多维粘弹性方程解的存在性
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摘要
本文分别用Galerkin法和位势井法研究了以下多维粘弹性方程在正定能量下和非正定能量下的初边值问题整体解的存在性，其中Q表示区间(0，1)(对初边值问题及周期边界问题)或(-∞，∞)(对初值问题)。
同时，对正定能量下的周期边界问题及初值问题也进行了研究，并在σ(s)∈C~1，σ(s)下方有界的条件下，得到了整体强解的存在和唯一性，而在其初值函数满足一定的光滑性条件下，得到强解的相应光滑性，对非正定能量下我们得到了解的真空隔离现象。
In this paper, multidimensional viscoelasticity equation is considered by Galerkin method and potential well under positive energy and nonpositive energy .WhereΩbelongs to (0,1) if it is initial boundary walue problem and periodic boundary problem or (-∞,∞) if it is initial value problem. And the existence of global solutions are established.
At the same time, periodic boundary problem and initial value problem of equation under positive energy are studied .Under the conditions ofσ(s)∈C~1,σ'(s) bounded below, we prove the existence and uniqueness of the global strong solutions. While the inatial value function smooth appropriate, we obtain the smooth theorems of the global strong solutions. Behaviour of vacuum isolating of solutions is studied under nonpositive enegy.
同时，对正定能量下的周期边界问题及初值问题也进行了研究，并在σ(s)∈C~1，σ(s)下方有界的条件下，得到了整体强解的存在和唯一性，而在其初值函数满足一定的光滑性条件下，得到强解的相应光滑性，对非正定能量下我们得到了解的真空隔离现象。
In this paper, multidimensional viscoelasticity equation is considered by Galerkin method and potential well under positive energy and nonpositive energy .WhereΩbelongs to (0,1) if it is initial boundary walue problem and periodic boundary problem or (-∞,∞) if it is initial value problem. And the existence of global solutions are established.
At the same time, periodic boundary problem and initial value problem of equation under positive energy are studied .Under the conditions ofσ(s)∈C~1,σ'(s) bounded below, we prove the existence and uniqueness of the global strong solutions. While the inatial value function smooth appropriate, we obtain the smooth theorems of the global strong solutions. Behaviour of vacuum isolating of solutions is studied under nonpositive enegy.
引文
[1] Boling Guo, Zhaohui lluo. The global attractor of the damped forced Ostrovsky equation[J].Journal of Mathematical Analysis and Applications, 2007,329:392-407P
[2] Wang Baoxiang,Guo Boling and Zhao Lifeng. The global well-posedness and spatial decay of solutions t'or the derivative complex Ginzburg-Landau equation in lll[J].Nonlinear Analysis,2004,57:1059-1076P
[3] Boling Guo, Rong Yuan. The Time-Periodic Solution to a 2D Generalized Oinzburg-Landau Equation[J].Journal of Mathematical Analysis and Applications, 2OO2, 266:186-199P
[4] Yongsheng Li, Boling Guo. Global Existence of Solutions to the Derivative 2D Ginzburg -Landau Equation[J].Journal of Mathematical Analysis and Applications, 2000,249:12-432P
[5] Boling Guo, Peicheng Zhu. Asymptotic Behavior of the Solution to the System for a Viscous Reactive Gas[J].Journal of Differential Equations, 1999,155:177-202P
[6] Guo Boling, Su Fengqiu. Global Weak Solution for the Landau-Lifshitz -Maxmell Equation in Three Space Dimensions[J]. Journal of Mathematical Analysis and Applications, 1997,211:326-346P
[7] Guo Boling, Li Yongsheng. Attrctor for Dissipative Klein-Gordon- Schrodinger Equations in R3[J].Journal of Differential Eouations. 1997.136:356-377P
[8] 郑镇汉，姚正安．系有集中质量的均匀与非均匀弹性杆的振动问题[J]．华中师范大学学报．2006(4)：531页，524-526页
[9] 赵红星，姚正安．具有超薄厚度的多孔介质中Stokes流的均匀化[J]．中山大学学报．2006(3)：45页，5-8页
[10] YaoZheng-an, Zhou Wenshu. AClassofSingular Nonlinear Diffusion Problem[J].Northeastern Mathematical Journai, 2006(1),22:1-4P
[11] Yao zheng-an, Zhou wenshu, Research Announcements On a Singular Diffusion Equation in Nondivergence Form[J]. Advances in Mathematics. 2005.34:503-505P
[12] 谭忠，姚正安．半线性椭圆方程的多解存在性[J]．数学进展．2002，31：343-354页
[13] 姚正安．广义双尺度收敛和具分层周期结构系数的偏微分方程的均匀化[J]．应用泛函分析学报．2000，4：10-19页
[14] JinshunZhang, Yongtang WuandXuemei Li. Quasi-periodic solution of the (2+1)-dimensional Boussinesq-Burgers soliton equation[J].Physica A: Statistical Mechanics and its Applications, 2003,319:213-232P
[15] YangZhijian. On local existence of solutions of initial boundary value problems for the "bad" Boussinesq-type equation[J].Nonlinear Analysis, 2002, 51:1259-1271P
[16] Zhijian Yang, Xia Wang. Blowup of solutions for the "bad" Boussinesq-type equation[J].Journal of Mathematical Analysis and Applications, 2003,285:282-298P
[17] Zhijian Yang, Xia Wang. Blow up of solutions for improved Boussinesq type equation[J].Journal of Mathematical Analysis and Applications, 2003,278:335-353P
[18] Zhijian Yang. Cauchy problem for quasi-linear wave equations with viscous damping[J].Journal of athematical Analysis and Applications, 2006,320:859-881P
[19] Yang Zhi jian. Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow[J]. Journal of Mathematical Analysis and Applications, 2006,313:197-217P
[20] Yang Zhi jian. Viscous solutions on some nonlinear wave equations [J]..Nonlinear Anlysis, 2005,63:e2607-e2619P
[21] Zhi jian Yang. Cauchy problem for quasi-linear wave equations withnonlinear damping and source terms [J]. Journal of Mathematical Analysis and Applications, 2004,300:218-243P
[22] Zhijian Yang,Guowang Chen.(Global existence of solutions for quasi-linear wave equations with viscous damping[J]. Journal of Mathematical Analysis and Applications, 2003,285:604-618P
[23] Yang Zhi jian.Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term[j]. Journal of Differential Equations, 2003, 187 : 520-540P
[24] Yang Zhi jian, Song Chang ming. Blow up of solutions for a class of quasilinear evolution equations[J]. Nonlinear Analysis, 1997, 28: 2017-2032P
[25] lshii,ll..Asymptotic stability and blowing up of solutions of some nonl inear equations[J]. Jour. Diff. Eqs. 1977, 26:291-319P
[26] Levine II.A..Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu_1=A(u)+ F(u)[J]. Arch. Rational. mech. Anal, 1973, 51:371-386P
[27] Sattinger, D. ll..On global solution of nonlinear hyperbolic equations [J]. Arch. Rat. Mech. Anal., 1968, 30:148-172P
[28] Tsutsumi, M., Existence and non-existence of global solutions for nonlinear equation[J], subl. Res. Inst. Math. Sci., 1972, 8:211-229P
[29]Mi Lsuhiro Nakao，Kosuke Ono．Existence of Global solutins to the Cauchy problem for the semi-linear dissipative wave equations[J]．Math．Z．，1993，214：325-342P
[30]Hu Maolin．G10bal S0luLion for the Quasilinear Wave Equatiotis with Viscosity[j]．Advances in Mathematics，2001，30(2)：123-132P
[31]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]．数学的实践与认以．2002，32(5)：808-812页
[32]张宏伟，呼青英．一类偶合非线性Klein-Gordon方程组的稳定集与不稳定集[J]．纯粹数学与应用数学．2002，18(3)：207-210页
[33]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页
[34]陈勇明.一类非线性波动方程解的爆破[J]．重庆工学院学报．2003，6
[35]R.A．Adams著：叶其孝等译．索伯列夫空间[M]．北京：人民教育出版社，1981
[36]王凡彬．广义神经传播型非线性拟双曲方程解的爆破和熄[J]．应用数学和力学．1996，7(11)：1039-1043页
[37]宋长明，任华国，高桂芳．一类非线性波动方程解的爆破问[J]．河南科学．2002，20(2)：111-113页
[38]查中伟．非线性发展方程混合问题解的Blow-up[J]．三峡大学学报．2002，24(3)：276-279页
[39]强晓风．两类双曲型方程解的爆破性质[J]．广东职业技术师范学院学报．2000(1)：16-19页
[40]张宏伟，陈国旺.一类非线性四阶波动方程的位势井方法[J]．数学物理学报．2003，23A(6)：758-768页
[41]刘亚成．半线性热方程整体解的存在性与非存在性[J]．数学年刊。1997，18A(1)：5-72页
[42]Nakao,M., L〃-estimates of solutions of some nonlinear degenerate diffusion equations, J. Math. Soc. Japan, 1985, 37(1) :23-30P
[43] Ryo lkehata and Takashi Suzuki. Stable and unstable sets for evolution equations of parabolic and hyperbolic type[J]. Iliroshima Math..J., 1996, 26:475-491P
[44] 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, 239:213-226P
[45] Jorge Alfredo Fsquivel-Avila.A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations [J]. Nonlinear Analysis, 2003, 52:1111-1127P
[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[J]．四川师范大学学报(自然科学版)．1995，18(1)：92-96页
[47] LiuYacheng. On potential wells and vacuum isolating of solutions for semilinear wave equations,Journal of Differential Equations, 2003, 192(1), 155-169P
[48] Liu Yacheng, Zhao Junsheng. Multidimensional viscoelasticity equations with nonlinear damping and source terms, Nonlinear Analysis, 2004, 56:851-865P
[49] Liu Yacheng, Zhao Junsheng. Nonlinear parabolic equations with critical initial conditions or Nonlinear Analysis, 2004, 58: 873-883P
[50] Liu Yacheng, Zhao.lunsheng. On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Analysis, 2006, In Press.
[51] J.A. Esquivel-Avila. The dynamics of a nonlinear wave equation, J. Math. Anal. Appl. 2003,279:135-150P
[52]杨林，王晓兰．一类非线性波动方程解的唯一性、光滑性[J]．云南大学学报(自然科学版)．2001，23(3)：166-168页
[53]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页
[54]杜心华．一类非线性波动方程混合问题整体解的存在唯一[J]．四川师范大学报(自然科学版)．1994，17(4)：35-42页
[55]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页
[56]闫春华，刘若慧，张杜霞．具阻尼的Klcin-Gordon方程组的整体解的存在性[J]．天中学刊．2001，16(2)：7-8页
[57]杨林．一类非线性波动方程解的存在性[J]．云南大学学报(自然科学版)．2001，23(2)：95-99页
[58]Liu Yacheng, On potential wells and vacuum isolating of solutions for semilinear wave equations, [J]. Jour. Diff. Eqs., 2003, 192(1) : 155-169P
[59]刘亚成，刘萍．关于位势井及其对强阻尼非线性波动方程的应用[J]．应用数学学报．2004，27(4)：133-146页
[60]Lions,.J.L.,Quelques methods de resolution des problemes aus limites non lineaires, Dunod, Pairs 1969
[61]Pao, C.V.,Nonlinear parabolic and elliptic equations, Plenum Press, New York and London, in 1992.
[62]Tokio Matsuyam.Quasilinear hyperbolic-hyperbolic singular perturbations with nonmonotone nonlinearity[J].Nonlinear Analysis,1999,35:589-607P
[2] Wang Baoxiang,Guo Boling and Zhao Lifeng. The global well-posedness and spatial decay of solutions t'or the derivative complex Ginzburg-Landau equation in lll[J].Nonlinear Analysis,2004,57:1059-1076P
[3] Boling Guo, Rong Yuan. The Time-Periodic Solution to a 2D Generalized Oinzburg-Landau Equation[J].Journal of Mathematical Analysis and Applications, 2OO2, 266:186-199P
[4] Yongsheng Li, Boling Guo. Global Existence of Solutions to the Derivative 2D Ginzburg -Landau Equation[J].Journal of Mathematical Analysis and Applications, 2000,249:12-432P
[5] Boling Guo, Peicheng Zhu. Asymptotic Behavior of the Solution to the System for a Viscous Reactive Gas[J].Journal of Differential Equations, 1999,155:177-202P
[6] Guo Boling, Su Fengqiu. Global Weak Solution for the Landau-Lifshitz -Maxmell Equation in Three Space Dimensions[J]. Journal of Mathematical Analysis and Applications, 1997,211:326-346P
[7] Guo Boling, Li Yongsheng. Attrctor for Dissipative Klein-Gordon- Schrodinger Equations in R3[J].Journal of Differential Eouations. 1997.136:356-377P
[8] 郑镇汉，姚正安．系有集中质量的均匀与非均匀弹性杆的振动问题[J]．华中师范大学学报．2006(4)：531页，524-526页
[9] 赵红星，姚正安．具有超薄厚度的多孔介质中Stokes流的均匀化[J]．中山大学学报．2006(3)：45页，5-8页
[10] YaoZheng-an, Zhou Wenshu. AClassofSingular Nonlinear Diffusion Problem[J].Northeastern Mathematical Journai, 2006(1),22:1-4P
[11] Yao zheng-an, Zhou wenshu, Research Announcements On a Singular Diffusion Equation in Nondivergence Form[J]. Advances in Mathematics. 2005.34:503-505P
[12] 谭忠，姚正安．半线性椭圆方程的多解存在性[J]．数学进展．2002，31：343-354页
[13] 姚正安．广义双尺度收敛和具分层周期结构系数的偏微分方程的均匀化[J]．应用泛函分析学报．2000，4：10-19页
[14] JinshunZhang, Yongtang WuandXuemei Li. Quasi-periodic solution of the (2+1)-dimensional Boussinesq-Burgers soliton equation[J].Physica A: Statistical Mechanics and its Applications, 2003,319:213-232P
[15] YangZhijian. On local existence of solutions of initial boundary value problems for the "bad" Boussinesq-type equation[J].Nonlinear Analysis, 2002, 51:1259-1271P
[16] Zhijian Yang, Xia Wang. Blowup of solutions for the "bad" Boussinesq-type equation[J].Journal of Mathematical Analysis and Applications, 2003,285:282-298P
[17] Zhijian Yang, Xia Wang. Blow up of solutions for improved Boussinesq type equation[J].Journal of Mathematical Analysis and Applications, 2003,278:335-353P
[18] Zhijian Yang. Cauchy problem for quasi-linear wave equations with viscous damping[J].Journal of athematical Analysis and Applications, 2006,320:859-881P
[19] Yang Zhi jian. Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow[J]. Journal of Mathematical Analysis and Applications, 2006,313:197-217P
[20] Yang Zhi jian. Viscous solutions on some nonlinear wave equations [J]..Nonlinear Anlysis, 2005,63:e2607-e2619P
[21] Zhi jian Yang. Cauchy problem for quasi-linear wave equations withnonlinear damping and source terms [J]. Journal of Mathematical Analysis and Applications, 2004,300:218-243P
[22] Zhijian Yang,Guowang Chen.(Global existence of solutions for quasi-linear wave equations with viscous damping[J]. Journal of Mathematical Analysis and Applications, 2003,285:604-618P
[23] Yang Zhi jian.Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term[j]. Journal of Differential Equations, 2003, 187 : 520-540P
[24] Yang Zhi jian, Song Chang ming. Blow up of solutions for a class of quasilinear evolution equations[J]. Nonlinear Analysis, 1997, 28: 2017-2032P
[25] lshii,ll..Asymptotic stability and blowing up of solutions of some nonl inear equations[J]. Jour. Diff. Eqs. 1977, 26:291-319P
[26] Levine II.A..Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu_1=A(u)+ F(u)[J]. Arch. Rational. mech. Anal, 1973, 51:371-386P
[27] Sattinger, D. ll..On global solution of nonlinear hyperbolic equations [J]. Arch. Rat. Mech. Anal., 1968, 30:148-172P
[28] Tsutsumi, M., Existence and non-existence of global solutions for nonlinear equation[J], subl. Res. Inst. Math. Sci., 1972, 8:211-229P
[29]Mi Lsuhiro Nakao，Kosuke Ono．Existence of Global solutins to the Cauchy problem for the semi-linear dissipative wave equations[J]．Math．Z．，1993，214：325-342P
[30]Hu Maolin．G10bal S0luLion for the Quasilinear Wave Equatiotis with Viscosity[j]．Advances in Mathematics，2001，30(2)：123-132P
[31]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]．数学的实践与认以．2002，32(5)：808-812页
[32]张宏伟，呼青英．一类偶合非线性Klein-Gordon方程组的稳定集与不稳定集[J]．纯粹数学与应用数学．2002，18(3)：207-210页
[33]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页
[34]陈勇明.一类非线性波动方程解的爆破[J]．重庆工学院学报．2003，6
[35]R.A．Adams著：叶其孝等译．索伯列夫空间[M]．北京：人民教育出版社，1981
[36]王凡彬．广义神经传播型非线性拟双曲方程解的爆破和熄[J]．应用数学和力学．1996，7(11)：1039-1043页
[37]宋长明，任华国，高桂芳．一类非线性波动方程解的爆破问[J]．河南科学．2002，20(2)：111-113页
[38]查中伟．非线性发展方程混合问题解的Blow-up[J]．三峡大学学报．2002，24(3)：276-279页
[39]强晓风．两类双曲型方程解的爆破性质[J]．广东职业技术师范学院学报．2000(1)：16-19页
[40]张宏伟，陈国旺.一类非线性四阶波动方程的位势井方法[J]．数学物理学报．2003，23A(6)：758-768页
[41]刘亚成．半线性热方程整体解的存在性与非存在性[J]．数学年刊。1997，18A(1)：5-72页
[42]Nakao,M., L〃-estimates of solutions of some nonlinear degenerate diffusion equations, J. Math. Soc. Japan, 1985, 37(1) :23-30P
[43] Ryo lkehata and Takashi Suzuki. Stable and unstable sets for evolution equations of parabolic and hyperbolic type[J]. Iliroshima Math..J., 1996, 26:475-491P
[44] 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, 239:213-226P
[45] Jorge Alfredo Fsquivel-Avila.A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations [J]. Nonlinear Analysis, 2003, 52:1111-1127P
[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[J]．四川师范大学学报(自然科学版)．1995，18(1)：92-96页
[47] LiuYacheng. On potential wells and vacuum isolating of solutions for semilinear wave equations,Journal of Differential Equations, 2003, 192(1), 155-169P
[48] Liu Yacheng, Zhao Junsheng. Multidimensional viscoelasticity equations with nonlinear damping and source terms, Nonlinear Analysis, 2004, 56:851-865P
[49] Liu Yacheng, Zhao Junsheng. Nonlinear parabolic equations with critical initial conditions or Nonlinear Analysis, 2004, 58: 873-883P
[50] Liu Yacheng, Zhao.lunsheng. On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Analysis, 2006, In Press.
[51] J.A. Esquivel-Avila. The dynamics of a nonlinear wave equation, J. Math. Anal. Appl. 2003,279:135-150P
[52]杨林，王晓兰．一类非线性波动方程解的唯一性、光滑性[J]．云南大学学报(自然科学版)．2001，23(3)：166-168页
[53]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页
[54]杜心华．一类非线性波动方程混合问题整体解的存在唯一[J]．四川师范大学报(自然科学版)．1994，17(4)：35-42页
[55]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页
[56]闫春华，刘若慧，张杜霞．具阻尼的Klcin-Gordon方程组的整体解的存在性[J]．天中学刊．2001，16(2)：7-8页
[57]杨林．一类非线性波动方程解的存在性[J]．云南大学学报(自然科学版)．2001，23(2)：95-99页
[58]Liu Yacheng, On potential wells and vacuum isolating of solutions for semilinear wave equations, [J]. Jour. Diff. Eqs., 2003, 192(1) : 155-169P
[59]刘亚成，刘萍．关于位势井及其对强阻尼非线性波动方程的应用[J]．应用数学学报．2004，27(4)：133-146页
[60]Lions,.J.L.,Quelques methods de resolution des problemes aus limites non lineaires, Dunod, Pairs 1969
[61]Pao, C.V.,Nonlinear parabolic and elliptic equations, Plenum Press, New York and London, in 1992.
[62]Tokio Matsuyam.Quasilinear hyperbolic-hyperbolic singular perturbations with nonmonotone nonlinearity[J].Nonlinear Analysis,1999,35:589-607P