关于位势井族及其对强阻尼非线性波动方程的应用
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摘要
本文研究以下强阻尼非线性波动方程的初边值问题其中Ω(?)R~n为有界域。f∈C，且f(u)u≥0。
首先，利用新定义的位势井族结合Galerkin方法对整体弱解的存在性进行研究，得到了新的弱解的存在条件。其次，利用一些重要的不等式如Holder不等式，Gronwall不等式等结合位势井族进一步对强解的存在性进行研究，得到了新的整体强解的存在条件及强解的唯一性。再次，在解的存在性基础上研究问题(1)-(3)在该族位势井的流之下的不变性，得到了解的真空隔离性质，即方程的所有解均在H_0~1(Ω)空间的一个小球的内部或一个大球的外部出现，而不会在中间的带形区域出现，形成一个无解区域称为真空隔离区域。最后，利用积分估计的方法研究了该问题解的渐近性质，并得到了较好的结果，使得解以指数形式趋于零。
This paper deals with the initial boundary value problem of a class of strongly damped wave equationsWhere,Ω(?)R~n is a bounded domain and f∈C with f(u)u≥0.
The existence of the weak solutions of the above problem is discussed by combining the Galerkin method and the family of new potential wells that defined in this paper, and the new existence terms are obtained. Besides, by utilyzing some important inequalities such as Holder inequality, Gronwall inequalities and the family of potetial wells, the existence of the strong solution of the problem is analyzed, and the new existence terms and uniqueness are gained then. Then, the invariance of the family of potential wells under the flow of (l)-(3) is reaserched, vacuum isolating property of the solutions is gained, that means all solutions of the equations may appear in the inside of a small ball or outside of a large ball, rather than the band-shape intervenient regine. Thus, a non-solution region called vacuum isolating region is formed. Finally, the asymptotic behaviors for solutions of the problem were studied by using integral estimate method. The rezults indicate that the solutions of the problems decay to zero according to the exponent of t.
首先，利用新定义的位势井族结合Galerkin方法对整体弱解的存在性进行研究，得到了新的弱解的存在条件。其次，利用一些重要的不等式如Holder不等式，Gronwall不等式等结合位势井族进一步对强解的存在性进行研究，得到了新的整体强解的存在条件及强解的唯一性。再次，在解的存在性基础上研究问题(1)-(3)在该族位势井的流之下的不变性，得到了解的真空隔离性质，即方程的所有解均在H_0~1(Ω)空间的一个小球的内部或一个大球的外部出现，而不会在中间的带形区域出现，形成一个无解区域称为真空隔离区域。最后，利用积分估计的方法研究了该问题解的渐近性质，并得到了较好的结果，使得解以指数形式趋于零。
This paper deals with the initial boundary value problem of a class of strongly damped wave equationsWhere,Ω(?)R~n is a bounded domain and f∈C with f(u)u≥0.
The existence of the weak solutions of the above problem is discussed by combining the Galerkin method and the family of new potential wells that defined in this paper, and the new existence terms are obtained. Besides, by utilyzing some important inequalities such as Holder inequality, Gronwall inequalities and the family of potetial wells, the existence of the strong solution of the problem is analyzed, and the new existence terms and uniqueness are gained then. Then, the invariance of the family of potential wells under the flow of (l)-(3) is reaserched, vacuum isolating property of the solutions is gained, that means all solutions of the equations may appear in the inside of a small ball or outside of a large ball, rather than the band-shape intervenient regine. Thus, a non-solution region called vacuum isolating region is formed. Finally, the asymptotic behaviors for solutions of the problem were studied by using integral estimate method. The rezults indicate that the solutions of the problems decay to zero according to the exponent of t.
引文
[1] Webb G F. Existence and Asymptotic Behaviour for a Strongly Damped Nonlinear Wave Equation. Canad.J.Math, 1980, 32: 631-643P
[2] 刘亚成，刘大成．方程u_(11)-α△u_1-△u=f(u)(α＞0)的初边值问题．华中理工大学学报．1988，6(6)：69-73页
[3] 刘亚成，王锋，刘大成．任意维数的强阻尼非线性波动方程(Ⅰ)-初边值问题．应用数学．1995，8(3)：262-266页
[4] G.Q. Xie and Y.M. Chen. A modified pulse-spectrum technique for solving inverse problems of two-dimensional elastic wave equation. Applied Numerical Mathematics, 1985(1.3):217-229P
[5] Dang Dinghai. On a wave equation with nonlinear strong damping. Nonlinear Analysis, 1990(15.11): 1005-1015P
[6] Peter W. Bates and Alfonso Castro. Existence and uniqueness for a variational hyperbolic system without resonance. Nonlinear Analysis, 1980, 9(4.6):1151-1156P
[7] Chen Caisheng. Weak solution for a fourth-order nonlinear wave equation. Journal of Southeast University, 2005(20):369-374P
[8] Sattinger. D H. On global solution of nonlinear hyperbolic equations. Arch. Rat. Mech.Anal, 1968(30):148-172P
[9] L.E.Payne and D.H.Sattinger. Saddle Points and Unstability of Nonlinear Hyperbolic Equations.Israel.J.Math, 1975,22:273-303P
[10] M.Tsutsumi. Existence and Non-existence of Global Solutions for Non-linear parabolic Equations. Publ.RIMS. Kyoto. Univ, 1972(73.8): 211-229P
[11] Ryo Ikehata and Takashi Suzuki. Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math.J, 1996(26): 475-491P
[12] Mitsuhiro Nakao, Kosuke Ono. Existence of Global solutions to the Cauchy problem for the semi-linear dissipative wave equations. Math.Z, 1993(214):325-342P
[13] Grozdena Todorova. Stable and Unstable Sets for the Cauchy Problem for a Nonlinear Wave Equation with Nonlinear Damping and Source Terms. Journal of Mathematical Analysis and Applica- tions, 1999(239):213-226P
[14] Jorge Alfredo Esquivel-Avila. A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations. Nonlinear Analysis, 2003 (52):1111-1127P
[15] 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．四川师范大学学报(自然科学版)．1995，18(1)：92-96页
[16] 张宏伟，呼青英．一类耦合非线性Klein-Gordon方程组的稳定集与不稳定集．纯粹数学与应用数学．2002,18(3)：207-210页
[17] 陈勇明，张正萍．一类非线性波动方程解的爆破．重庆工学院学报．2003，12(17.6)：38-40页
[18] Fujita H. On the blowing up of solutions of the Cauchy problem for u_1=△u+△u~(1+α). J.Fac.Sci.,Univ.Tokyo Sect.L, 1996 (13): 109-124P
[19] Levine H A. The role of critical exponents in blow-up theorems. SIAM Rew,1990 (32): 262-288P
[20] Souplet Ph. Blow-up in non-local reaction-diffusion equations. SIAM J.Math.Anal, 1998(29):1301-1334P
[21] 张宏伟，陈国旺．一类非线性四阶波动方程的位势井方法．数学物理学报．2003,23A(6)：758-768页
[22] Garcia Azorero.J.E and Peral Alonso L. Hardy inequalities and some critical elliptic and parabolic problems. J. of Diff.Equs, 1990 (144):441-476P
[23] Nakao M. L~p -estimates of solutions of some nonlinear degenerate diffusion equations. J.Math.Soc.Japan, 1985(37.1):23-30P
[24] 蒋良军．高维强阻尼非线性波动方程整体解的渐近理论．四川师范大学学报．2004，11(27)：592-596页
[25] 尚亚东．一类强阻尼非线性波动方程解的blow-up．烟台大学学报．2002．(15)：90-96页
[26] 孙淑珍．四阶强阻尼非线性波动方程的整体W~(k，p)解．吉林大学自然科学学报．2000，1(1)：18-22页
[27] 尚亚东．两类强阻尼非线性波动方程解的blow-up．工程数学学报．2000，5(17.2)：65-70页
[28] Ryo Ikehata．Global existence of solutions for semilinear damped wave equation in 2-D exterior domain．J．Differential Equation,2004(200):53-68P
[29] Zhijian Yang and Guowang Chen．Global existence of solutions for quasi-linear wave equations with viscous damping．J．Math．Anal. Appl， 2003(285):604-618P
[30] Fang Shao-mei,Guo Boling. Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations. Applied Mathematics and Mechanics,2003,7(24.6):673-683P
[31] 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．应用数学学报．2004(27.4)：133-146页
[32] 刘亚成，徐润章，刘萍．关于位势井及其对u_(11)-△u-△u_1-△u_(11)=f(u)的应用．(待发表)
[33] Liu Yacheng. On potential wells and vacuum isolating of solutions of Nonlinear Parabolic Equations. Publ.RTMS, 1972/73 (8): 211-229P
[34] 王元明，徐军祥编著．索伯列夫空间讲义．南京：东南大学出版社，2003．8
[35] R．A．Adams著；叶其孝等译．索伯列夫空间．北京：人民教育出版社，1981
[36] J．L．Lions著．郭柏灵，汪礼扔 译．非线性边值问题的一些解法．广州：中山大学出版社，1992
[37] 陈恕行，洪家兴编著．偏微分方程的近代方法．上海：复旦大学出版社，1988
[38] 李大潜，陈韵梅著．非线性发展方程。北京：科学出版社，1989
[39] 陈宁．人口问题中具广义Ginzbur-Landau型方程解的渐进性和Blow-up．生物数学学报．2005(20.3)：307-314页
[40] Steven Paul Levandosky. Decay Estimates for Fourth-Order Wave Equations. Journal of Differential Equations, 1998 (143): 360- 413P
[41] Van Horssen W T. An asymptotic theory for a class of initial boundary value problems for weakly nonlinear wave equations with an application to a model of the galloping cscillation of overhead transmission lines.SIAM.J.Appl.Math, 1988, (48.6) : 1243 -1277P
[42] Lai S Y. The asymptotic theory of solutions for a perturbed telegraph wave equation and its application. Appl. Math Mech, 1997, (7) : 656-662P
[43] Blota C.J,Van Der Burgh. Validity of approximations for time periodic solutions of a forced nonlinear hyperbolic differential equation. Appl Anal,1994,52 (1-4): 155-176P
[44] Asskura F. Existence of a global solution to a semilinear wave equation with slowly decreasing initial data in three space dimensions. Comm P D E, 1986,13 (11) :1459-1487P
[45] Lai S Y, Mu C L. The asymptotic theory of initial value problems for semilinear wave equations in three space dimensions. Appl Math JCU, 1997 (12B): 321-332P
[46] 杨从军．一类拟线性退化抛物型方程的初边值问题．数学学报．1995，38(1)：134-139页
[47] Hu Maolin. Global solution for the Quasilinear Wave Equations with Viscosity. Advances in Mathematics, 2001,30 (2): 123-132P
[48] 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．厦门大学学报(自然科学版)．2002，41(4)：418-422页
[49] 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．数学的实践与认识．2002，32(4)：658-663页
[50] Fila M. Boundedness of global solutions of nonlinear diffusion equations. J.of Diff.Equs, 1992 (98) : 226-240P
[51] Aguilar J.A.and Peral I. On some nonlinear parabolic equations, preprint U.A.M.1997
[52] G.Boling. Initial boundary value problem for one class of system of multidimensional inhomogeneous GBBM equation. J.Chin Ann of Math. 1987 (B8) :266-238P
[53] S.Yadong. Blow-up of solutions of initial boundary problem for a class of evolution equations with nonlinear strong dissipipation term. Journal of Xinyang Teachers College (Natural Science Edition) , 2002,15 (2) : 136-140P
[54] J.M.Ball. Remarks on blow-up and nonexistence theorems for non-linear evolution equations. Quart.J.Math. 1977 (28): 473-486P
[55] 王耀东 著．偏微分方程的L~2理论．北京：北京大学出版社，1982
[56] 郭柏灵，庞小峰．孤立子．北京：北京科学出版社，1978
[57] Steven Levandosky. Stablility and Instability of Fourth-Order Solitary Waves. Journal of Dynamics and Differential Equations, 1998,10(1): 151-188P
[58] Mitsuhiro 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
[2] 刘亚成，刘大成．方程u_(11)-α△u_1-△u=f(u)(α＞0)的初边值问题．华中理工大学学报．1988，6(6)：69-73页
[3] 刘亚成，王锋，刘大成．任意维数的强阻尼非线性波动方程(Ⅰ)-初边值问题．应用数学．1995，8(3)：262-266页
[4] G.Q. Xie and Y.M. Chen. A modified pulse-spectrum technique for solving inverse problems of two-dimensional elastic wave equation. Applied Numerical Mathematics, 1985(1.3):217-229P
[5] Dang Dinghai. On a wave equation with nonlinear strong damping. Nonlinear Analysis, 1990(15.11): 1005-1015P
[6] Peter W. Bates and Alfonso Castro. Existence and uniqueness for a variational hyperbolic system without resonance. Nonlinear Analysis, 1980, 9(4.6):1151-1156P
[7] Chen Caisheng. Weak solution for a fourth-order nonlinear wave equation. Journal of Southeast University, 2005(20):369-374P
[8] Sattinger. D H. On global solution of nonlinear hyperbolic equations. Arch. Rat. Mech.Anal, 1968(30):148-172P
[9] L.E.Payne and D.H.Sattinger. Saddle Points and Unstability of Nonlinear Hyperbolic Equations.Israel.J.Math, 1975,22:273-303P
[10] M.Tsutsumi. Existence and Non-existence of Global Solutions for Non-linear parabolic Equations. Publ.RIMS. Kyoto. Univ, 1972(73.8): 211-229P
[11] Ryo Ikehata and Takashi Suzuki. Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math.J, 1996(26): 475-491P
[12] Mitsuhiro Nakao, Kosuke Ono. Existence of Global solutions to the Cauchy problem for the semi-linear dissipative wave equations. Math.Z, 1993(214):325-342P
[13] Grozdena Todorova. Stable and Unstable Sets for the Cauchy Problem for a Nonlinear Wave Equation with Nonlinear Damping and Source Terms. Journal of Mathematical Analysis and Applica- tions, 1999(239):213-226P
[14] Jorge Alfredo Esquivel-Avila. A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations. Nonlinear Analysis, 2003 (52):1111-1127P
[15] 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．四川师范大学学报(自然科学版)．1995，18(1)：92-96页
[16] 张宏伟，呼青英．一类耦合非线性Klein-Gordon方程组的稳定集与不稳定集．纯粹数学与应用数学．2002,18(3)：207-210页
[17] 陈勇明，张正萍．一类非线性波动方程解的爆破．重庆工学院学报．2003，12(17.6)：38-40页
[18] Fujita H. On the blowing up of solutions of the Cauchy problem for u_1=△u+△u~(1+α). J.Fac.Sci.,Univ.Tokyo Sect.L, 1996 (13): 109-124P
[19] Levine H A. The role of critical exponents in blow-up theorems. SIAM Rew,1990 (32): 262-288P
[20] Souplet Ph. Blow-up in non-local reaction-diffusion equations. SIAM J.Math.Anal, 1998(29):1301-1334P
[21] 张宏伟，陈国旺．一类非线性四阶波动方程的位势井方法．数学物理学报．2003,23A(6)：758-768页
[22] Garcia Azorero.J.E and Peral Alonso L. Hardy inequalities and some critical elliptic and parabolic problems. J. of Diff.Equs, 1990 (144):441-476P
[23] Nakao M. L~p -estimates of solutions of some nonlinear degenerate diffusion equations. J.Math.Soc.Japan, 1985(37.1):23-30P
[24] 蒋良军．高维强阻尼非线性波动方程整体解的渐近理论．四川师范大学学报．2004，11(27)：592-596页
[25] 尚亚东．一类强阻尼非线性波动方程解的blow-up．烟台大学学报．2002．(15)：90-96页
[26] 孙淑珍．四阶强阻尼非线性波动方程的整体W~(k，p)解．吉林大学自然科学学报．2000，1(1)：18-22页
[27] 尚亚东．两类强阻尼非线性波动方程解的blow-up．工程数学学报．2000，5(17.2)：65-70页
[28] Ryo Ikehata．Global existence of solutions for semilinear damped wave equation in 2-D exterior domain．J．Differential Equation,2004(200):53-68P
[29] Zhijian Yang and Guowang Chen．Global existence of solutions for quasi-linear wave equations with viscous damping．J．Math．Anal. Appl， 2003(285):604-618P
[30] Fang Shao-mei,Guo Boling. Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations. Applied Mathematics and Mechanics,2003,7(24.6):673-683P
[31] 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．应用数学学报．2004(27.4)：133-146页
[32] 刘亚成，徐润章，刘萍．关于位势井及其对u_(11)-△u-△u_1-△u_(11)=f(u)的应用．(待发表)
[33] Liu Yacheng. On potential wells and vacuum isolating of solutions of Nonlinear Parabolic Equations. Publ.RTMS, 1972/73 (8): 211-229P
[34] 王元明，徐军祥编著．索伯列夫空间讲义．南京：东南大学出版社，2003．8
[35] R．A．Adams著；叶其孝等译．索伯列夫空间．北京：人民教育出版社，1981
[36] J．L．Lions著．郭柏灵，汪礼扔 译．非线性边值问题的一些解法．广州：中山大学出版社，1992
[37] 陈恕行，洪家兴编著．偏微分方程的近代方法．上海：复旦大学出版社，1988
[38] 李大潜，陈韵梅著．非线性发展方程。北京：科学出版社，1989
[39] 陈宁．人口问题中具广义Ginzbur-Landau型方程解的渐进性和Blow-up．生物数学学报．2005(20.3)：307-314页
[40] Steven Paul Levandosky. Decay Estimates for Fourth-Order Wave Equations. Journal of Differential Equations, 1998 (143): 360- 413P
[41] Van Horssen W T. An asymptotic theory for a class of initial boundary value problems for weakly nonlinear wave equations with an application to a model of the galloping cscillation of overhead transmission lines.SIAM.J.Appl.Math, 1988, (48.6) : 1243 -1277P
[42] Lai S Y. The asymptotic theory of solutions for a perturbed telegraph wave equation and its application. Appl. Math Mech, 1997, (7) : 656-662P
[43] Blota C.J,Van Der Burgh. Validity of approximations for time periodic solutions of a forced nonlinear hyperbolic differential equation. Appl Anal,1994,52 (1-4): 155-176P
[44] Asskura F. Existence of a global solution to a semilinear wave equation with slowly decreasing initial data in three space dimensions. Comm P D E, 1986,13 (11) :1459-1487P
[45] Lai S Y, Mu C L. The asymptotic theory of initial value problems for semilinear wave equations in three space dimensions. Appl Math JCU, 1997 (12B): 321-332P
[46] 杨从军．一类拟线性退化抛物型方程的初边值问题．数学学报．1995，38(1)：134-139页
[47] Hu Maolin. Global solution for the Quasilinear Wave Equations with Viscosity. Advances in Mathematics, 2001,30 (2): 123-132P
[48] 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．厦门大学学报(自然科学版)．2002，41(4)：418-422页
[49] 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．数学的实践与认识．2002，32(4)：658-663页
[50] Fila M. Boundedness of global solutions of nonlinear diffusion equations. J.of Diff.Equs, 1992 (98) : 226-240P
[51] Aguilar J.A.and Peral I. On some nonlinear parabolic equations, preprint U.A.M.1997
[52] G.Boling. Initial boundary value problem for one class of system of multidimensional inhomogeneous GBBM equation. J.Chin Ann of Math. 1987 (B8) :266-238P
[53] S.Yadong. Blow-up of solutions of initial boundary problem for a class of evolution equations with nonlinear strong dissipipation term. Journal of Xinyang Teachers College (Natural Science Edition) , 2002,15 (2) : 136-140P
[54] J.M.Ball. Remarks on blow-up and nonexistence theorems for non-linear evolution equations. Quart.J.Math. 1977 (28): 473-486P
[55] 王耀东 著．偏微分方程的L~2理论．北京：北京大学出版社，1982
[56] 郭柏灵，庞小峰．孤立子．北京：北京科学出版社，1978
[57] Steven Levandosky. Stablility and Instability of Fourth-Order Solitary Waves. Journal of Dynamics and Differential Equations, 1998,10(1): 151-188P
[58] Mitsuhiro 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