1998 Fiscal Year Annual Research Report

非線形発展方程式と楕円型方程式の研究

Research Project

Project/Area Number	09440070
Research Institution	Waseda University
Principal Investigator	大谷光春早稲田大学, 理工学部, 教授 (30119656)
Co-Investigator(Kenkyū-buntansha)	坂口茂愛媛大学, 理学部, 助教授 (50215620) 林仲夫東京理科大学, 理学部, 助教授 (30173016) 石井仁司東京都立大学, 理学部, 教授 (70102887) 田中和永早稲田大学, 理工学部, 助教授 (20188288) 山田義雄早稲田大学, 理工学部, 教授 (20111825)
Keywords	非線形 / 楕円型方程式 / 放物型方程式 / 分散型方程式 / 漸近挙動 / 発展方程式
Research Abstract	計画調書の研究目的にかかげた目標に関する成果が,着実にあがりつつある.以下にその主なものを挙げる. 方程式 (E)_1 -Δu=\|u\|^<q-2>u x∈Ω, u(x)=0 x∈∂Ω に対して,次の定理を得た. 1) 「Ω=R^N＼B_R,B_R={x∈R^N;\|x\|【less than or equal】R},2^<q<+∞ (2^は,ソボレフ型埋蔵HH^1_0(Ω)⊂L^q(Ω)の臨界指数)とするとき,(E)_1はH^1(Ω)∩L^q(Ω)に属する(球対称)非自明解をもつ(大谷・橋本_哲).」1<q【less than or equal】2^の場合には,非自明解が存在しないことが既に知られており(大谷・橋本_貴),有界領域に対する既知の結果(1<q<2^のき存在,2^【less than or equal】q<+∞のとき非存在)との双対性から,解の存在が予想されていたが,この長年の未解決問題が肯定的に解決された意義は極めて大きい. 2) Ω=Ω_d×R^<N-d>,Ω_dをd一次元円環領域とする.非自明解の存在に関して,今まで解明されていなかった場合2^<q【less than or equal】N_dが解決された:(i)2^<q<N_dの場合は,解が存在し,(ii)q=N_dの場合は,解が存在しない.即ち,既存の結果とあわせると「(i)1<q<N_dのとき存在(ii)N_d【less than or equal】qのとき非存在」となり,この問題に対する完全な解答が得られたことになる.この事実から,「領域のd次元対称性は,実効的次元を(d-1)次元だけ減ずる効果をもたらす」ことが結論づけられた. (3) 方程式 (E)_2 -Δu+u=α(x)\|u\|^<q-2>u+f(x) x∈IR^N, 2<q<2^ 0<a(x),\|a(x)-1\|Ce^<λ\|x\|>,λ>0 に対して,\|\|f\|\|_<H-1(R^N)>が十分小さければ,(E)_2は少なくとも2つの正値解をもつ.(田中・足達)この結果は,従来の結果のa(・)に対する仮定を大幅に緩めたものになっている.さらに\|\|f\|\|_<H-1>-0としたときの解の漸近挙動も決定されている. この他に,坂口による準線形放物型方程式の解の空間変数に関する導関数の零点の(時間)漸近的振る舞いに関する研究,林による臨界指数の非線形性を有する非線形Schrodinger方程式の解の漸近挙動に関する研究,山田によるreactiondiffusion systemに対するGlobal Attractorの研究などの興味深い結果が得られている.

Research Products

(14 results)

All Other

All Publications (14 results)

[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)
[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)
[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)
[Publications] S.Hashimoto & M.Otani: "Elliptic equations with singnlarity on the foundary" Differential and Inteqial Equations. (to appear).
[Publications] S.Takeuti & Y.Yamada: "A Symptotic properties of a reaction-diffasion equation with degenerate p-laplacian" Nonlinear Analysis, Theory, Methods & Applications. (to appear).
[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).
[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)
[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).
[Publications] S.Adadni & K.Tanaka: "Tradinger type inequalities in IR^N and their bont exponents" Proc.Amer.Math.Soc.(to appear).
[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)
[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)
[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)
[Publications] S.Sakaguchi & T.Suzuki: "Interior imperfeat iguition connd occur on a rat of positive measure" Arch.Rational Mech.Aual.142. 143-153 (1998)
[Publications] R.Magnanini & S.Sakaguchi: "Spatial artical points not moving alay the heat flow II:The cent rosymmetric care" Math.Z. (to appear).

1998 Fiscal Year Annual Research Report

非線形発展方程式と楕円型方程式の研究

Principal Investigator

大谷 光春 早稲田大学, 理工学部, 教授 (30119656)

Research Products

[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)

[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)

[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)

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

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

[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).

[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)

[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).

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

[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)

[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)

[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)

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

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

大谷光春早稲田大学, 理工学部, 教授 (30119656)