよりもより複雑な multi peak パターンを極限とする解の族の構成に成功した. また 1 次元の特異摂動問題についても従来考察されていない高エネルギーをもつ解のクラスに関する研究においても Felmer 氏, Maritinez 氏との共同研究により,その極限の energy limit function を用いた特徴付け, energy limit function のみたす極限方程式の導出, 極限方程式の解に対応する存在結果を得ることができた.

• 2001

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変分的手法により非線型微分方程式の解の存在に関する研究を行った. 今年度は次のような研究成果が得られた.まず非線型楕円型方程式に関しては, R^N における方程式 -\Delta u = g(u) のエネルギー最小解の Mountain Pass Theorem による特徴づけに関して研究を行った. 従来 g(u)/u の単調性の仮定の下でこのような特徴付けは示されていたが,この様な仮定なしでも成立することを示した. またこの特徴付けの応用として漸近的に線型のオーダーをもつ R^N における非線型楕円型方程式の解の存在が得られた. (以上は L.Jeanjean 氏との共同研究である).ハミルトン系に関しては 2 体問題をモデルとする特異性をもつハミルトン系に対する周期軌道の存在問題を考察し, 特異点集合 S が体積をもつ場合に特異性が V(q)\sim -1/\dist(q,S)^\alpha (\alpha\in (0,2)) と非常に弱い場合にもその存在を示し, 従来よく用いられる strong force と呼ばれるクラスとは異なるポテンシャルのクラスにおいても non-collision 周期軌道の存在が保証されることを示した. (この結果は足達慎二氏との共同研究である).

• 2000

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変分的手法により空間非一様性をもつ楕円型方程式 -\Delta u = f(x,u) の解の存在問題を研究し, 特異摂動問題において進歩を見ることができた.方程式が空間変数 x に依存する場合, 解のプロファイルは依存しないときと比べると非常に複雑となる. 例えば空間次元が 1 の場合, 方程式が x に依存しなければ, すべての解は空間周期的となり, 比較的簡単な解構造をもつ. しかし方程式が x に依存すると, たとえ 1 次元であっても複雑なプロファイルをもつ解が現れることが一般に期待される.このような解の存在のメカニズムを理解することを目標とし, 本年は特異摂動の設定の下で研究を行い, 相転移問題に関連する状況において, 空間次元が 1 の時, 界面を伴う解の変分的構成を行った. このような問題の解は無限次元の関数空間上定義された氾関数の critical point を求める問題として定式化される. 従来 Lianupov-Schmidt 法等により有限次元空間上定義された氾関数に対する問題に帰着され研究されてきたが, ここではより直接的な有限次元への帰着法を見いだすことにより, 1 点に集中する多重界面をもつ解等の存在を非常に広いクラスの方程式に対して示すことができた.ここで用いられている方法は 1 次元非線型 Sch\"odinger 方程式 -\epsilon^2\Delta u +V(x)u ＝u^p にも適用でき, ポテンシャル V(x) が極小値をとる x において集中する sign-changing bump をもつ解を構成できる (del Pino 氏, Felmer 氏との共同研究, preprint).

• 1999

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変分的手法により非線型楕円型方程式の解の存在問題、Hamilton系の研究を行った。特に本年度はRNにおける非線型楕円形方程式の正値解の存在、多重度に関して進歩があった。　具体的にはRNにおける非線型楕円型方程式－Δu＋u＝a(x)up＋f(x)　in RN　　　　　　　　　(＊)正値解の存在、多重度を扱った。特に解が方程式に非常にデリケートに依存していることを示す次のような存在結果を得た。係数a(x)はコンパクト集合を除いて１、コンパクト集合上で０と１の間の値をとる連続関数とする。このとき非常に小さい、しかし０でない正の関数f(x)に対して、(＊)は少なくとも４つの正値解をもつ。そのうち２つの解はf(x)に連続的に依存しf(x)→0すると－Δu＋u＝a(x)upの解に収束する。しかし他のふたつはf(x)→0としても－Δu＋u＝a(x)upの解に強収束せず、無限遠にエネルギーの中心部が平行移動してゆく解ω(・－yf)(｜yf｜→∞　as f(x)→0)としてのプロファイルをもつ。(Calculus of Variations and Partial Differential Equationsより出版予定)。通常、微分方程式の研究においては方程式の係数等に解が連続的に依存する場合が研究されているが、(＊)のような簡単な方程式であっても微小な摂動f(x)により不連続な依存性が現れることを示しており、興味深いと思われる。また同様のアイデアに基づいたもう一編の論文では、より一般的な非線型項g(x,u)を扱いCao、Jeanjean、Hirano、Zhouらの結果を拡張している。(Nonlinear Analysis:T.M.A.より出版予定)。　またHamilton系に関してはmulti-bump solutionと呼ばれる記号力学系に対応した複雑な解軌道の構成をNehari多様体を用いた変分的な方法により行った。

• 1998

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変分的手法により非線形微分方程式の解の存在問題の研究を行った。特に本年度は(i)２体問題型のハミルトン型に対する非有界軌道の存在、(ii)２体問題型のハミルトン型に対する周期軌道の存在と関連するnon-compact 多様体上の閉測地線の存在(iii)Moser-Trudinger 型の不等式の最良指数についての研究を行った。(i) ハミルトン型に関して２体問題型のポテンシャル〈I〉Ｖ(q) 〈/I〉～－1／｜〈I〉q 〈/I〉｜〈SUP〉α〈/SUP〉（α＞０）に対して無限から来て無限に飛びさる軌道の変分的な構成を考え、与えられた〈I〉Ｈ〈/I〉＞０をtotal energy としてもち、さらに与えられた入射角、出射角をもつ軌道の存在を空間次元〈I〉Ｎ〈/I〉に関する制限なしでstrong force 条件（α＞２）の下で示した。ここで空間次元が２のときは回転数を有効に利用することができ比較的容易に証明はなされるが、〈I〉Ｎ〈/I〉≧３の場合は異なりＲ〈SUP〉N〈/SUP〉＼｛０｝上のループ空間のtopology に関する考察が必要不可欠となることに注意して頂きたい。なお〈I〉Ｈ〈/I〉＝０のときは古典力学における放流軌道に対応し、非常に興味ある問題であるが、その存在は今後の課題としたい。(ii) またハミルトン系の周期解に関しても２体問題型ポテンシャル〈I〉Ｖ（ｑ）〈/I〉～～－1／｜〈I〉q 〈/I〉｜〈SUP〉α〈/SUP〉を考え、α＝２の場合を扱った。α＝２の場合はいわゆるstrong force とweak force の境界の場合でありprescribed energy problem は今までほとんど研究されていない。Total energy が０の周期解の存在を特に考え、その存在を示した。また関連する問題としてnon-compact Riemman 多様体（〈I〉Ｓ〈SUP〉n〈/SUP〉〈/I〉×Ｒ、〈I〉ｇ〈/I〉）上の閉測地線の存在を示した。(iii) 楕円形型方程式に関してはMoser-Trudinger 型の不等式の最良指数について研究を行い、Ogawa, ozawa により導入されたスケール不変なMoser-Trudinger 型の不等式は有界領域の場合と異なり最良指数を達成しないことを示した。

• 1997

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研究成果は（ⅰ）非線型楕円型方程式に関する存在結果および（ⅱ）ハミルトン系に対する周期軌道の存在問題に関する結果の２つに大別される。　まず非線型楕円型方程式に関してはRNにおけるscaler field equation -Δu+V(x)u=up, u∈H1(RN)の生値解について考察した。ここでは、ポテンシャルV(x)がxについて周期的な場合を主な対象とした。このような場合、ground stateと呼ばれるenergyが最小の解ω0 (x)が存在することはconcentration-compactness method等によりよく知られている。本研究では、より高いenergy levelの生値解の存在について考察し、ω0 (x)の重ね合わせた形の解u(x)～∑Nj=1ω0 (x-lj),(│lI-lj│>>1)の存在をポテンシャルV(x)に対する適当な条件の下で示した。非線型方程式に対する解の重ね合わせの方法はごく最近変分的に見直され、注目を集めているが、その存在のための条件は具体的にcheckしにくいものであった。ここではそのような具体例を初めて与えている。　次にハミルトン系の周期解に関しては、ポテンシャルV(q)が特異性V(q)=-1/│q│aをもつ場合に与えられたtotal energy Hをもつ周期解の存在について研究を行った。　この問題においては、特異性の指数αが重要な役割を果たすことがわかっている。02のときはH>0に対してのみ解の存在が期待でき、それぞれの場合に研究が行われている。ここではα=2の場合を考え、H=0に対する周期解の存在を考察した。α>2あるいは0<α<2の場合と異なり、α=2の場合は摂動に対して極めてsensitiveであるが周期解の存在を保証する十分広い摂動のクラスを求めるのに成功した。また対応する状況下でnon-compact Riemann多様体上の閉測地線の存在を議論した。研究成果の発表1998　(with A. Amrosetti) On Keplerian N-body type problems, in“Nonlinear Analysis and continuum mechanics”(G. Battazo, G.P. Galdi, E. Lanconelli, P. Pacci ed.) Springer, 1998, 15-25.1997　Multiple positive solutions for some nonlinear elliptic systems, Topological Methods in Nonlinear Analysis, 10 (1997), 15-45.

### Overseas Activities

• 変分的手法による非線型問題の研究

2011.04
-
2012.03

フランス   コンテ大学

チリ   チリ大学

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