一类非线性四阶波动方程的初边值问题
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摘要
对于一类非线性四阶波动方程的初边值问题：其中Ω(?)R~n为边界充分光滑的有界区域。
研究了其整体弱解的存在性、唯一性、光滑性和爆破性质。所得的四个主要结果如下：
1、运用Galerkin方法结合势井理论构造稳定集证明了：
定理(存在性)：设则问题(0.1)-(0.3)存在整体弱解u满足：
2、若p满足更强的条件时，运用能量方法结合不等式技巧证明了：
定理(唯一性)：设则(0.1)-(0.3)的整体弱解是唯一的。
3、运用Galerkin方法、稳定集和不等式技巧证明了：
定理(光滑性)：设则问题(0.1)-(0.3)存在唯一整体弱解
西南交通大学硕士研究生学位论文
第11页
，，满足:
。。L旬(o，T:H了(。)。万‘(。))
u，。犷(0 .T:H了(卿)
u，，任乙‘(0，了:L，(。))
4、运用凸性分析方法结合势井理论构造不稳定集证明了:
定理(.破):邵。。V，u，任LZ(。)，E(o) 的局部解，则存在有限常数了，使得当l峥产时成立!。.2
厂(。)范数意义下在有限时刻发生blow一uP.
分二，即u在
For the initial-boundary value problem of a kind of nonlinear fourth-order wave equations:
where Rn is bounded domain with sufficiently smooth boundary.
What studied in this paper are the global existence, uniqueness, smoothness and blow-up of the weak solutions of (0.1) - (0.3). Our four main results are stated as follows:
1 By using the Galerkin method and constructing stable set
according to the potential well theory, It is proved:
Theorem (existence): Let
. Then problem (0.1)-(0.3) has global weak solutions u satisfying:
2 If p satisfies appropriately stronger condi tions, by using the energy method and the trick of inequality, It is proved:
Theorem (uniqueness): Let
. Then the global weak solution
of problem (0.1)-(0.3) is unique.
3 By using the Galerkin method, stable set and trick of inequality, It is proved:
Theorem (smoothness) : Let
. Then the problem (0.1)-(0.3) has unique global weak solution u satisfying:
4 By the convexity method and constructing unstable set according to the potential well theory, It is proved:
Theorem (blow up): Let u, u is the
local solution of problem (0.1) ?0.3), The there exists a finite constant T, such that. u blows up in f ini te
time under the L2() norm.
研究了其整体弱解的存在性、唯一性、光滑性和爆破性质。所得的四个主要结果如下：
1、运用Galerkin方法结合势井理论构造稳定集证明了：
定理(存在性)：设则问题(0.1)-(0.3)存在整体弱解u满足：
2、若p满足更强的条件时，运用能量方法结合不等式技巧证明了：
定理(唯一性)：设则(0.1)-(0.3)的整体弱解是唯一的。
3、运用Galerkin方法、稳定集和不等式技巧证明了：
定理(光滑性)：设则问题(0.1)-(0.3)存在唯一整体弱解
西南交通大学硕士研究生学位论文
第11页
，，满足:
。。L旬(o，T:H了(。)。万‘(。))
u，。犷(0 .T:H了(卿)
u，，任乙‘(0，了:L，(。))
4、运用凸性分析方法结合势井理论构造不稳定集证明了:
定理(.破):邵。。V，u，任LZ(。)，E(o)
厂(。)范数意义下在有限时刻发生blow一uP.
分二，即u在
For the initial-boundary value problem of a kind of nonlinear fourth-order wave equations:
where Rn is bounded domain with sufficiently smooth boundary.
What studied in this paper are the global existence, uniqueness, smoothness and blow-up of the weak solutions of (0.1) - (0.3). Our four main results are stated as follows:
1 By using the Galerkin method and constructing stable set
according to the potential well theory, It is proved:
Theorem (existence): Let
. Then problem (0.1)-(0.3) has global weak solutions u satisfying:
2 If p satisfies appropriately stronger condi tions, by using the energy method and the trick of inequality, It is proved:
Theorem (uniqueness): Let
. Then the global weak solution
of problem (0.1)-(0.3) is unique.
3 By using the Galerkin method, stable set and trick of inequality, It is proved:
Theorem (smoothness) : Let
. Then the problem (0.1)-(0.3) has unique global weak solution u satisfying:
4 By the convexity method and constructing unstable set according to the potential well theory, It is proved:
Theorem (blow up): Let u, u is the
local solution of problem (0.1) ?0.3), The there exists a finite constant T, such that. u blows up in f ini te
time under the L2() norm.
引文
[1] An Lianjun, Peirce A. A weakly nonlinear analysis of elaseto-plastic microstructure models. SIAM J.App.Math.,1995,55: 136～155.
[2] Banks H.T., Gilliam D.S.. Shubov V. I., Global solvability for damped abstract nonlinear hyperbolic systems. Diff. Integ. Equa. 1997,10:309-332.
[3] Love A. H.. A Treatise on the Mathematical Theory of Elaslicity. Dover, New York. 1964.
[4] Makhankov V.G...Dynamics of classical soliton. Phys. Rep. Rev. Section Phys. Lett. (Section C), 1978, 35C:1-128.
[5] Christiansen P.L., Lomdahl P.S, Muto V. On a Toda lattice model with a transversal degree of freedom. Nonlinearity, 1990, 4:477-501.
[6] Warnecke S. G.., Uber das homogene Dirichlet-Problem bei nichtlinearn partiellen Diffenrential gleichungen vom Typ der Boussinesq-Gleichung. Math. Meth. In the Appl. Sci., 1987, 9:493-519.
[7] Steven Paul Levandosky. Decay Estimates for Fourth-Order Wave Equatios[J]. Journal of Differential Equations,1998,143:360-413.
[8] Steven Levandosky. Stability and Instability of Fourth-Order Solitary Waves[J].Journalof Dynamics and Differential Equations,1998,10(1): 151-188.
[9] Juha Berkovits. On the bifurcation of large amplitude solutions for a system of wave and beam equations [J]. Nonlinear Analysis 2003,52:343-354.
[10] A.S.Ackleh,H.T.Banks and G.A.Pinter. A Nonlinear Beam Equation [J]. Applied Mathematics Letters,2002,15:381-387.
[11] K.Balachandran,J.Y.Park and I.H.Jung. Existence of Solutions of Nonlinear Extensible Beam Equation [J]. Mathematical and Computer Modelling,2002,36:747-754.
[12] Jia-Quan Liu. Free vibrations for an asymmetric beam equation [J]. Nonlinear Analysis,2002,51:487-497.
[13] Thomas Bartsch, Yanheng Ding. Periodic solutions of superlinear beam and membrane equations with perturbations from symmetry [J]. Nonlinear Analysis,2001,44:727-747.
[14] Alberto Aosio. A geometrical nonlinear correction to the Timoshenko beam equation [J]. Nonlinear Analysis,2001,47:729-740.
[15] S.A.Avdonin,N.G.Medhin,T.L.Sheronova. Identification of a piecewise constant coefficient in the beam equation [J]. Journal of Computational and Applied Mathematics,2000,114:11-21.
[16] S.M.Choo,S.K.Chung. Finite element Galerkin solutions for the nonplanar oscillatory beam equations [J]. Applied Mathematics and Computation,2000,114:279-301.
[17] S. M. Choo, S. K. Chung. Fintite difference approximate solutions for the strongly damped extensible beam equations [J]. Applied Mathematics and Computation, 2000, 112: 11-32.
[18] Tokio Matsuyama. Quasilinear hyperbolic-hyperbolic singular perturbations with nonmonotone nonlinearity[J]. Nonlinear Analysis, 1999, 35: 589-607.
[19] G. Perla Menzala. The Beam Equation as a Limit of a 1-D Nonlinear
Von Kármán Model [J]. Applied Mathematics Letters, 1999, 12:47-52
[20] S. M. Choo and S.K. Chung. L~2-Error Estimate for the Strongly Damped Extensible Beam Equations [J]. Appl. Math. Lett. 1998, 11 (6):101-107.
[21] Q-Heung Choi and Tachsun Jung. A Semilinear Beam Equation with Nonconstant Loads [J]. Nonlinear Aanlysis, Theory, Methods & Applications, 1997, 30 (8) :5515-5525.
[22] 马如云．一类四阶周期边值问题的可解性[J]．数学物理学报，1995，15(3)：315-318．
[23] 黄发伦，黄永忠．相应于具阻尼的Euler-Bernoulli梁方程的Co半群[J]．中国科学A辑，1992，OOA(2)：122-133．
[24] 丁彦恒．关于非线性梁方程的一点注记[J]．数学学报，1990，33(2)：172-181．
[25] 任瑞芳，荆付翠，武洁琼．变速度反馈梁振动发展系统的存在性[J]．山西大学学报(自然科学版)，2003，26(1)：9—12．
[26] 尹蝓，裴莲淑，金正国．非线性横梁方程的振动问题[J]．延边大学学报(自然科学版)，2001，27(1)：1-5．
[27] 唐秋林，耿建生，吴美云．一维非线性梁振动方程周期解的存在性[J]．南通工学院学报，2001，17(3)：51-53．
[28] 张宏伟．非线性高阶发展方程中的几个问题[D]．郑州大学，2002．
[29] J．L．Lions著．郭柏灵，汪礼衱译．非线性边值问题的一些解法[M]．广州：中山大学出版社，1992．
[30] 陈恕行，洪家兴 编著．偏微分方程的近代方法[M]．上海：复旦大学出版社，1988．
[31] 王耀东著．偏微分方程的L~2理论[M]．北京：北京大学出版社，1982．
[32] 李大潜，陈韵梅 著．非线性发展方程[M]．北京：科学出版社，1989．
[33] D.H.Sattinger. On Global Solution of Nonlinear Hyperbolic Equations[J]. Arch.Rat.Mech.Anal.,1968,30:148-172.
[34] L.E:Payne and D.H.Sattiner. Saddle Points and Unstabiliyt of Nonlinear Hyperboic Equatons[J]. Israel J.Math, 1975,22:273-303.
[35] M.Tsutsumi. Existence and Nonexistence of Global Solutions for Non-linearParabolicEqutions[J]. Publ.RIMS,KyotoUniv. 1972/73,8:211-229.
[36] H.A.Levine and R.A.Smith Ames. A Potential Well Theory for the Heat Equation with a Nonlinear Boundary Condition[J]. Math.Meth. in the Appl.Sci. 1987,9:127-136.
[37] Mitsuhiro Nakao,Kosuke Ono. Existence of global solutions to the Cauchy problem for the semi-linear dissipative wave equations[J]. Math.Z. 1993,214:325-342.
[38] Ryo Ikehata and Takashi Suzuki. Stable and unstable sets for evolution eqations of parabolic and hyperbolic type[J]. Hiroshima Math.J.,1996,26: 475-491.
[39] Ryo Ikehata. Some Remarks on the Wave Equations with Nonlinear Damping and Source Terms[J]. Nonlinear Analysis,Theory, Methods & Applications,1996,10(27):l165-1175.
[40] Kosuke Ono. Blowing up and Global Existence of Solutions for some Degenerate Nonlinear Wave Equations with Some Dissipation[J]. Nonlinear Analysis, Theory, Methods & Applications, 1997,7(30):4449-4457.
[41] Kosuke Ono. Global Existence,Decay, and Blowup of Solutions for Some
Mildy Degenerate Nonlinear Kirchhoff Strings[J]. Journal of Differential Equations, 1997,137:273-301.
[42] Tokio Matsuyam. Quasilinear hyperbolic-hyperbolic singular perturbations with nonmonotone nonlinearity[J]. Nonlinear Analysis,1999,35:589-607.
[43] Grozdena Todorova. Stable and Unstable Sets for the Cauchy Problem for a Nonlinear Wave Equation with Nonlinear Damping and Source Terms[J]. Journal of Mathematical Analysis and Applications,1999,239: 213-226.
[44] Jorge Alfredo Esquivel-Avila. A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations[J]. Nonlinear Analysis,2003.52:1111-1127.
[45] 杜心华．一类非线性波动方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)，1994，17(4)：35-42．
[46] 杨晗．一类非线性热传导方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)，1995，18(1)：92-96
[47] 杨从军．一类拟线性退化抛物型方程的初边值问题[J]．数学学报，1995，38(1)：134-139．
[48] Hu Maolin. Global Solution for the Quasilinear Wave Equations with Viscosity[J]. Advances in Mathematics,2001,30(2): 123-132.
[49] 闫春华，刘若慧，张桂霞．具阻尼的Klein-Gordon方程组整体解的存在性[J]．天中学刊，2001，16(2)：7-8．
[50] 杨林．一类非线性波方程解的存在性[J]．云南大学学报(自然科学版)，2001，23(2)：95-99．
[51] 杨林，王晓兰．一类非线性波方程解的唯一性、光滑性[J]．云南大学学
报(自然科学版)，2001，23(3)：166-168．
[52] 谭忠．具有特殊扩散过程的反应扩散方程[J]．数学年刊，22A：5(2001)，597-606．
[53] 李庆霞．一类非线性双曲方程的局部解存在性[J]．数学研究，2002，35(2)：175-180．
[54] 李庆霞，谭忠．具有耗散和阻尼的Kirchhoff型方程的整体解的存在性[J]．厦门大学学报(自然科学版)，2002，41(4)：418-422．
[55] 呼青英，张宏伟．一类非线性双曲方程整体弱解的存在性与非存在性[J]．数学研究，2002，35(1)：72-78．
[56] 张宏伟，呼青英．具阻尼的Klein-Gordon方程组整体解的存在性、衰减性和爆破性[J]．数学理论与应用，2002，22(2)：34-38，77．
[57] 杨宏志，呼青英．一类非线性发展方程组整体解的存在性和不存在性[J]．数学的实践与认识，2002，32(4)：658-663．
[58] 杨宏志，张宏伟．具阻尼的非线性波动方程的稳定集与不稳定集[J]．数学的实践与认识，2002，32(5)：808-812．
[59] 张宏伟，呼青英．一类耦合非线性Klein-Gordon方程组的稳定集和不稳定集[J]．纯粹数学与应用数学，2002，18(3)：207-210．
[60] 王培林．具有Neumann边界及临界Sobolev指数的半线性抛物方程[J]．厦门大学学报(自然科学版)，2003，42(2)：144-147．
[61] 陈勇明．一类非线性波方程解的爆破[J]．重庆工学院学报，2003，6．
[62] R．A．Adams著；叶其孝等译．索伯列夫空问[M]．北京：人民教育出版社，1981．
[2] Banks H.T., Gilliam D.S.. Shubov V. I., Global solvability for damped abstract nonlinear hyperbolic systems. Diff. Integ. Equa. 1997,10:309-332.
[3] Love A. H.. A Treatise on the Mathematical Theory of Elaslicity. Dover, New York. 1964.
[4] Makhankov V.G...Dynamics of classical soliton. Phys. Rep. Rev. Section Phys. Lett. (Section C), 1978, 35C:1-128.
[5] Christiansen P.L., Lomdahl P.S, Muto V. On a Toda lattice model with a transversal degree of freedom. Nonlinearity, 1990, 4:477-501.
[6] Warnecke S. G.., Uber das homogene Dirichlet-Problem bei nichtlinearn partiellen Diffenrential gleichungen vom Typ der Boussinesq-Gleichung. Math. Meth. In the Appl. Sci., 1987, 9:493-519.
[7] Steven Paul Levandosky. Decay Estimates for Fourth-Order Wave Equatios[J]. Journal of Differential Equations,1998,143:360-413.
[8] Steven Levandosky. Stability and Instability of Fourth-Order Solitary Waves[J].Journalof Dynamics and Differential Equations,1998,10(1): 151-188.
[9] Juha Berkovits. On the bifurcation of large amplitude solutions for a system of wave and beam equations [J]. Nonlinear Analysis 2003,52:343-354.
[10] A.S.Ackleh,H.T.Banks and G.A.Pinter. A Nonlinear Beam Equation [J]. Applied Mathematics Letters,2002,15:381-387.
[11] K.Balachandran,J.Y.Park and I.H.Jung. Existence of Solutions of Nonlinear Extensible Beam Equation [J]. Mathematical and Computer Modelling,2002,36:747-754.
[12] Jia-Quan Liu. Free vibrations for an asymmetric beam equation [J]. Nonlinear Analysis,2002,51:487-497.
[13] Thomas Bartsch, Yanheng Ding. Periodic solutions of superlinear beam and membrane equations with perturbations from symmetry [J]. Nonlinear Analysis,2001,44:727-747.
[14] Alberto Aosio. A geometrical nonlinear correction to the Timoshenko beam equation [J]. Nonlinear Analysis,2001,47:729-740.
[15] S.A.Avdonin,N.G.Medhin,T.L.Sheronova. Identification of a piecewise constant coefficient in the beam equation [J]. Journal of Computational and Applied Mathematics,2000,114:11-21.
[16] S.M.Choo,S.K.Chung. Finite element Galerkin solutions for the nonplanar oscillatory beam equations [J]. Applied Mathematics and Computation,2000,114:279-301.
[17] S. M. Choo, S. K. Chung. Fintite difference approximate solutions for the strongly damped extensible beam equations [J]. Applied Mathematics and Computation, 2000, 112: 11-32.
[18] Tokio Matsuyama. Quasilinear hyperbolic-hyperbolic singular perturbations with nonmonotone nonlinearity[J]. Nonlinear Analysis, 1999, 35: 589-607.
[19] G. Perla Menzala. The Beam Equation as a Limit of a 1-D Nonlinear
Von Kármán Model [J]. Applied Mathematics Letters, 1999, 12:47-52
[20] S. M. Choo and S.K. Chung. L~2-Error Estimate for the Strongly Damped Extensible Beam Equations [J]. Appl. Math. Lett. 1998, 11 (6):101-107.
[21] Q-Heung Choi and Tachsun Jung. A Semilinear Beam Equation with Nonconstant Loads [J]. Nonlinear Aanlysis, Theory, Methods & Applications, 1997, 30 (8) :5515-5525.
[22] 马如云．一类四阶周期边值问题的可解性[J]．数学物理学报，1995，15(3)：315-318．
[23] 黄发伦，黄永忠．相应于具阻尼的Euler-Bernoulli梁方程的Co半群[J]．中国科学A辑，1992，OOA(2)：122-133．
[24] 丁彦恒．关于非线性梁方程的一点注记[J]．数学学报，1990，33(2)：172-181．
[25] 任瑞芳，荆付翠，武洁琼．变速度反馈梁振动发展系统的存在性[J]．山西大学学报(自然科学版)，2003，26(1)：9—12．
[26] 尹蝓，裴莲淑，金正国．非线性横梁方程的振动问题[J]．延边大学学报(自然科学版)，2001，27(1)：1-5．
[27] 唐秋林，耿建生，吴美云．一维非线性梁振动方程周期解的存在性[J]．南通工学院学报，2001，17(3)：51-53．
[28] 张宏伟．非线性高阶发展方程中的几个问题[D]．郑州大学，2002．
[29] J．L．Lions著．郭柏灵，汪礼衱译．非线性边值问题的一些解法[M]．广州：中山大学出版社，1992．
[30] 陈恕行，洪家兴 编著．偏微分方程的近代方法[M]．上海：复旦大学出版社，1988．
[31] 王耀东著．偏微分方程的L~2理论[M]．北京：北京大学出版社，1982．
[32] 李大潜，陈韵梅 著．非线性发展方程[M]．北京：科学出版社，1989．
[33] D.H.Sattinger. On Global Solution of Nonlinear Hyperbolic Equations[J]. Arch.Rat.Mech.Anal.,1968,30:148-172.
[34] L.E:Payne and D.H.Sattiner. Saddle Points and Unstabiliyt of Nonlinear Hyperboic Equatons[J]. Israel J.Math, 1975,22:273-303.
[35] M.Tsutsumi. Existence and Nonexistence of Global Solutions for Non-linearParabolicEqutions[J]. Publ.RIMS,KyotoUniv. 1972/73,8:211-229.
[36] H.A.Levine and R.A.Smith Ames. A Potential Well Theory for the Heat Equation with a Nonlinear Boundary Condition[J]. Math.Meth. in the Appl.Sci. 1987,9:127-136.
[37] Mitsuhiro Nakao,Kosuke Ono. Existence of global solutions to the Cauchy problem for the semi-linear dissipative wave equations[J]. Math.Z. 1993,214:325-342.
[38] Ryo Ikehata and Takashi Suzuki. Stable and unstable sets for evolution eqations of parabolic and hyperbolic type[J]. Hiroshima Math.J.,1996,26: 475-491.
[39] Ryo Ikehata. Some Remarks on the Wave Equations with Nonlinear Damping and Source Terms[J]. Nonlinear Analysis,Theory, Methods & Applications,1996,10(27):l165-1175.
[40] Kosuke Ono. Blowing up and Global Existence of Solutions for some Degenerate Nonlinear Wave Equations with Some Dissipation[J]. Nonlinear Analysis, Theory, Methods & Applications, 1997,7(30):4449-4457.
[41] Kosuke Ono. Global Existence,Decay, and Blowup of Solutions for Some
Mildy Degenerate Nonlinear Kirchhoff Strings[J]. Journal of Differential Equations, 1997,137:273-301.
[42] Tokio Matsuyam. Quasilinear hyperbolic-hyperbolic singular perturbations with nonmonotone nonlinearity[J]. Nonlinear Analysis,1999,35:589-607.
[43] Grozdena Todorova. Stable and Unstable Sets for the Cauchy Problem for a Nonlinear Wave Equation with Nonlinear Damping and Source Terms[J]. Journal of Mathematical Analysis and Applications,1999,239: 213-226.
[44] Jorge Alfredo Esquivel-Avila. A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations[J]. Nonlinear Analysis,2003.52:1111-1127.
[45] 杜心华．一类非线性波动方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)，1994，17(4)：35-42．
[46] 杨晗．一类非线性热传导方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)，1995，18(1)：92-96
[47] 杨从军．一类拟线性退化抛物型方程的初边值问题[J]．数学学报，1995，38(1)：134-139．
[48] Hu Maolin. Global Solution for the Quasilinear Wave Equations with Viscosity[J]. Advances in Mathematics,2001,30(2): 123-132.
[49] 闫春华，刘若慧，张桂霞．具阻尼的Klein-Gordon方程组整体解的存在性[J]．天中学刊，2001，16(2)：7-8．
[50] 杨林．一类非线性波方程解的存在性[J]．云南大学学报(自然科学版)，2001，23(2)：95-99．
[51] 杨林，王晓兰．一类非线性波方程解的唯一性、光滑性[J]．云南大学学
报(自然科学版)，2001，23(3)：166-168．
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