强阻尼非线性波动方程组解的研究
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摘要
本文运用了Potential Well方法研究了一类强阻尼波动方程组初边值问题在Rn整体解的存在性,定义了该方程组的位势深度d,运用庞加莱-索伯列夫嵌入定理证明了位势深度d>0,再通过构建适当的能量函数E(t),稳定集和不稳定集,当初始值属于不稳定集时,解在有限时间爆破和解的衰减估计.第一章介绍了预备知识.第二章讨论了一类强阻尼波动方程组初边值问题在Rn解的存在性.第三章研究了一类强阻尼波动方程组初边值问题在Rn解的爆破,第四章介绍了一类强阻尼波动方程组初边值问题在Rn解的一致衰减估计.
The global existence of solution for a class of strongly damped wave equations with initial-boundary value problem is studied using Potential Well method. The Potential depth of this problem are defined,and by using Poincare-Sobolev embedding theorem, it is proved that the potential depth is a positive number. The paper introductes the energy function,the stable and unstable sets,we prove that the existence of global solutions under some conditions.it is also show that if the initial data belongs to the unstable set,the solution is blowed up in finite time,the decay behavior of solution is disscussed.In chapter one,We introduce basic knowledge;In chapter two,We introduce the global existence of solution for the strongly damped wave equations with initial-boundary value problem is studied.In chapter three,We introduce the global blow up of solution for the strongly damped wave equations with initial-boundary value problem is studied.In chapter four,We introduce the decay behavior of solution for the strongly damped wave equations with initial-boundary value problem is studied.
The global existence of solution for a class of strongly damped wave equations with initial-boundary value problem is studied using Potential Well method. The Potential depth of this problem are defined,and by using Poincare-Sobolev embedding theorem, it is proved that the potential depth is a positive number. The paper introductes the energy function,the stable and unstable sets,we prove that the existence of global solutions under some conditions.it is also show that if the initial data belongs to the unstable set,the solution is blowed up in finite time,the decay behavior of solution is disscussed.In chapter one,We introduce basic knowledge;In chapter two,We introduce the global existence of solution for the strongly damped wave equations with initial-boundary value problem is studied.In chapter three,We introduce the global blow up of solution for the strongly damped wave equations with initial-boundary value problem is studied.In chapter four,We introduce the decay behavior of solution for the strongly damped wave equations with initial-boundary value problem is studied.
引文
[1]Webb G F.Existence and A symptotic Behavior for a Strongly A damped Nonlinear Wave Equation [J]. Cannd J Math,1980,32:631-643.
[2]Dang D A.On the Strong Damped Wave Equation [J]. SIAM J Math Anal, 1988,19:1409-1418.
[3]刘亚成,刘大成.方程utt-u-Aut=f(u)的初边值问题[J].华中理工大学学报,1988,16：169-173.
[4]刘亚成.任意维数的强阻尼非线性波动方程初边值问题[J].应用数学,1995,8(3)：262-266.
[5]Massatt P.Limiting Behavior for Strongly Damped Nonlinear Wave Equations [J].J DiffEgs,1983,48:334-349.
[6]扬志坚,陈国旺.一类四阶拟线性波动方程初边值问题的整体解[J].数学物理学报,1995,15(增刊)：1-9.
[7]H.A.Levine,Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=Au+F(u)[J],Trans. Amer. Math. Soc.1974. 192:1-21.
[8]Claudianor O.Alves.On existence, uniform decay rates and blow up for so-lutions of systems of nonlinear wave equations with damping and source terms[J], Discrete Contin.. Dyn. Syst.2009.3:585-608.
[9]Huang wenyi,Zhang jian.Global solutions and finite time blow up for wave equations with both strong and nonlinear damping terms [J], Math. Appl 2008.21(4):787-793.
[10]Sattinger D.H. On Global Solution Of Nonlinear Hyperbolic Equations.Arch .Rat.Mech.Anal.,1968.30:148-172
[11]Paul H.Rabinowitz.Minimax methods in critical point theory with applica-tions to differential equations.[M].published by the American Mathematical Society,1986
[12]J.W.Cholewa T.Dlotko. Global attractors in abstract parabolic prob-lems[M], London mathematical society.2000.
[13]R.A.A.dams,索伯列夫空间[M],人民教育出版社,1981.
[14]Roger Teman, Infinite-Dimensional Dynamical Systems In Mechanics And Physics.[M] New york:Springer,1977.
[15]Esquivel-Avila,Qualitative analysis of a nonlinear wave equation [J], Discrete Contin. Dyn. Syst.10(2004).787-804..
[16]K.Ramdani,T.Takahashi,and M.Tucsnak.Uniformly exponentially stable ap-proximations for a class of second order evolution equations-application to LQR problem[J], ESAIM Control Optim. Calc. Var.,13(3):503-527,2007.
[17]K. J.Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calculus of Variations,[J] 22(2005):483-494.
[18]Hiroshi Uesaka.Oscillation or nonosillation property for semilinear wave equations [J], Journal of Computational and Applied Mathematics,2004; 164-165; 23-730.
[19]Galaktionov V A,Pohoza,ev S I.Blow-up and critical exponents for nonlinear hyperbolic equations[J], Nonlinear Analysis,2003,53:453-466.
[20]P.Sintzoff. Symmetry of solutions of a semilinear elliptic equation with un-bounded coefficients, Diff. Int. Eqs.,16(2003):769-786.
[21]陈勇明,杨晗,一类具耗散项的非线性四阶波动方程解的爆破[J],四川大学学报(自然科学版),2008.45(3)：459-462.
[22]张建,工开端,相互作用波整体解的不存在性[J],四川师范大学学报(自然科学版),1994.17(3)：10-13.
[23]A.Munch and A.F.Pazoto.Uniform stabilization of a viscous numerical approx-imation for a locally damped wave eqution. ESAIM Control Optim[J], Calc. Var.,13(2):265-293(electronic),2007.
[24]K.Ramdani,T.Takahashi,and M.Tucsnak.Uniformly exponentially stable ap-proximations for a class of second order evolution equations-application to LQR problem[J], ESAIM Control Optim. Calc. Var.,13(3):503-527,2007..
[25]A.Ambrosetti, J.Garicia Azorero and I.Peral Alonso, Multiplicity results for some nonlinear elliptic equations, J. Functional Analysis,137(1996):219-242.
[26]L.R.Tcheugoue Tebou and E.Zuazua.Uniform boundary stabilization of the finite difference space discretization of the 1-d wave eqution[J], Adv. Comput. Math.,26(1-3):337-365,2007..
[27]R.Ikehata and T.Suzuki.Stable and unstable sets for evolution equations of parabolic and hyperbolic type.Hiroshima Math.J.,26:475-491,1996.
[28]S.Messaoudi and B.Said-Houari.Global non-existence of solutions of a class of wave equations with non-linear damping and source terms.Math.Methods Appl.Sci.,27:1687-1696,2004.
[29]Z.Yang.Existence and asymptotic behavior of solutions for a claas of quasilinear evolution equations with non-linear damping and source terms.Math.Meth.Appl.Sci.,25:795-814,2002.
[30]F.Gazzola and M.Squassina.Global solutions and finite time blow up for damped semilinear wave equations [J],Ann. I.H.Poincare.2006.23:185-207.
[2]Dang D A.On the Strong Damped Wave Equation [J]. SIAM J Math Anal, 1988,19:1409-1418.
[3]刘亚成,刘大成.方程utt-u-Aut=f(u)的初边值问题[J].华中理工大学学报,1988,16：169-173.
[4]刘亚成.任意维数的强阻尼非线性波动方程初边值问题[J].应用数学,1995,8(3)：262-266.
[5]Massatt P.Limiting Behavior for Strongly Damped Nonlinear Wave Equations [J].J DiffEgs,1983,48:334-349.
[6]扬志坚,陈国旺.一类四阶拟线性波动方程初边值问题的整体解[J].数学物理学报,1995,15(增刊)：1-9.
[7]H.A.Levine,Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=Au+F(u)[J],Trans. Amer. Math. Soc.1974. 192:1-21.
[8]Claudianor O.Alves.On existence, uniform decay rates and blow up for so-lutions of systems of nonlinear wave equations with damping and source terms[J], Discrete Contin.. Dyn. Syst.2009.3:585-608.
[9]Huang wenyi,Zhang jian.Global solutions and finite time blow up for wave equations with both strong and nonlinear damping terms [J], Math. Appl 2008.21(4):787-793.
[10]Sattinger D.H. On Global Solution Of Nonlinear Hyperbolic Equations.Arch .Rat.Mech.Anal.,1968.30:148-172
[11]Paul H.Rabinowitz.Minimax methods in critical point theory with applica-tions to differential equations.[M].published by the American Mathematical Society,1986
[12]J.W.Cholewa T.Dlotko. Global attractors in abstract parabolic prob-lems[M], London mathematical society.2000.
[13]R.A.A.dams,索伯列夫空间[M],人民教育出版社,1981.
[14]Roger Teman, Infinite-Dimensional Dynamical Systems In Mechanics And Physics.[M] New york:Springer,1977.
[15]Esquivel-Avila,Qualitative analysis of a nonlinear wave equation [J], Discrete Contin. Dyn. Syst.10(2004).787-804..
[16]K.Ramdani,T.Takahashi,and M.Tucsnak.Uniformly exponentially stable ap-proximations for a class of second order evolution equations-application to LQR problem[J], ESAIM Control Optim. Calc. Var.,13(3):503-527,2007.
[17]K. J.Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calculus of Variations,[J] 22(2005):483-494.
[18]Hiroshi Uesaka.Oscillation or nonosillation property for semilinear wave equations [J], Journal of Computational and Applied Mathematics,2004; 164-165; 23-730.
[19]Galaktionov V A,Pohoza,ev S I.Blow-up and critical exponents for nonlinear hyperbolic equations[J], Nonlinear Analysis,2003,53:453-466.
[20]P.Sintzoff. Symmetry of solutions of a semilinear elliptic equation with un-bounded coefficients, Diff. Int. Eqs.,16(2003):769-786.
[21]陈勇明,杨晗,一类具耗散项的非线性四阶波动方程解的爆破[J],四川大学学报(自然科学版),2008.45(3)：459-462.
[22]张建,工开端,相互作用波整体解的不存在性[J],四川师范大学学报(自然科学版),1994.17(3)：10-13.
[23]A.Munch and A.F.Pazoto.Uniform stabilization of a viscous numerical approx-imation for a locally damped wave eqution. ESAIM Control Optim[J], Calc. Var.,13(2):265-293(electronic),2007.
[24]K.Ramdani,T.Takahashi,and M.Tucsnak.Uniformly exponentially stable ap-proximations for a class of second order evolution equations-application to LQR problem[J], ESAIM Control Optim. Calc. Var.,13(3):503-527,2007..
[25]A.Ambrosetti, J.Garicia Azorero and I.Peral Alonso, Multiplicity results for some nonlinear elliptic equations, J. Functional Analysis,137(1996):219-242.
[26]L.R.Tcheugoue Tebou and E.Zuazua.Uniform boundary stabilization of the finite difference space discretization of the 1-d wave eqution[J], Adv. Comput. Math.,26(1-3):337-365,2007..
[27]R.Ikehata and T.Suzuki.Stable and unstable sets for evolution equations of parabolic and hyperbolic type.Hiroshima Math.J.,26:475-491,1996.
[28]S.Messaoudi and B.Said-Houari.Global non-existence of solutions of a class of wave equations with non-linear damping and source terms.Math.Methods Appl.Sci.,27:1687-1696,2004.
[29]Z.Yang.Existence and asymptotic behavior of solutions for a claas of quasilinear evolution equations with non-linear damping and source terms.Math.Meth.Appl.Sci.,25:795-814,2002.
[30]F.Gazzola and M.Squassina.Global solutions and finite time blow up for damped semilinear wave equations [J],Ann. I.H.Poincare.2006.23:185-207.