Kubo Hideo
Faculty of Science Mathematics Mathematics | Professor |
Last Updated :2025/02/13
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Educational Organization
- Bachelor's degree program, School of Science
- Master's degree program, Graduate School of Science
- Doctoral (PhD) degree program, Graduate School of Science
■Career
Career
Educational Background
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■Research activity information
Papers
- Global solvability for nonlinear wave equations with singular potential
Vladimir Georgiev, Hideo Kubo
Journal of Differential Equations, 375, 514, 537, Elsevier BV, Dec. 2023, [Peer-reviewed]
English, Scientific journal - On the Rellich Type Inequality for Schrödinger Operators with Singular Potential
V. Georgiev, H. Kubo
“Harmonic Analysis and Partial Differential Equations”, M. Ruzhansky, J. Wirth (eds.), Trends in Mathematics, Springer Nature Switzerland., 77, 89, 2022, [Peer-reviewed]
English - On the Cauchy Problem for the Nonlinear Wave Equation with Damping and Potential
M. Kato, H. Kubo
“Harmonic Analysis and Partial Differential Equations”, M. Ruzhansky, J. Wirth (eds.), Trends in Mathematics, Springer Nature Switzerland., 45, 61, 2022, [Peer-reviewed]
English, In book, In this note, we study the Cauchy problem for the nonlinear wave equation with damping and potential terms. The aim of this study is to generalize the result in Georgiev et al. (J. Differ. Equ. 267(5):3271–3288, 2019) into two directions. One is to relax the condition which characterizes the behavior of the coefficient of the damping term at spatial infinity as in (6). The other is to treat the slowly decreasing initial data. The decaying rate of the data affects the global behavior of the solutions even if the nonlinear exponent lies in the super-critical regime (see Theorem 5 below). - Blow-up for Strauss type wave equation with damping and potential
Wei Dai, Hideo Kubo, Motohiro Sobajima
Nonlinear Analysis: Real World Applications, 57, 103195, Feb. 2021, [Peer-reviewed]
English, Scientific journal - Blow-up phenomena of semilinear wave equations and their weakly coupled systems
Masahiro Ikeda, Motohiro Sobajima, Kyouhei Wakasa
Journal of Differential Equations, 267, 9, 5165, 5201, Elsevier BV, Oct. 2019, [Peer-reviewed]
English, Scientific journal - Modification of the vector-field method related to quadratically perturbed wave equations in two space dimensions
H. Kubo
"Advanced Studies in Pure Mathematics 81, 2019 Asymptotic Analysis for Nonlinear Dispersive and Wave Equations", 81, 139, 172, 2019, [Peer-reviewed]
English - Critical exponent for nonlinear damped wave equations with non-negative potential in 3D
V. Georgiev, H. Kubo, K. Wakasa
J. Differential Equations, 267, 3271, 3288, 2019, [Peer-reviewed]
English, Scientific journal - Beckner type of the logarithmic Sobolev and a new type of Shannon's inequalities and an application to the uncertainty principle
H. Kubo, T. Ogawa, T. Suguro
Proceedings of the American Mathematical Society, Vol. 147 (4), 4, 1511, 1518, American Mathematical Society (AMS), 2019, [Peer-reviewed]
English, Scientific journal - Localization of innexins in the antennae of the Japanese carpenter ant, Camponotus japonicus and its putative involvement in the chemosensory mechanism for nestmate-nonnestmate discrimination
Tatsuya Uebi, Yusuke Takeichi, Kouji Yasuyama, Naoyuki Miyazaki, Kazuyoshi Murata, Satoshi Kurihara, Eichi Takaya, Hideo Kubo, Toshiaki Omori, Mamiko Ozaki
CHEMICAL SENSES, 43, 5, E142, E142, OXFORD UNIV PRESS, Jun. 2018, [Peer-reviewed]
English - Putative Neural Network Within an Olfactory Sensory Unit for Nestmate and Non-nestmate Discrimination in the Japanese Carpenter Ant: The Ultra-structures and Mathematical Simulation.
Yusuke Takeichi, Tatsuya Uebi, Naoyuki Miyazaki, Kazuyoshi Murata, Kouji Yasuyama, Kanako Inoue, Toshinobu Suzaki, Hideo Kubo, Naoko Kajimura, Jo Takano, Toshiaki Omori, Ryoichi Yoshimura, Yasuhisa Endo, Masaru K Hojo, Eichi Takaya, Satoshi Kurihara, Kenta Tatsuta, Koichi Ozaki, Mamiko Ozaki
Frontiers in cellular neuroscience, 12, 310, 310, 310, 2018, [Peer-reviewed], [Invited], [International Magazine]
English, Scientific journal, Ants are known to use a colony-specific blend of cuticular hydrocarbons (CHCs) as a pheromone to discriminate between nestmates and non-nestmates and the CHCs were sensed in the basiconic type of antennal sensilla (S. basiconica). To investigate the functional design of this type of antennal sensilla, we observed the ultra-structures at 2D and 3D in the Japanese carpenter ant, Camponotus japonicus, using a serial block-face scanning electron microscope (SBF-SEM), and conventional and high-voltage transmission electron microscopes. Based on the serial images of 352 cross sections of SBF-SEM, we reconstructed a 3D model of the sensillum revealing that each S. basiconica houses > 100 unbranched dendritic processes, which extend from the same number of olfactory receptor neurons (ORNs). The dendritic processes had characteristic beaded-structures and formed a twisted bundle within the sensillum. At the "beads," the cell membranes of the processes were closely adjacent in the interdigitated profiles, suggesting functional interactions via gap junctions (GJs). Immunohistochemistry with anti-innexin (invertebrate GJ protein) antisera revealed positive labeling in the antennae of C. japonicus. Innexin 3, one of the five antennal innexin subtypes, was detected as a dotted signal within the S. basiconica as a sensory organ for nestmate recognition. These morphological results suggest that ORNs form an electrical network via GJs between dendritic processes. We were unable to functionally certify the electric connections in an olfactory sensory unit comprising such multiple ORNs; however, with the aid of simulation of a mathematical model, we examined the putative function of this novel chemosensory information network, which possibly contributes to the distinct discrimination of colony-specific blends of CHCs or other odor detection. - An RNAi Screen for Genes Involved in Nanoscale Protrusion Formation on Corneal Lens in Drosophila melanogaster.
Ryunosuke Minami, Chiaki Sato, Yumi Yamahama, Hideo Kubo, Takahiko Hariyama, Ken-Ichi Kimura
Zoological science, 33, 6, 583, 591, Dec. 2016, [Peer-reviewed], [Domestic magazines]
English, The "moth-eye" structure, which is observed on the surface of corneal lens in several insects, supports anti-reflective and self-cleaning functions due to nanoscale protrusions known as corneal nipples. Although the morphology and function of the "moth-eye" structure, are relatively well studied, the mechanism of protrusion formation from cell-secreted substances is unknown. In Drosophila melanogaster, a compound eye consists of approximately 800 facets, the surface of which is formed by the corneal lens with nanoscale protrusions. In the present study, we sought to identify genes involved in "moth-eye" structure, formation in order to elucidate the developmental mechanism of the protrusions in Drosophila. We re-examined the aberrant patterns in classical glossy-eye mutants by scanning electron microscope and classified the aberrant patterns into groups. Next, we screened genes encoding putative structural cuticular proteins and genes involved in cuticular formation using eye specific RNAi silencing methods combined with the Gal4/UAS expression system. We identified 12 of 100 candidate genes, such as cuticular proteins family genes (Cuticular protein 23B and Cuticular protein 49Ah), cuticle secretion-related genes (Syntaxin 1A and Sec61 ββ subunit), ecdysone signaling and biosynthesis-related genes (Ecdysone receptor, Blimp-1, and shroud), and genes involved in cell polarity/cell architecture (Actin 5C, shotgun, armadillo, discs large1, and coracle). Although some of the genes we identified may affect corneal protrusion formation indirectly through general patterning defects in eye formation, these initial findings have encouraged us to more systematically explore the precise mechanisms underlying the formation of nanoscale protrusions in Drosophila. - ON THE POINTWISE DECAY ESTIMATE FOR THE WAVE EQUATION WITH COMPACTLY SUPPORTED FORCING TERM
Hideo Kubo
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 14, 4, 1469, 1480, AMER INST MATHEMATICAL SCIENCES, Jul. 2015, [Peer-reviewed]
English, Scientific journal, In this paper we derive a new type of pointwise decay estimates for solutions to the Cauchy problem for the wave equation in 2D, in the sense that one can diminish the weight in the time variable for the forcing term if it is compactly supported in the spatial variables. As an application of the estimate, we also establish an improved decay estimate for the solution to the exterior problem in 2D. - On the exterior problem for nonlinear wave equations with small initial data
Hideo Kubo
NONLINEAR DYNAMICS IN PARTIAL DIFFERENTIAL EQUATIONS, 64, 281, 288, MATH SOC JAPAN, 2015, [Peer-reviewed]
English, International conference proceedings, The aim of this note is to give an overview concerning the mixed problem for a system of nonlinear wave equations with small and smooth initial data. In particular, we are interested in the three and two space dimensional case. - Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions
Hideo Kubo
JOURNAL OF DIFFERENTIAL EQUATIONS, 257, 8, 2765, 2800, ACADEMIC PRESS INC ELSEVIER SCIENCE, Oct. 2014, [Peer-reviewed]
English, Scientific journal, In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions, assuming that the initial data is small and smooth. We establish the same type of lower bound of the lifespan for the problem as that for the Cauchy problem, despite of the weak decay property of the solution in two space dimensions. (C) 2014 Elsevier Inc. All rights reserved. - Global existence for quadratically perturbed massless Dirac equations under the null condition
S. Katayama, H. Kubo
Fourier Analysis: Pseudo-Differential Operators,Time-Frequency Analysis and Partial Differential Equations(edited by M. Ruzhansky and V. Turunen), 253, 262, 2014, [Peer-reviewed] - ALMOST GLOBAL EXISTENCE FOR EXTERIOR NEUMANN PROBLEMS OF SEMILINEAR WAVE EQUATIONS IN 2D
Soichiro Katayama, Hideo Kubo, Sandra Lucente
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 12, 6, 2331, 2360, AMER INST MATHEMATICAL SCIENCES, Nov. 2013, [Peer-reviewed]
English, Scientific journal, The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition. - GLOBAL EXISTENCE FOR EXTERIOR PROBLEMS OF SEMILINEAR WAVE EQUATIONS WITH THE NULL CONDITION IN 2D
Hideo Kubo
EVOLUTION EQUATIONS AND CONTROL THEORY, 2, 2, 319, 335, AMER INST MATHEMATICAL SCIENCES, Jun. 2013, [Peer-reviewed]
English, Scientific journal, In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions under some geometric restriction on the obstacle. We prove a global existence result for the problem with small and smooth initial data, provided that the nonlinearity is taken to be cubic and satisfies the null condition. - LOWER BOUND OF THE LIFESPAN OF SOLUTIONS TO SEMILINEAR WAVE EQUATIONS IN AN EXTERIOR DOMAIN
Soichiro Katayama, Hideo Kubo
JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS, 10, 2, 199, 234, WORLD SCIENTIFIC PUBL CO PTE LTD, Jun. 2013, [Peer-reviewed]
English, Scientific journal, We consider the Cauchy-Dirichlet problem for semilinear wave equations in a three space-dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hormander where the Cauchy problem was treated. We show that our estimate is sharp at least for radially symmetric case. - Opal Films with Dome-Shaped Structures Fabricated by Hot Embossing
Hiroshi Fudouzi, Takahiko Hariyama, Yumi Yamahama, Shinya Yoshioka, Daisuke Ishii, Ken-ichi Kimura, Hideo Kubo, Masatsugu Shimomura, Yoshihiro Uodu
KOBUNSHI RONBUNSHU, 70, 5, 227, 231, SOC POLYMER SCIENCE JAPAN, May 2013, [Peer-reviewed]
Japanese, Scientific journal, Hierarchical microstructures in beetle epidermis layers enable a wide variety of structural colors. However, conventional artificial mimicking techniques are limited, mainly, to multi-layer film formation or 3D colloidal crystal array assemblies on flat substrates. In this paper, we propose a new method applying hot embossing using a metal mesh pressed onto the self-assembled opal film on a plastic substrate. Firstly, opal films composed of 0.2 mu m polystyrene colloids and infilling elastic silicone polymer were coated on flat polyvinyl chloride (PVC) sheets. Then the surface of the opal film was thermally deformed by pressing a micro structured mold into it above the glass transition temperature. Micro-spectroscopic analysis revealed that the spectral reflection from the tip of a convex shaped dome was the same as the one from the flat area under a tilting angle of 25 degrees. Our new method is expected to contribute to the fabrication of future complex biomimetic structural color elements. - Generalized wave operators for a system of semilinear wave equations in three space dimensions
Hideo Kubo, Koji Kubota
HOKKAIDO MATHEMATICAL JOURNAL, 42, 1, 81, 111, HOKKAIDO UNIV, DEPT MATHEMATICS, Feb. 2013, [Peer-reviewed]
English, Scientific journal, This paper is concerned with the final value problem for a system of semi-linear wave equations. The main issue is to solve the problem when the nonlinearity is of a long-range type. By assuming that the solution is spherically symmetric, we shall show global solvability of the final value problem around a suitable final state, and hence, the generalized wave operator and long range-scattering operator can be constructed. - Global existence and blow-up for wave equations with weighted nonlinear terms in one space dimension
Hideo KUBO, Ayako OSAKA, Muhammet YAZICI
Interdisciplinary Information Sciences, 19, 2, 143, 148, Tohoku University, 2013, [Peer-reviewed]
English, We consider the initial value problem for wave equations with weighted nonlinear terms in one space dimension. Under the assumption that the initial data and nonlinearity are odd functions, we are able to show global existence of small amplitude solutions. We also prove that symmetric assumptions on the initial data are necessary to obtain the global solution, by showing a blow-up result. - Parameter identification problem for a parabolic equation - application to the Black-Scholes option pricing model
Yury M. Korolev, Hideo Kubo, Anatoly G. Yagola
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, 20, 3, 327, 337, WALTER DE GRUYTER & CO, Sep. 2012, [Peer-reviewed]
English, Scientific journal, We consider an inverse problem of parameter identification for a parabolic equation. The underlying practical example is the reconstruction of the unknown drift in the extended Black-Scholes option pricing model. Using a priori information about the unknown solution (i.e. its Lipschitz constant), we provide a solution to this non-linear ill-posed problem, as well as an error estimate. Other types of a priori information may be used (for example, monotonicity and/or convexity of the unknown solution). - Lower bounds for the lifespan of solutions to nonlinear wave equations in elasticity
Hideo Kubo
Progress in Mathematics, 301, 187, 212, Springer Basel, 2012, [Peer-reviewed]
English, In book, In this paper we study the lifespan of solutions to nonlinear wave equations in elasticity with small initial data. Main step of our argument is to construct a good approximate solution. A natural choice of the approximation seems to be the leading term of solutions to the free elastic wave equation. However, it does not satisfy the nonlinear elastic wave equation in a suitable sense. For this reason, we modify the approximation by adding a higher-order term. Then, we are able to obtain a lower bound of the lifespan which is expressed in terms of initial data and a coefficient in the nonlinearity. - The Rate of Convergence to the Asymptotics for the Wave Equation in an Exterior Domain
Soichiro Katayama, Hideo Kubo
FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 53, 3, 331, 358, KOBE UNIV, DEPT MATHEMATICS, Dec. 2010
English, Scientific journal, In this paper we consider the mixed problem for the wave equation exterior to a non-trapping obstacle in odd space dimensions. We derive a rate of the convergence of the solution for the mixed problem to its asymptotic profile, which is written as a solution for the Cauchy problem. As a by-product, we are able to find out the radiation field of solutions to the mixed problem in terms of the scattering data. - A remark on long range effect for a system of semilinear wave equation
Hideo Kubo, Motoharu Takaki
Rendiconti dell'Istituto di Matematica dell'Universita di Trieste, 2010 - On the large time behavior of solutions to semilinear system of the wave equation
Soichiro Katayama, Hideo Kubo
Proceedings of the 5th International ISSAC congress, 2009 - An alternative proof of global existence for nonlinear wave equations in an exterior domain
Soichiro Katayama, Hideo Kubo
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 60, 4, 1135, 1170, MATH SOC JAPAN, Oct. 2008, [Peer-reviewed]
English, Scientific journal, The aim of this article is to present a simplified proof of a global existence result for systems of nonlinear wave equations in an exterior domain. The novelty of our proof is to avoid completely the scaling operator which would make the argument complicated in the mixed problem, by using new weighted pointwise estimates of a tangential derivative to the light cone. - Asymptotic behavior of solutions to semilinear systems of wave equations
Soichiro Katayama, Hideo Kubo
Indiana University Mathematics Journal, 57, 1, 377, 400, 2008
English, Scientific journal, We consider the Cauchy problem for a class of systems of semilinear wave equations, which is closely connected to the weak null condition and Alinhac's condition. We show that the energy of some global solutions to these systems grows to infinity as time tends to infinity and consequently these solutions never approach any free solutions. Indiana University Mathematics Journal ©. - Decay estimates of a tangential derivative to the light cone for the wave equation and their application
Soichiro Katayama, Hideo Kubo
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 39, 6, 1851, 1862, SIAM PUBLICATIONS, 2008
English, Scientific journal, We consider wave equations in three space dimensions and obtain new weighted L-infinity-L-infinity estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the Lorentz boost operators. - Asymptotic behavior of solutions to semilinear wave equations with dissipative structure
Hideo Kubo
Discrete and Continuous Dynamical Systems, Supplement, 602-613, 2007 - Uniform decay estimates for the wave equation in an exterior domain
KUBO H.
Asymptotic analysis and singularities, 31-54, 31, 54, Math. Soc. Japan, 2007 - Note on weighted Strichartz estimates for Klein-Gordon equations with potential
Kubo Hideo, Lucente Sandra
Tsukuba journal of mathematics, 31, 143-173, 1, 143, 173, Institute of Mathematics, University of Tsukuba, 2007
Japanese - Existence and asymptotic behavior of radially symmetric solutions to a semilinear hyperbolic system in odd space dimensions
Hideo Kubo, Koji Kubota
CHINESE ANNALS OF MATHEMATICS SERIES B, 27, 5, 507, 538, SHANGHAI SCIENTIFIC TECHNOLOGY LITERATURE PUBLISHING HOUSE, Sep. 2006
English, Scientific journal, This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t --> -infinity in the energy norm, and to show it has a free profile as t --> +infinity. Our approach is based on the work of [11]. Namely we use a weighted L-infinity norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper. - Large time behavior of solutions to semilinear systems of wave equations
H Kubo, K Kubota, H Sunagawa
MATHEMATISCHE ANNALEN, 335, 2, 435, 478, SPRINGER, Jun. 2006
English, Scientific journal, This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that "modified" free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non-existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough. - Blowup for systems of semilinear wave equations in two space dimensions
Hideo Kubo, Masahito Ohta
Hokkaido Mathematical Journal, 35, 3, 697, 717, 2006
English, Scientific journal, We consider semilinear systems of wave equations with multiple propagation speeds and find out the critical order of the nonlinearity which characterizes large time behavior of small amplitude solutions to the system by establishing blowup results. We also evaluate the lifespan of the solution in terms of the size of the initial data from above and below. We underline that not only the order of the nonlinearity but also the way of coupling among unknowns in it has a major effect on the lifespan. © 2006 by the University of Notre Dame. All rights reserved. - Large-time behavior of solutions for a nonlinear system of wave equations
Hideo Kubo
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 63, 5-7, E2279, E2287, PERGAMON-ELSEVIER SCIENCE LTD, Nov. 2005, [Peer-reviewed]
English, Scientific journal, This note deals with the initial value problem for a nonlinear system of wave equations. First we show the global existence of small amplitude solutions to the system. Then we study large-time behavior of the global solutions. In some cases we find different behavior of the solution from the free solution. Such a behavior is characterized as a solution to a certain inhomogeneous wave equations. (C) 2005 Elsevier Ltd. All rights reserved. - On Systems of Semi linear Wave Equations with Unequal Propagation Speeds in Three Space Dimensions
Hideo Kubo, Masahito Ohta
FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 48, 1, 65, 98, KOBE UNIV, DEPT MATHEMATICS, Apr. 2005
English, Scientific journal, In this paper we study coupled systems of semilinear wave equations and derive sharp conditions for the small data global existence and blowup for the system. The way of the interaction in the nonlinearities plays an important role to determine the condition. We focus on the case where propagation speeds also come into play. The discrepancy of the speeds is actually essential in Theorem 3.1 for instance. Moreover, in some cases we have different conclusion for the same non-linearity according to the order of them. To handle such cases, we modify the argument presented by F. John [12]. - On the global behavior of classical solutions to coupled systems of semilinear wave equations, in “New trends in the theory of hyperbolic equations”
Hideo Kubo, Masahito Ohta
Operator Theory Adv. and Appl., Birkh¨auser Verlag, 159, 113, 211, 2005, [Peer-reviewed] - GLOBAL SOLVABILITY FOR SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH MULTIPLE SPEEDS IN TWO SPACE DIMENSIONS
Akira Hoshiga, Hideo Kubo
DIFFERENTIAL AND INTEGRAL EQUATIONS, 17, 5-6, 593, 622, KHAYYAM PUBL CO INC, May 2004, [Peer-reviewed]
English, Scientific journal, In this paper we deal with systems of nonlinear wave equations in two space dimensions. When the system has common propagation speeds and cubic nonlinearity, the small data global existence result was obtained by Katayama [9], provided that the cubic part of Taylor's expansion for the nonlinearity satisfies the so-called null condition. The aim of this paper is to extend the result to the case where the system has multiple speeds of propagation. To realize this, we make use of a kind of Hardy's inequality given in Lemma 2.2 below, which creates the loss of decay but only with respect to (1 + parallel to x vertical bar - c(i)t vertical bar). Thus we are able to absorb such a loss by means of the decay estimates in Proposition 4.2 below. - On point-wise decay estimates for the wave equation and their applications in "Dispersive Nonlinear Problems in Mathematical Physics"
KUBO Hideo
Quaderni di Matematica, Seconda Universit´a di Napoli, 15, 2004, [Peer-reviewed] - Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions
H Kubo, K Tsugawa
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 9, 2, 471, 482, AMER INST MATHEMATICAL SCIENCES, Mar. 2003
English, Scientific journal, In this paper, we treat the coupled system of wave equations whose nonlinearities are \u\(pj)\v\(qj) and propagation speeds may be different from each other. We study the lower bounds of p(j) and q(j) to assure the global existence of a class of small amplitude solutions which includes self-similar solutions. The exponent of self-similar solutions plays crucial role to find the lower bounds. Moreover, we prove that the discrepancy of propagation speeds allow us to bring them down. Conversely, if such conditions for the global existence do not hold, then no self-similar solution exists even for small initial data. - On the small data global existence and scattering for systems of semilinear wave equations
Hideo Kubo
Hyperbolic problems and related topics, 219, 234, 2003, [Peer-reviewed] - Coupled system of semilinear wave equations
Hideo Kubo
Lecture Notes of Seminario Interdisciplinare di Matematica, 75, 85, 2003, [Peer-reviewed] - Seattering for systems of semilinear wave educations with diffrent speeds of propagation(共著)
H. Kubo, K.Kubota
Adv. Difference Equations, 7, 441-468/,, 2002 - Time-local well-posedness of (1 + 2)-dimensional wave-map type equations and the null condition(On well-posedness and regularity of solutions to partial differential equations)
Hideo Kubo
数理解析研究所講究録, 1284, 16, 31, 2002 - Weighted decay estimates for the wave equation
P D'Ancona, Georgiev, V, H Kubo
JOURNAL OF DIFFERENTIAL EQUATIONS, 177, 1, 146, 208, ACADEMIC PRESS INC, Nov. 2001, [Peer-reviewed]
English, Scientific journal, In this work we study weighted Sobolev spaces in R-n generated by the Lie algebra of vector fields (1 + \x \ (2))(1/2)partial derivative (x1), j = 1,..., n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in R-n. As an application we derive weighted Ll estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math. 119, 1291-1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces. (C) 2001 Academic Press. - Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions - Dedicated to Professor Kiyoshi Mochizuki on the occasion of his 60th birthday
H Kubo, K Kubota
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 53, 4, 875, 912, MATH SOC JAPAN, Oct. 2001, [Peer-reviewed]
English, Scientific journal, We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in R-n x R, where n = 2, 3. As an application of the estimate, we study the asymptotic behavior as t --> +/-infinity of solutions u(x, t) and v(x, t) to a system of semilinear wave equations: partial derivative (2)(t)u - Deltau = \v\(p), partial derivative (2)(t)v - Deltav = \u\(q) in R-n x R, where (n + 1)/(n - 1) < p <less than or equal to> q with n = 2 or n = 3. More precisely, it is known that there exists a critical curve Gamma = Gamma (p, q, n) = 0 on the p-q plane such that, when Gamma > 0, the Cauchy problem for the system has a global solution with small initial data and that, when Gamma less than or equal to 0, a solution of the problem generically blows up in finite time even if the initial data are small. In this paper, when Gamma > 0, we construct a global solution (u(x, t), v(x, t)) of the system which is asymptotic to a pair of solutions to the homogeneous wave equation with small initial data given, as t --> -infinity, in the sense of both the energy norm and the pointwise convergence. We also show that the scattering operator exists on a dense set of a neighborhood of 0 in the energy space. - Supercritical semilinear wave equation with non-negative potential
Georgiev, V, C Heiming, H Kubo
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 26, 11-12, 2267, 2303, MARCEL DEKKER INC, 2001
English, Scientific journal, We prove a weighted L-infinity estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential. - Small data blowup for systems of semilinear wave equations with different propagation speeds in three space dimensions
H Kubo, M Ohta
JOURNAL OF DIFFERENTIAL EQUATIONS, 163, 2, 475, 492, ACADEMIC PRESS INC, May 2000, [Peer-reviewed]
English, Scientific journal, We consider the Cauchy problem for a system of semilinear wave equations with small initial data and critical nonlinearity. As for a class of systems of quasilinear wave equations with critical nonlinearity, the small data global existence has been well developed for the case when the propagation speeds are distinct. In contrast with the quasilinear case, we show that the critical small data blowup occurs for the semilinear case, even if the propagation speeds are different from each other. (C) 2000 Academic Press. - Weighted Strichartz estimate for the wave equation
P D'Ancona, Georgiev, V, H Kubo
COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, 330, 5, 349, 354, EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER, Mar. 2000, [Peer-reviewed]
English, Scientific journal, In this work we study weighted Sobolev spaces in R-n generated by the Lie algebra of vector fields
(1 + \x\(2))(1/2)partial derivative(xj), j = 1,..,n.
Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in R-n. As an application we derive weighted L-q estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established in [6] and establish global existence result for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS. - Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition
A Hoshiga, H Kubo
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 31, 3, 486, 513, SIAM PUBLICATIONS, Mar. 2000, [Peer-reviewed]
English, Scientific journal, This paper deals with the Cauchy problems of nonlinear hyperbolic systems in two space dimensions with small data. We assume that the propagation speeds differ from each other and that nonlinearities are cubic. Then it will be shown that if the nonlinearities satisfy the null condition, there exists a global smooth solution. To prove this kind of claim, one usually makes use of the generalized differential operators Omega(ij), S, and L-i, which will be introduced in section 1. But it is difficult to adopt the operators L-i = x(i)partial derivative(t) + t partial derivative x(i) to our problem, because they do not commute with the d'Alembertian whose propagation speed is not equal to one. We succeed in taking L-i away from the proof of our theorem. One can apply our method to a scalar equation; hence L-i are needless in this kind of argument. - Global existence and blow-up of the classical solutions to systems of semilinear wave equations in three space dimensions
Hideo Kubo, Masahito Ohta
Rend. Is-tit. Mat. Univ. Trieste, 31 suppl. 2, 145, 168, 2000, [Peer-reviewed] - Critical exponent for wave equation with potential
V. Georgiev, C. Kerller, Hideo Kubo
Rend. Istit. Mat. Univ. Trieste, 31 suppl. 2, 103, 127, 2000, [Peer-reviewed] - Weighted Strichartz esti-mate for the wave equation and low regularity solutions
Piero D’Ancona, Vladimir Georgiev, Hideo Kubo
Rend. Istit. Mat. Univ. Trieste, 31 suppl. 2, 51, 61, 2000, [Peer-reviewed] - Critical blowup for systems of semilinear wave equations in low space dimensions
H Kubo, M Ohta
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 240, 2, 340, 360, ACADEMIC PRESS INC, Dec. 1999, [Peer-reviewed]
English, Scientific journal, We consider the Cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data. Del Santo ct al. ["Geometric Optics and Related Topics" (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117-140, Birkhauser, Boston, 1997] have studied the existence and nonexistence of global classical solutions of the Cauchy problem except for the critical case. In this paper we study the critical case, and we show the nonexistence of global classical solutions and also give the upper bounds of the life span. (C) 1999 Academic Press. - Chauchy problem of nonlinear wave equations with small and smooth initial data (Harmonic Analysis and Nonlinear Partial Differential Equations)
Hideo Kubo
RIMS Kokyuroku, 1102, 91, 111, Kyoto University, Jun. 1999
English - Asymptotic behaviors of radially symmetric solutions of □u = |u|p for super critical values p in even space dimensionsn
Hideo Kubo, Kôji Kubota
Japanese Journal of Mathematics, 24, 2, 191, 256, 1998, [Peer-reviewed]
English, Scientific journal - Slowly decaying solutions for semilinear wave equations in odd space dimensions
H Kubo
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 28, 2, 327, 357, PERGAMON-ELSEVIER SCIENCE LTD, Jan. 1997, [Peer-reviewed]
English, Scientific journal - Asymptotic behaviors of radially symmetric solutions of □u = |u|p for super critical values p in high dimensions(非線形発展方程式とその応用)
H. Kubo, K. Kubota
数理解析研究所講究録, 966, 88, 94, Kyoto University, 1996
English - On the critical decay and power for semilinear wave equations in odd space dimensions
Hideo Kubo
Discrete and Continuous Dynamical Systems, 2, 2, 173, 190, Southwest Missouri State University, 1996, [Peer-reviewed]
English, Scientific journal, In this paper we study global behaviors of solutions of initial value problem to wave equations with power nonlinearity. We shall derive space-time decay estimates according to decay rates of the initial data with low regularity (in classical sense). Indeed we can control L∞-norm of a solution in high dimension, provided the initial data are radially symmetric. This enables us to construct a global solution under suitable assumptions and to obtain an optimal estimate for a lifespan of a local solution. - Asymptotic behaviors of radial solutions to semilinear wave equations in odd space dimensions
Hideo Kubo, K\\^oji Kubota
Hokkaido Math.J., 1995 - Asymptotic behaviors of radially symmetric solutions of ❯ u=|u|p for super critical values p in odd space dimensions
Hideo Kubo, Kôji Kubota
Hokkaido Mathematical Journal, 24, 2, 287, 336, 1995, [Peer-reviewed]
English, Scientific journal, We study asymptotic behaviors as t→±∞ of solutions to the nonlinear wave equation utt–Δ u=|u|p(p>
1) in x∈ ℝn, ∞<
t<
∞ for p larger than a critical value p0(n) . These asymptotic behaviors guarantee the existence of the scattering operator. We prove the radially symmetric small solutions exist and are asymptotic to the solutions of the homogeneous wave equations, provided n is odd and n≥ 5. © 1995, Hokkaido University. All rights reserved. - ASYMPTOTIC-BEHAVIOR OF SOLUTIONS TO SEMILINEAR WAVE-EQUATIONS WITH INITIAL DATA OF SLOW DECAY
H KUBO
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 17, 12, 953, 970, JOHN WILEY & SONS LTD, Sep. 1994, [Peer-reviewed]
English, Scientific journal, Some useful and remarkable property are derived from a representation formula of a radially symmetric solution to the Cauchy problem for a homogeneous wave equation in odd space dimensions. These properties provide us with enough information to consider the semilinear case, namely, the associated integral equation with the problem will be considered on a weighted L(infinity)-space. This formulation enables us to deal with the problem for slowly decaying initial data. - Blow-up for semilinear wave equations with initial data of slow decay in low space dimensions
久保 英夫
Differential and Integral Equations, 7, 315, 321, 1994, [Peer-reviewed]
Other Activities and Achievements
- Lower bound of the lifespan of solutions to nonlinear elastic wave equation (Regularity and Singularity for Geometric Partial Differential Equations and Conservation Laws)
Kubo Hideo, RIMS Kokyuroku, 1845, 33, 58, Jul. 2013
Kyoto University, English - 26pXR-10 Multilinear operators: a natural extension of the Hirota's bilinear operator
Endo Kentaro, Kubo Hideo, Toda Kouichi, Meeting abstracts of the Physical Society of Japan, 68, 1, 336, 336, 26 Mar. 2013
The Physical Society of Japan (JPS), Japanese - A remark on long range effect for a system of semilinear wave equation
Hideo Kubo, Motoharu Takaki, Rendiconti dell'Istituto di Matematica dell'Universita di Trieste, 2010 - The rate of convergence to the asymptotics for the wave equation in an exterior domain
Soichiro Katayama, Hideo Kubo, Funkcialaj Ekvacioj, 53, 3, 331, 358, 2010
In this paper we consider the mixed problem for the wave equation exterior to a non trapping obstacle in odd space dimensions. We derive a rate of the convergence of the solution for the mixed problem to its asymptotic profile, which is written as a solution for the Cauchy problem. As a by-product, we are able to find out the radiation field of solutions to the mixed problem in terms of the scattering data., English - On the large time behavior of solutions to semilinear system of the wave equation
Soichiro Katayama, Hideo Kubo, Proceedings of the 5th International ISSAC congress, 2009 - An alternative proof of global existence for nonlinear wave equations in an exterior domain
Soichiro Katayama, Hideo Kubo, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 60, 4, 1135, 1170, Oct. 2008
The aim of this article is to present a simplified proof of a global existence result for systems of nonlinear wave equations in an exterior domain. The novelty of our proof is to avoid completely the scaling operator which would make the argument complicated in the mixed problem, by using new weighted pointwise estimates of a tangential derivative to the light cone., MATH SOC JAPAN, English - An elementary proof of global existence for nonlinear wave equations in an exterior domain
Soichiro Katayama, Hideo Kubo, J. Math. Soc. Japan, 2008 - Decay estimates of a tangential derivative to the light cone for the wave equation and their
Soichiro Katayama, Hideo Kubo, SIAM J. Math. Anal., 39, 6, 1851, 1862, 2008 - Asymptotic behavior of solutions to semilinear systems of wave equations,
Soichiro Katayama, Hideo Kubo, Indiana Univ. Math. J., 2008 - Blow-up for nonlinear wave equations with multiple speeds (Evolution Equations and Asymptotic Analysis of Solutions)
Kubo Hideo, Ohta Masahito, RIMS Kokyuroku, 1358, 77, 97, Feb. 2004
Kyoto University, English - Supercritical semilinear wave equation with non-negative potential
Georgiev, V, C Heiming, H Kubo, COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 26, 11-12, 2267, 2303, 2001
We prove a weighted L-infinity estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential., MARCEL DEKKER INC, English - Small data blowup for systems of semilinear wave equations with different propagation speeds in three space dimensions (vol 163, pg 475, 2000)
H Kubo, M Ohta, JOURNAL OF DIFFERENTIAL EQUATIONS, 168, 2, 477, 477, Dec. 2000
ACADEMIC PRESS INC, English, Others - Asymptotic behaviors of radial solutions to semilinear wave equations in odd space dimensions
Hideo Kubo, Kôji Kubota, Hokkaido Mathematical Journal, 24, 1, 9, 51, 1995
English
Books and other publications
- 多変数の微積分とベクトル解析 (新・数理/工学ライブラリ 応用数学 3)
神保 秀一, 久保 英夫
数理工学社, 10 Sep. 2020, 4864810680, 176, Japanese, [Joint work] - The role of metrics in the theory of partial differential equations, Advanced Studies in Pure Mathematics, 85
Y. Giga, N. Hamamuki, H. Kuroda, T. Ozawa, H. Kubo
Mathematical Society of Japan, 2020, 9784864970907, 543p, English, Scholarly book, [Joint editor] - RIMS Kôkyûroku Bessatsu B70 "Harmonic Analysis and Nonlinear Partial Differential Equations"
Hideo Takaoka, Hideo Kubo
Research Institute for Mathematical Sciences Kyoto University, Apr. 2018, 166, [Joint editor] - RIMS Kôkyûroku Bessatsu B65 "Harmonic Analysis and Nonlinear Partial Differential Equations"
Hideo Kubo, Hideo Takaoka
Research Institute for Mathematical Sciences Kyoto University, May 2017 - RIMS Kôkyûroku Bessatsu B60 "Harmonic Analysis and Nonlinear Partial Differential Equations"
Hideo Kubo, Mitsuru Sugimoto
Research Institute for Mathematical Sciences Kyoto University, Dec. 2016, 212, [Joint editor] - RIMS Kôkyûroku Bessatsu B56 "Harmonic Analysis and Nonlinear Partial Differential Equations"
Hideo Kubo, Mitsuru Sugimoto
Research Institute for Mathematical Sciences Kyoto University, Apr. 2016, 215, [Joint editor] - RIMS Kôkyûroku Bessatsu B49 "Harmonic Analysis and Nonlinear Partial Differential Equations"
Hideo Kubo, Mitsuru Sugimoto
Research Institute for Mathematical Sciences Kyoto University, Apr. 2014, 137 - Hokkaido Math. J. vol.37
Hideo Kubo, Hiroyuki Takamura, Special Issue “Nonlinear Wave Equations”
Hokkaido University, 2008, [Joint editor] - "New trends in the theory of hyperbolic equations", Oper. Theory Adv. Appl.
Hideo Kubo, Masahito Ohta, On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations
BirkhäuserVerlag, 2005, 159, 113-211, [Contributor] - Dispersive Nonlinear Problems in Mathematical Physics
KUBO Hideo, On point-wise decay estimates for the wave equation and their applications
2004, 123-148, [Contributor]
Lectures, oral presentations, etc.
- Global existence and blow-up for nonlinear wave equations with inverse-square potential
Hideo Kubo
The 24th Northeastern Symposium on Mathematical Analysis, 21 Feb. 2023, English, Invited oral presentation
20 Feb. 2023 - 21 Feb. 2023, Sendai, Japan, [Invited], [International presentation] - 重み付きRellich 型不等式とその応用
Hideo Kubo
非線型偏微分方程式と走化性, 29 Nov. 2022, Japanese, Invited oral presentation
29 Nov. 2022 - 01 Dec. 2022, 北九州市, Japan, [Invited], [Domestic Conference] - 逆二乗冪型ポテンシャルを伴う非線型波動方程式の解析 (PartⅡ)
Hideo Kubo
第43回発展方程式若手セミナー, 06 Sep. 2022, Japanese, Invited oral presentation
05 Sep. 2022 - 07 Sep. 2022, [Invited] - 逆二乗冪型ポテンシャルを伴う非線型波動方程式の解析 (PartⅠ)
Hideo Kubo
第43回発展方程式若手セミナー, 05 Sep. 2022, Japanese, Invited oral presentation
05 Sep. 2022 - 07 Sep. 2022, [Invited] - On the Rellich type inequality for Schrödinger operators with potential of inverse-square type
Hideo Kubo
Mathematical Analysis of Nonlinear Dispersive and Wave Equations, 25 Aug. 2022, English, Invited oral presentation
24 Aug. 2022 - 26 Aug. 2022, Tokyo, Japan, [Invited], [International presentation] - Global existence for semilinear wave equations with potential of inverse-square type
Hideo Kubo
応用解析研究会, 23 Jul. 2022, English, Invited oral presentation
23 Jul. 2022 - 23 Jul. 2022, 東京都, Japan, [Invited] - On the nonlinear wave equation with lower order terms
Hideo Kubo
Seminar of Applications of Differential Equations in Sciences, 22 Dec. 2021, English
22 Dec. 2021 - 22 Dec. 2021, [Invited] - 低階項を伴う非線型波動方程式の初期値問題について
久保英夫
東京大学解析学火曜セミナー, 16 Nov. 2021, 東京大学, Japanese
東京都, Japan, [Invited] - On the effect of slowly decreasing initial data for nonlinear wave equations with damping and potential in the scaling critical regime
Hideo Kubo
13th ISAAC Congress 2021, 03 Aug. 2021, English
02 Aug. 2021 - 06 Aug. 2021, Ghent University, Belgium, [Invited], [International presentation] - On the semilinear wave equation with lower order terms
久保英夫
第37回 九州における偏微分方程式研究集会, 27 Jan. 2020
[Invited] - 非線型波動方程式に対する幾何学的および双対的アプローチ (Part I)
久保英夫
第9回室蘭非線形解析研究会, 11 Jan. 2020
[Invited] - Bio-inspired mathematical model of an effective integration of information
KUBO Hideo
第80回応用物理学会秋季学術講演会, 21 Sep. 2019, 公益社団法人 応用物理学会
札幌市, [Invited], [International presentation] - Asymptotic behavior for the nonlinear damped wave equation with a positive potential
KUBO Hideo
信州大学偏微分方程式研究集会, 28 Jun. 2019, 信州大学
松本市, [Invited] - ルールダイナミクスの適応性について
KUBO Hideo
On the activation of adaptive filters by the self-organization, 23 May 2019 - Critical exponent for nonlinear damped wave equations with non-negative potential in 3D
KUBO Hideo
偏微分方程式セミナー, 26 Apr. 2019, 北海道大学
北海道大学 - 波動方程式に対する重み付きエネルギー評価とその周辺
久保 英夫
感応寺山セミナー2019, 19 Jan. 2019
[Domestic Conference] - On the metric perturbation for semilinear wave equations
KUBO Hideo
SEMINARIO DI EQUAZIONI ALLE DERIVATE PARZIALI, 13 Dec. 2018, Università di Pisa, English
Pisa, [Invited], [International presentation] - Global existence for nonlinear damped wave equations with a potential
KUBO Hideo
第14回非線型の諸問題, 11 Sep. 2018
[Invited] - Remark on Kolmogorov's superposition theorem
KUBO Hideo
RIMS共同研究「Mathematical Analysis of Self-Organization with Constraints」, 16 May 2018 - Global existence for nonlinear damped wave equations with potential
KUBO Hideo
Zhejiang-Hokudai Workshop, 28 Mar. 2018
[Invited], [International presentation] - On the exterior problem for systems of nonlinear wave equations with multiple speeds
KUBO Hideo
Workshop on Nonlinear Wave Equations, Apr. 2017, Fudan University - Asymptotic behavior of solutions to quasilinear wave equations with dissipative structure
KUBO Hideo
7th Euro-Japanese Workshop on Blow-up, Sep. 2016, The Mathematical Research and Conference Center
Będlewo - On the local smoothing for the Dirac equation
KUBO Hideo
10th International ISAAC Congress, Aug. 2015, University of Macau - On the exterior problem for the wave equation with critical nonlinearity in 2D
KUBO Hideo
Analysis of Relativistic and Non-Relativistic models in Quantum Mechanics, Apr. 2014, University of Roma - On the null condition for nonlinear massless Dirac Equations in 3D
KUBO Hideo
Fourier Analysis and Pseudo-Differential Operators, Jun. 2012, Aalto University - Generalized wave operator for a system of nonlinear wave equations
KUBO Hideo
7th International ISAAC Congress, Jul. 2009, Imperial College London - Lifespan for nonlinear wave equations in an exterior domain
KUBO Hideo
SEMINARIE, Analyse numeric et E.D.P., Mar. 2009, Universite Paris-Sud - Large time behavior of solutions to semilinear wave equations with dispersive structure
KUBO Hideo
FRG/JAMI workshop “Nonlinear Dispersive Equations", Mar. 2007, Johns Hopkins University - Global and almost global existence for wave equations on unbounded domains
KUBO Hideo
6'eme Conf'erence Internationale AIMS, “Systemes Dynamiques, Equations Differentielles et Applications", Jun. 2006, Universite de Poitiers - 非線形波動方程式に対する散乱作用素の一つの構成法
久保 英夫
ENCOUNTER with MATHEMATICS “第31回スペクトル・散乱理論", Oct. 2004, 中央大学 - 波動方程式の解の時空評価と非線型摂動への応用
久保 英夫
日本数学会函数方程式論特別講演, Sep. 2003, 千葉大学 - On the small data global existence and scattering for systems of semilinear wave equations
KUBO Hideo
Hyperbolic Problems and Related Topics, Sep. 2002
Cortona - Global existence to nonlinear wave equations with a potential in three dimensions
KUBO Hideo
微分方程式の総合的研究, Dec. 2000, 東京大学 - Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition
KUBO Hideo
微分方程式の総合的研究, Dec. 1997, 大阪大学
Research Themes
- 非線形消散波動方程式の一般論の構築と宇宙論および流体力学への応用
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
Apr. 2022 - Mar. 2027
高村 博之, 若杉 勇太, 加藤 正和, 佐々木 多希子, 久保 英夫, 津田谷 公利, 若狭 恭平
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Tohoku University, Coinvestigator, 22H00097 - 強双曲型方程式において弱零条件の果たす役割の解明
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Apr. 2019 - Mar. 2024
久保 英夫, 加藤 正和, 津田谷 公利, 若狭 恭平, Yordanov Borislav
本研究の目的は、アインシュタイン方程式をプロトタイプとする強双曲型方程式に対する
非線型摂動について、その安定性を弱零条件として特徴付けることである。その目的を達成するために、当該年度においては、アインシュタイン方程式を初期値問題として扱うための枠組みついての検討を詳細に亘って行った。具体的には、アインシュタイン方程式を扱う座標系を時間的座標軸が常に時間的であるように選ぶことによって得られる3+1形式に着目した。この定式化は数値相対論の分野で標準的に用いられているものである。まず、時空を空間的超平面によってスライスし、ラプス関数とシフトベクトルにより座標系を張る。アインシュタイン方程式の共変性に由来するゲージに関する自由度により、この様な座標系を採用しても一般性を失うことはない。この座標系においてローレンツ計量の3+1分解を行い、この分解に従ってアインシュタイン方程式を書き下すと、時間に依存しない拘束条件(ハミルトン拘束条件、運動量拘束条件)と時間発展する空間的超曲面の外的曲率に関する双曲型の方程式が得られる。これらの方程式系はADM形式と呼ばれるが、時間発展する方程式を導く際に、アインシュタイン・テンソルを表に出さず、リッチ・テンソルで表示されたアインシュタイン方程式を用いると数学的に扱いやすくなることが知られている。しかし、このADM形式において得られる方程式系は弱双曲型であり、初期値の微小摂動に関して時間大域的な安定性に問題があった。その困難を克服するために導入されたのがBSSN形式であり、実際、方程式系は強双曲型となり、アダマールの意味で適切となる。こうした理由から、我々はアインシュタイン方程式のBSSN形式を解析の対象とした。
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, Principal investigator, 19H01795 - New development of mathematical theory of turbulence by collaboration of the nonlinear analysis and computational fluid dynamics
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
May 2016 - Mar. 2021
小薗 英雄, 三浦 英之, 久保 英夫, 木村 芳文, 芳松 克則, 前川 泰則, 隠居 良行, 金田 行雄, 小池 茂昭
1. 尺度不変な斉次Besov空間における定常Navier-Stokes方程式の解の存在と正則性について
n次元空間において,与えられた外力$f \in \dot B^{-3+\frac np}_{p, q}$ が十分小さければ,$u \in B^{-1+\frac np}_{p, q}$なる定常Navier-Stokes 方程式の解$u$ が一意的に存在することを証明した.ただし,1 ≦ p < ≦, 1 ≦ q ≦ ∞ である.応用として,定常Navier-Stokes 方程式に対する自己相似解が得られる.証明方法は,斉次Besov 空間$\dot B^s_{p, q}$, s>0 におけるHoelder型Leibnitz 規則と,n/p-s を指標とする埋め込み定理である.尚,鶴見により,仮定 1 ≦p < n かつ s>0 は最良であることが明らかにされた.
2. Navier-Stoke流の影響下におけるKeller-Segel方程式系に対する時間大域的解の存在及び有限時間爆発の指標
全平面領域における細胞性粘菌の密度$n$が,速度場 u を持つNavier-Stoke方程式に従う非圧縮性粘性流体の影響下にある場合を記述するKeller-Segel方程式系を,尺度不変な関数空間で考察した.まず,初期値$n_0 \in L^1(R^2)$, $u_0 \in L^2(R^2)$ が十分小さければ,時間大域的な古典解n, uが一意的に存在することを証明した.手法は線形熱半群の L^p-L^q 型評価とその摂動による.更に解が有限時刻で爆発する指標を,u_0 に何ら仮定を課すことなくn_0 のL^1における大きさで表現した.この指標は流体の影響がない場合の$\|n_0\|_{L^1(R^2)}$ の閾値 8π を含むものである.また爆発時刻 T における挙動を考察した.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Waseda University, Coinvestigator, 16H06339 - Toward the integrated dynamics that connect evolutionary economics and engineering
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Jul. 2016 - Mar. 2019
Kubo Hideo, NISHIBE makoto
Our personal behavior is conducted by the custom and routine of ourselves based on our experience and/or success stories. But in some cases, we are forced to change the usual strategy due to the change of the external circumstances. We studied such an adaptation to the stimuli from the outside world should not be designed by the top-down mechanism but by the bottom-up mechanism, in the framework of Mathematical analysis. In particular, we analyzed the immediate adaptation by following the way of information processing used by insects.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, Principal investigator, 16KT0015 - Advanced Analysis on Evolving Patterns in Nonlinear Phenomena Driven by Singular Structure
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
May 2014 - Mar. 2019
GIGA Yoshikazu
We prove the existence and the uniqueness of a solution and clarify its behavior for evolution equations mainly nonlinear diffusion equations describing evolution of patterns and shapes like crystal growth phenomena. We introduce new notions of a solution which allows shape with singularities for equations having singular structure. We thus establish foundation of mathematical analysis which easily describes real phenomena. Based on these fundamental results, we are able to numerically calculate phenomena which had been difficult to calculate, for example, phenomena of colliding spirals on surfaces of crystals.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), The University of Tokyo, Coinvestigator, 26220702 - An investigation of symmetries in the geometric structure and existence of global solutions to nonlinear dispersive wave equations
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Apr. 2013 - Mar. 2018
Takaoka Hideo, KUBO Hideo, NAKANISHI Kenji, TSUGAWA Kotaro
In this study, I have developed the local and global well-posedness for the initial value problem related to the nonlinear Schrodinger equations in which dispersion effect and nonlinear interaction effect are incorporating. Using the Fourier analysis, I separated the solution into two parts; non-resonant and resonant oscillation parts, which have different in nature and distinguish nonuniformity part of solutions. For the nonlinear Schrodinger equations both with derivative in nonlinearities and on a sphere domain, I improved the local well-posedness for large function spaces. Moreover, I showed that there exists exchange of energy between Fourier modes. In the research process, I observed the estimation of energy exchange between different Fourier modes, due to the contribution in the nonlinear interaction.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Coinvestigator, 25287022 - Biomimetics based on the functional structure and the formation process of organisms
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)
Jun. 2012 - Mar. 2017
HARIYAMA Takahiko
In order to embody the material design based on the surface structure of organisms and to develop the energy saving production process, we focused on the moth eye structures and the structural colors of several organisms; 1. Fabrication of high brightness surface structure by self-organizing method, 2. Observation of morphogenesis of organisms for industrialization, 3. To discover the meaning of the organism's "not precise structure but precise function", we organized research teams of different fields including mathematics, physics, biology, chemistry and engineering, and clarified their functions.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area), Hamamatsu University School of Medicine, Coinvestigator, 24120004 - Mathematical Theory of turbulence by the method of modern analysis and computational science
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
May 2012 - Mar. 2017
KOZONO HIDEO, OZAWA Tohru
The challenging problem on global well-posedness of the Navier-Stokes equations had been so fully investigated that several remarkable results are obtained. Furthermore, our DNS of the uniformly isotropic turbulence is still by far the larger computational performance so that we could deal with the turbulent fluid with the high Reynolds number without any error of the experiment and indeterminacy. Our study has been based on the DNS of such a world highest standard and we could succeed to overcome difficulty of turbulence with the high Reynolds number. In this way, our research projects have developed the modern mathematical analysis, the applied mathematics, computational science and hydrodynamics and hopefully will lead the relevant subjects to the world-wide level.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Waseda University, Coinvestigator, 24224003 - Global behavior for nonlinear wave
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Apr. 2012 - Mar. 2016
Kubo Hideo, KATAYAMA Soichiro, TAKAMURA Hiroyuki, HOSHIGA Akira, NAKAMURA Makoto, DOI Kazuyuki
Our research is concerned with equations which describe the way of propagation of waves. More precisely, we study the nonlinear effect produced by the interaction among waves, as well as the effect coming from the structure of the space-time in which the wave exists. For instance, when there exists an obstacle in the space, we are able to show that the way of propagation of waves are similar to that for the case where there exists no obstacle. In particular, it is a big progress to solve this kind of problem in two space dimensional case.
Moreover, we also study the way of propagation of waves in the space-time equipped with the metric which describes the expanding universe. By considering the property of function which represents the wave in detail, we rigorously proved the following intuitive image: the wavelength of the waves become long, so that the waves are stabilized in such an expanding universe model.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, Principal investigator, 24340024 - Quantum stochastic analysis - Transforms and spectral analysis
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Apr. 2011 - Mar. 2015
OBATA Nobuaki, FUKUIZUMI Reika, HASEGAWA Takehisa, SEGAWA Etsuo, KUBO Hideo, HIAI Fumio, SUZUKI Kanako
For the development of quantum stochastic analysis we focused on 'quantum white noise calculus' from analytic aspect and 'spectral analysis of complex networks' from algebraic aspect. We aimed at the establishment of the mathematical fundamentals and the paradigm for collaborating with other research fields for applications. By means of quantum white noise calculus, the Bogoliubov transform and the Girsanov transform are characterized by the white noise differential equations of new types. A quantum probabilistic method is applied to the spectral analysis of digraphs such as Manhattan product. The phase transition of various dynamics on networks is studied in detail with the help of numerical computation. New statistical properties of quantum walks on graphs such as localization are obtained by generalizing the existing method of spectral analysis.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Tohoku University, Coinvestigator, 23340027 - Analysis of differential equations on graphs.
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
2011 - 2013
TRUSHIN Igor, KUBO Hideo, MOCHIZUKI Hiyoshi
In this project we consider Schrodinger operators on noncompact graphs which consist of some infinite rays and compact part attached. Spectral and scattering problems on graphs arise as simplified models in mathematics, physics, chemistry and engineering when one considers the propagation of waves of different natures in thin, tube-like domains. We study scattering direct and inverse problems which are important in applied physics. (1)We treat an inverse scattering problem on a graph with an infinite ray and a loop joined at one point. Reconstruction procedure is presented.(2)We consider Schrodinger operators on noncompact star-shaped graphs including some finite rays. We show that our spectral representation formula provides the time dependent formulation of the scattering theory. The scattering operator is constructed in the configuration space, and then is related to the scattering matrix in the momentum space. Corresponding inverse scattering problem is investigated.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tohoku University, Coinvestigator, 23540181 - Theory of global well-posedness on the nonlinear partial differential equations
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
2008 - 2012
KOZONO Hideo, YANAGIDA Eiji, ISHIGE Kazuhiro, NAKAMURA Makoto, KUBO Hideo, KANEDA Yukio, ISHIHARA Takashi, YOSHIMATSU Katsunori, KAGEI Yoshiyuki, EI Shinichro
We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Coinvestigator, 20224013 - On the limiting amplitude principle for the exterior problem of the wave equation
Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
2010 - 2011
KUBO Hideo
Large time behavior of solutions to the exterior problem whose boundary value is oscillating in time is considered. It can be expressed as a product of a time periodic function with the same period as the boundary value and the resonance of the correspo nding Helmholtz equation. For the radially symmetric case, the existence of the resonance is actually proved. In conclusion, the limiting amplitude principle for the exterior problem of the wave equation with a periodic boundary value was formulated.
Japan Society for the Promotion of Science, Grant-in-Aid for Challenging Exploratory Research, Tohoku University, Principal investigator, 22654017 - Research of global behavior of classical solutions for quasilinear Wave equations
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
2008 - 2010
HOSHIGA Akira, KATAYAMA Soichiro, KUBO Hideo, KUROKAWA Yuki
In this study, we succeeded in the classification of the sufficient conditions (null-conditions) for the global existence of the classical solutions to the system of nonlinear wave equations in 2 and 3 space dimensions, according to the type of the nonlinear terms (Null-form type, Non-resonance type and Nonlinear dissipation type). As development of the research, we also obtained precise evaluations to the lifespan of the classical solutions for the first order hyperbolic PDE systems with multiple propagation speeds and for the nonlinear elastic wave equations.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 20540206 - Phase Space Analysis of Partial Differential Equations
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
2007 - 2010
NISHITANI Tatsuo, HAYASHI Nakao, DOI Shinichi, SUGIMOTO Mitsuru, SUNAGAWA Hideaki, KUBO Hideo, TAKUWA Hideki, UMEDA Tomio, IWASAKI Chisato, HOSHIRO Toshihiko, FUJIIE SETSURO, TOMITA Naohito
Much progress has been achieved on linear hyperbolic Cauchy problem, on precise asymptotic behaviors of solutions to nonlinear dissipative and wave equations and on semi-classical resonances, by local and global phase space analysis, in deep cooperation with all research members through annual international meeting. We have also successfully supported young mathematicians to acquire the techniques of phase space analysis by annual instructive conference.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Osaka University, Coinvestigator, 19204013 - On study of evolution equations with hyperbolic properties
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
2007 - 2010
HAYASHI Nakao, NISHITANI Tatsuo, DOI Shinichi, KUBO Hideo
We studied nonlinear Schrodinger equations, nonlinear Klein-Gordon equations and their systems. Time decay and asymptotic behavior of solutions were shown. We applied these results to show existence of wave or modified wave operators. In the case of critical nonlinearities, it was shown that main terms of solutions can be represented through nonlinearities clearly.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Osaka University, Coinvestigator, 19340030 - On the asymptotic behavior of solution to systems of nonlinear wave equations of long range type
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
2005 - 2007
KUBO Hideo, HAYASHI Nakao, MATSUMURA Akitaka
The aim of this research is to study the asymptotic behavior of wave functions perturbed by the influence of the nonlinearity and characterize such asymptotic behavior. It is known that if the influence of the nonlinearity is too strong, then the wave function diverges in a finite time. On the other hand, if the influence of the nonlinearity is weak, then the wave function exists globally in time and it tends to a wave function which is free from the nonlinear perturbation in the sense of the energy as time goes to infinity.
In this research, we treat the intermediate case, namely, we are interested in the case where the perturbed wave function exists globally in time, but it does not tend to any free wave function as time goes to infinity. In order to consider such nonlinear perturbation, our first task is to find nonlinear wave equations which admit global in time solutions whose asymptotic behavior may differ from any solution to the corresponding homogeneous wave equations. Then the next step is to show that its asymptotic behavior is actually different from the free solution. As for these problems, we seemed to find several examples of such nonlinear perturbation. For some examples, the asymptotic behavior of the wave function is better compared with that of the free solution. On the other hand, it is worse than that of the free solution for the other examples. Such difference is determined by a quantity which is computed from the order and the coefficients of the nonlinearity.
In the former case, the asymptotic profile is given by a second iterate of the free solution. On the other hand, in the latter case, the asymptotic profile is closely related to the radiation field for the free solution. We obtain a suitable ordinary differential equation whose solution gives the modification of the free radiation field.
In conclusion, the nonlinear perturbation of long range type is complicated and contains a full of variety to produce different kinds of asymptotic behavior.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Osaka University, Principal investigator, 17540157 - On study of partial differential equations describing natural phenomena
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
2003 - 2005
HAYASHI Nakao, NISHITANI Tatsuo, DOI Shin-ichi, MATSUMURA Akitaka, KUBO Hideo, SUGIMOTO Mitsuru
1,P.I.Naumkin and I studied the Burgers equation with pumping and showed a existence in time of solutions and asymptotic behavior of solutions by using a suitable transformation and the structure of nonlinear term.
2,E.I.Kaikina and I studied the KdV equations in a half line with 0 boundary value at the origin. Airy function is oscillating rapidly in the left hand side and decaying exponentially in the right hand side. We showed asymptotics of solutions to the KdV equation by making use of this property.
3,E.I.Kaikina, P.I.Naumkin and I studied nonlinear complex dissipative equations with sub-critical nonlinearities and showed a solution is stable in the neighborhood of a self similar, solution.
4,P.I.Naumkin, Shimomura, Tonegawa and I did a joint work on nonlinear Schredinger equations with cubic nonlinearities. It was known that there exists a modified wave operator under some geometric assumptions on the final data. We succeeded to remove a strong geometric assumption by finding a new way to get a second approximate solution of the problem.
5,E.I.Kaikina, P.I.Naumkin and I studied nonlinear damped wave equations with super-critical or critical nonlinearities. In the previous works, it was known that a global existence theorem holds in space dimension is less than 5. We improved this result for any space dimension by using the weighted Sobolev spaces and estimates of solutions linear problem. Furthermore, in the critical case we showed asymptotics of solutions. The result implies the decay order in time of solutions is higher than that of solutions to linear problem. We obtained the results by using the method we found in the study of nonlinear dissipative equations
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Osaka University, Coinvestigator, 15204009 - Mathematical analysis of interface problems in mathematical physics
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
2002 - 2004
SHIMIZU Senjo, SHIBATA Yoshihiro, KIKUCHI Koji, HOSHIGA Akira, ADACHI Shinji, NAKAJIMA Toru
In this research, we consider the Stokes equation with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equation. We analyzed this problem by the following procedure : (1) Analysis of the resolvent problem (2) Generation of Analytic semigroups (3) L_p-L_q estimates
(1)Obtained is the L_p estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in R^n. The result has been obtained by Grubb and Solonnikov by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from theirs. The core of my approach is to estimate the solutions in the whole space and half-space case. We apply the Fourier multiplier theorem to solution of the model problems.
(2)First we introduce the Helmholtz decomposition. Then we delete pressure trem and reduce to the problem only including velocity vector. Then we generated analytic semigroup to this reduced Stokes equation.
(3)We obtained local energy decay estimates and L_p-L_q estimates of the solutions to the Stokes equation with Neumann boudary condition. Comparing with the non-slip (Dirichlet) boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null net force at the boundary.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 14540171 - 摂動型波動方程式に対する重みつき時空評価に関する研究
Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
2002 - 2004
久保 英夫
本研究の目的は波動方程式においてその線型部分が摂動された方程式を解析し,その解の挙動と摂動項がない方程式の解のそれとの違いを調べることである.時間と空間がある意味で対等であるという波動方程式の性質から,その解の挙動は時間変数と空間変数の混在した形の減衰評価によって,より良く近似されると考えられる.そこで,重みつき時空評価がどのような形で摂動型波動方程式の解について成り立つか考察した.
まず,ポテンシャル項による摂動のある場合に重みつき時空評価を摂動のない場合と同様な形で導いた.しかし,質量項がない場合にはポテンシャルが無限遠方で十分速く減衰しているという仮定が必要であり,他方,質量項のある場合にはポテンシャルの減衰をそれ程必要としない代わりに最終的な評価は微分の損失を含んでいる.前者の評価式は更に非線型問題への応用が可能である.この様な評価を導くために散乱理論・フーリエ積分作用素・補間空間論などの理論を用いた.
また,非線型項による摂動による影響が重みつき時空評価にどのように影響するかについても調べた.非線形項の次数が高ければ,小さな解に対して摂動のない解が満たすのと同様の重みつき時空評価が得られた.このような評価式は,伝播速度の異なる非線型波動方程式系を解析するのにも有効である.更に,空間2次元の問題を扱うとき,時空評価からルベーグ空間における評価を導くことによって,より広いクラスの非線型項に対して時間大域解の存在を示すことが出来ることが分かった.
Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), Principal investigator, 14740114 - Analysis of gradient flow equations and Lagrange equations of action integrals associated to quasiconvex functionals
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
2002 - 2003
KIKUCHI Koji, KUBO Hideo, SHIMIZU Senjo, NEGORO Akira, NAKAJIMA Toru, HOSHIGA Akira, OHTA Masahito
This research was projected in order to investigate the following problems. 1.Constructing gradient flows associated to typical quasiconvex functionals, 2.Study in Lagrange equations of action integrals associated to typical quasiconvex functionals, 3.Discovering phenomena that show differences between convex and quasiconvex functions. During the term of the project the head investigator, Kikuchi, attended various conferences and discussed with specialists in related research areas. In the second year Workshop on spectral theory and differential operators was held at Fudan University, Shanghai, China, and the head investigator attended this conference, anounced his recent result and gathered information. Other investigators also attended various conferences held in Japan or abroad and gathered recent information. Thereby following research results are obtained. The most progresses are obtained in Problem 2. Linear application is investigated for a Lagrange equation of an action integrals associated to a functional that corresponds a value of the integral of F(Du(x)) for a function u, and several results are obtained in case that F is quasiconvex and linear growth. Before obtaining this result, it is obtained for the same equation that a sequence of approximate solutions to this equation constructed by Rothe's method converges to a function and that, if it satisfies the energy conservation law, it is a weak solution in the space of BV functions. This is already established for convex cases, and now it is successfully established for quasiconvex cases. Related to Problem 3, the problem requires a different observation from that in convex cases. So far, energy inequality is obtained by the use of the convexity of the functional, and hence this method is not available in quasiconvex cases. Instead our constructiong approximate solutions elementwisely makes it possible to obtain energy inequality. This seems to be a large difference between convex and quasiconvex functions. In research related to Problem 1, although constructing a gradient flow is not complete, it is sucseeded to find an identity in the process of constructing approximate solutions, which should be a key for our destination.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 14540202 - 非線型波動におけるStrauss予想のある一般化
Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A)
2000 - 2001
久保 英夫
本研究では、べき乗型の非線型項をもつ波動方程式の初期値問題を一般次元において扱った。台コンパクトな初期値については、その大きさがある意味で十分小さければ、ある臨界指数が存在して、非線型項の原点近傍でのオーダーがその臨界指数よりも真に大きいとき、時間大域的に弱解が存在することが知られていた。また、初期値が球対称の場合には、その無限遠方での減衰度に関する臨界オーダーのあることが知られていた。ここでは、球対称とは限らない一般の初期値に対して大域可解性を示すことを目標とし、ほぼ満足のいく結果が得られた。
証明の要点は次の2点である。一つは、斉次波動方程式に対する初期値問題の解を適切な重みつきルベーグ空間で評価できたこと。それには、初期値が属する空間として、通常の微分作用素だけではなく、ローレンツ群に付随するリー代数を表現するベクトル場も加えた微分作用素から生成される重みつきソボレフ空間を採用したことが決め手となった。もう一つは、非斉次波動方程式に対する初期値問題の解について、非斉次項が各時刻において台コンパクトであるという仮定のもとに得られていた評価を一般の場合に拡張したことである。そのために、スケーリングの議論を適用し、非斉次項の空間無限遠方での適当な可積分性の仮定のもと、必要な不等式を導くことができた。
以上の準備のもと、よく知られた手順に従って、小さな初期値に対して時間大域解の存在を証明された。結果として、初期値が予想される臨界オーダーより真に速く減衰していればよいことを、初期値の属する空間の性質から結論することができる。
Japan Society for the Promotion of Science, Grant-in-Aid for Encouragement of Young Scientists (A), Shizuoka University, Principal investigator, 12740105 - Research in evolution equations related to variational problems
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
2000 - 2001
KIKUCHI Koji, KUBO Hideo, SHIMIZU Senjo, NEGORO Akira, OHTA Masahito, HOSHIGA Akira
This research was projected in order to investigate the following problems. 1. Constructing gradient flows for various variational problems in, for example, nonlinear elasticity, 2. Bifurcation phenomena for gradient flow equations, 3. Hyperbolic equations related to deformation of elasticity and to area functional, 4. Application of the method of discrete Morse semiflow to the theory of Schrodinger equations, 5. Relation between blowup solutions and the method of discrete Morse semiflow. In the first year of this project World Congress of Nonlinear Analysts which is held once in each four years was held and hence the head investigator, Kikuchi, and another investigator, Ohta, attended this congress and gathered some recent information related to this project. In the second year Czechoslovak International Conference on Differential Equations and Their Applications was held and the head investigator attended this conference, anounced his recent result and gathered information. Besides each investigators attended various conferences held in Japan or abroad, announced each results and gattered recent information. Thereby following research results have been obtained. The most progresses are obtained in problems 1 and 3. The result related to 1 is that a gradient flow can be consructed when a quasiconvex functional satisfies some coersiveness condition. Furthermore, though the form of equation is restrictive, it turns out that a gradient flow for some quasiconvex functional can be constructed even if it does not satisfy such a coersiveness condition. The result related to 3 is that Dirichle condition for the equation of motion of vibrating membrane should be weaker than the usual weak formulation (that the trace vanishes). This result is obtined by applying a result in direct variational method to the theory of evolution equations, what is the most feature of this research project. Some facts related to Problem 4 are also obtained. It is confident that some new theories related to 2 and 5 will also be developed. But by now frames of these works have not yet been obtained. It should be expected in the future.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 12640205 - Research on properties of Markov processes governed by the pseudo-differential operators with variable orders and application of the m to nonlinear analysis
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
1998 - 1999
NEGORO Akira, KUBO Hideo, KIKUCHI Koji, TAKANO Masaru
As is well known, under suitable conditions, it has been shown that there exist pure jump type Markov processes governed by Levy. generating operators with degenerate Levy mesures. So we would like to know what conditions these Markov processes have their transition densities under. Recently, by using MALLIAVIN calculus, Kunita has constructed transition densities of these Markov proceses in some class. So, we tried to adapt the pseudo-differential operators theory for this problem and restricted our study to the case that the supports of Levy measures degenerated into mutualy independent d lines for each x in RィイD1dィエD1. Cosequently, we have got that Markov processes governed the following generators, L have transition densities. The L is
<>
where θィイD2jィエD2(x) (j = 1, 2,…, d) are smooth RィイD1dィエD1-valued functions with bounded derivatives on RィイD1dィエD1 and satisfy |θィイD2jィエD2(x)|=1(j = 1, 2,…, d). Putting Θ(x)=(θィイD21ィエD2(x), θィイD22ィエD2(x), …, θィイD2dィエD2(x)), we assume that the eigenvalues of Θ(x)*Θ(x) are unifomly bounded to the below. And also, α is a constant satisfying 1 < α < 2 and nィイD2jィエD2(x,y) (j = 1,…, d) are smooth funcutions with bounded derivatives satisfying usual coditions. Now, we are rounding off the above work. We regret to say that we were able to have no result about the relation between nolinear differential operators and stochastic processes. But while we were studing this problem, we had the following results.
(1) A one dimensional hyperbolic equation uィイD2ttィエD2 - uィイD2xxィエD2 = 0 is treated under a free boundary condition uィイD32(/)XィエD3-uィイD32(/)tィエD3=QィイD12ィエD1. The existance and the uniqueness of a classical solution is established loccaly.
(2) A weak solution to some forth order nonlinear parabolic equation is constructed by the method of time semidisceretization. A technique of geometoric measure theory is employed in order to obtain to obtain the convergence of the nonlinear terms.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 10640159 - 非線型波動方程式に関する研究
1996
Competitive research funding