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Nonlinear Evolution Equations and Elliptic Equations

Research Project

Project/Area Number 09440070
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionWaseda University

Principal Investigator

OTANI Mitsuharu  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (30119656)

Co-Investigator(Kenkyū-buntansha) ISHII Hitoshi  Tokyo Metropolitan University, Faculty of Science, Professor, 理学部, 教授 (70102887)
TANAKA Kazunaga  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
YAMADA Yoshio  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20111825)
SAKAGUCHI Shigeru  Ehime University, Faculty of Science, Associate Professor, 理学部, 助教授 (50215620)
SUZUKI Takashi  Osaka University, Faculty of Science, Professor, 理学研究科, 教授 (40114516)
林 仲夫  東京理科大学, 理学部, 助教授 (30173016)
堤 正義  早稲田大学, 理工学部, 教授 (70063774)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥11,700,000 (Direct Cost: ¥11,700,000)
Fiscal Year 1999: ¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 1998: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 1997: ¥4,200,000 (Direct Cost: ¥4,200,000)
KeywordsPartial Differential Equatiion / Elliptic Equations / Parabolic equations / Calculus of Variation / 非線形 / 分散型方程式 / 漸近挙動 / 発展方程式 / 非線形楕円型方程式 / 非線形放物型方程式 / 臨界ソボノフ指数
Research Abstract

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 … More 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. Less

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (61 results)

All Other

All Publications (61 results)

  • [Publications] Satoshi HASHIMOTO and Mitsuharu OTANI: "Elliptic Equations with Singularity on the Boundary"Differential and Integral Equations. vol.12 no.3. 339-349 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Mitsuharu OTANI and Yoshie SUGIYAMA: "C^∞-solutions of generalized porous medium equations"Proceedings of the Conference on "Nonlinear Partial Differential Equations and Applications". ed.by G.Boiling World Scientific. 62-70 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takahiro HASHIMOTO and Mitsuharu OTANI: "Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains"Houston J. Math.. 23. 267-290 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Takeuchi and Yoshio Yamada: "Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian"to appear in Nonlinear Anal. TMA..

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Ichikawa and Yoshio Yamada: "Some remarks on global solutions to quasilinear parabolic system with cross-diffusion"to appear in Funkcial. Ekvac..

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Adachi and Kazunaga TANAKA: "Four positive solutions for the semilinear elliptic equation : -△u+u^p=a(x)u+f(x) in R^N"to appear in Calculus of Variations and Partial Differential Equations.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Adachi and Kazunaga TANAK: "Trudinger type inequalities in R^N and their best exponents"to appear in Proc. Amer. Math. Soc..

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A.KODA and Takashi SUZUKI: "A note on the blow-up pattern for a parabolic equation."J. Math. Tokushima Univ.. vol.32. 19-25 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R.IKEHATA and Takashi SUZUKI: "Semilinear parabolic equations involving critical Sobolev exponent : local and asymptotic behavior of solutions"to appear in Integral and Differential Equations.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shigeru SAKAGUCHI and Takashi SUZUKI: "Interior imperfect ignition cannot occur on a set of positive measure"Arch. Rational Mech. Anal.. vol.142. 143-153 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Rolando MAGNANINI and Shigeru SAKAGUCHI: "Spatial critical points not moving along the heat flow II : The centrosymmetric case Corrigendum,"Math. Z. vol. 230. vol.230. 695-712 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Rolando MAGNANINI and Shigeru: "Spatial critical points not moving along the heat flow II : The centrosymmetric case Corrigendum,"Math. Z. vol. 232. vol.232. 389 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Satoshi HASHIMOTO and Mitsuharu OTANI: "Elliptic Equations with Singularity on the Boundary"Differential and Integral Equations. vol. 12, No, 3. 339-349 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Mitsuharu OTANI and Yoshie SUGIYAMA: "CィイD1∞ィエD1-solutions of generalized porous medium equations"proceedings of the Conference on "Nonlinear Partial Differential Equations and Applications" ed. by Gue Boling and Yang Dadi, World Scientific. 62-70 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takahiro HASHIMOTO and Mitsuharu OTANI: "Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains"Houston J. Math.. vol. 23. 267-290 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Takeuchi and Yoshio Yamada: "Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian"Nonlinear Anal. TMA. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Ichikawa and Yoshio Yamada: "Some remarks on global solutions to quasilinear parabolic system with cross-diffusion"Funkcial. Ekvac. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Adachi and Kazunaga TANAKA: "Four positive solutions for the semilinear elliptic equation : -Δu + uィイD1pィエD1 = a (x)u+f (x) in RィイD1nィエD1"Calculus of Variations and Partial Differential Equations. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Adachi and Kazunaga TANAKA: "Trudinger type inequalities in RィイD1nィエD1 and their best exponents"Proc. Amer. Math. Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. KODA and Takashi SUZUKI: "A note on the blow-up pattern for a parabolic equation"J. Math. Tokushima Univ.. vol. 32. 19-25 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. IKEHATA and Takashi SUZUKI: "Semilinear parabolic equations involving critical Sobolev exponent : local and asymptotic behavior of solutions"Integral and Differential Equations. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shigeru SAKAGUCHI and Takashi SUZUKI: "Interior imperfect ignition cannot occur on a set of positive measure"Arch. Rational Mech. Anal.. vol1. 142. 143-153 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Rolando MAGNANINI and Shigeru SAKAGUCHI: "Spatial critical points not moving along the heat flow II : The centrosymmetric case Corrigendum"Math. Z. vol1. 230 vol1. 232 389. 695-712 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Satoshi HASHIMOTO & Mitsuharu OTANi: "Elliptic Equations with Singularity on the Boundary"Differential and Integral Equations. Vol.12 NO.3. 339-349 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] A.Yoshida and Y.Yamada:: "Global attractivity of coexistence states for a certain class of reaction diffusion systems with 3×3 cooperative matrices"Adv.Math.Sci.Appl.. Vol.9. 695-706 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.KABEYA & Kazumaga TANAKA: "Uniqueness of positive radial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition"Comm.Partial Differential Equations. Vol.24. 563-598 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] D.Chopp,L.C.Evays & Hitoshi IsHii: "Waiting time effects for Gauss curvature flows"Indiana Univ.Math.J.. Vol.48. 311-334 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.NAGAI,T.SENBA & Takashi Suzuki: "Chemtactic collapse in a parabolic system of mathematical biology"Hiroshima Math.J.. (to appear). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Shigeru SAKAGUCHI: "When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?"J.Analyse Math.. Vol.78. 219-243 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] E.B.Chan & M.Otani et al.: "Approximation for the first eigenralues of some nonlinear elliptic operators in Banach Spaces" Advances in Mathematical Sciences and Applications. 8. 273-283 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Otani & Y.Sugiyama: "C^∞-solutions of generalized porous medium equations" Proceedings of the Conference on “Nonlinear Partial Differential Equations and Applications" ed.by G.Boling & Y.Dudi. World Suimtific. 62-70 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Otani & M.Shinba: "Dynamical systems and time-dependent subdifferential operadons" Proceedings of International Conference on “Dynamical Systems and Differential Equations "ed.by.S.C.HV, South Missouristate Univ.(press). 153-161 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Hashimoto & M.Otani: "Elliptic equations with singnlarity on the foundary" Differential and Inteqial Equations. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Takeuti & Y.Yamada: "A Symptotic properties of a reaction-diffasion equation with degenerate p-laplacian" Nonlinear Analysis, Theory, Methods & Applications. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Yoshida & Y.Yamada: "Global attractivity of coexistenea states for a certain dess of reaction diffasion systems with 3×3 comperative mutricss" Advenes in Mathematical Sciences and Applications. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Ambrosetti & K.Tanaka: "On Keplerian N-body type problems" “Nonlinear Analysis and continuum mechanies" ed.by G.Puttazo np.Galdi et al. (springer). 15-25 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Tanaka: "Periodic solutions for singular Hami Horian systems and closed gcodesito on new compair Riemennian memipold." Ann.Inst.H.Poincare Anal.Non Lineaire. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Adadni & K.Tanaka: "Tradinger type inequalities in IR^N and their bont exponents" Proc.Amer.Math.Soc.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Hayashi & P.I.Naumkin: "Asymptotics in large time of sdutions to nonlinear S chrodinger and Hartree equations" Amer.J.Math.120. 369-389 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Hayashi & P.I.Naumkin: "Large time asymptotics of sdutions to the genetalized Benjamin one equation" Trans.Amer.Math.Soc.351. 109-130 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Hayashi, E.I.Kaikina & P.I.Naumkin: "Large time behavior of solutions to generaliged denirative nonlinear Sdinodirger equation" Disorate and Continuono Dynamical Systems. 5. 93-106 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Sakaguchi & T.Suzuki: "Interior imperfeat iguition connd occur on a rat of positive measure" Arch.Rational Mech.Aual.142. 143-153 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] R.Magnanini & S.Sakaguchi: "Spatial artical points not moving alay the heat flow II:The cent rosymmetric care" Math.Z. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M. Otani-H. Inoue: "Periodic problems for heat convection equations in noncylindrical domains" Funkcialaj Ekracioj. 40. 19-39 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M. Otani-T. Hashimoto: "Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains" Houston J. Math. 23. 267-290 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M. Otani-T. Hashimoto: "Nonexistence of positive solutions for some quasilinear elliptic equations in striolike domains" Discrete and Continuous Dynamical Systems. 3. 565-578 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M. Tsutsumi: "Local Existence and uniqueness of viscosity solutions for U_t=U△U-γ|▽u|^2+f(t,u)" Advances in Mathematical Sciences and Appl.7. 55-68 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M. Tsutsumi: "The Meissner effect and the Ginzburg-Landau equations in the presence of an applied magnetic field" J. Math. Phys.38. 3046-3054 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M. Tsutsumi: "Regional blow up of solutions to the initial-boundary value problem for U_t=U^δ(△u+u)" Proceedings of the Royal Society of Ed.127A. 871-887 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] A. Ambrosetti-K. Tanaka: "On Keplerian N-body type problems" Nonlinear analysis and continuum me chanics. Springer. 15-25 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K. Tanaka: "Multiple positive solutions for some nonlinear elliptic systems" Topological methods in nonlinear analysis. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] N. Hayashi-P.I. Naumkin: "Asymptotic behavior in time of solutions to the derivative nonlinear Schrodinger equation revisited" Discrete and Continuous Dynamical Systems. 3. 383-400 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] N. Hayashi-K. Kato: "Analyticity in time and smoothing effect of solutions to nonlinear Schrodinger equations" Commun. Math. Phys. 184. 273-300 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] A. de Bouard, N. Hayashi & J.C. Saut: "Global existence of small solutions to a relativistic nonlinear Schrodinger equation" Commun. Math. Phys. 189. 73-105 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] N. Hayashi-H. Hirata: "Local existence in time of small solutions to the elliptic-hyperbol Darey-Stewartson system in the usual Sobolev space" Proceeding of the Edinburgh Math. Society. 40. 563-581 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] N. Hayashi-K. Kato: "Global existence of small analytic solutions to Schrodinger equations with quadratic nonlinearity" Commun. In P.D.E. 22. 773-798 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] R. Magnanini-S. Sakaguchi: "The spatial critical points not moving along the heat flow" J. Analyse Math. 71. 237-261 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S. Sakaguchi: "Spatial critical points of nonnegative solutions of the evolution p-Laplacian equation : the fast diffi case" Differential Integral Equations. 10. 1049-1063 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S. Sakaguchi-T. Suzuki: "Nonexistence of solutions for a degenerate parabolic equation describing imperfect ignition" Nonlinear Anal.31. 665-669 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] H. Ishii-M. Arisawa: "Some properities of ergodic attractors for controlled dynamical systems" Discrete Contin. Dynam. Systems. 4・1. 43-54 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] H. Ishii, M. Arisawa-P.L. Lions: "Un cotre-exemple pour la theorie du controle ergodique en dimension infinie" C.R. Acad. Sci. Paris Ser I Math.325・1. 37-71 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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