两类非线性数学物理模型方程的初边值问题
|
摘要
本文致力于研究两类非线性数学物理模型方程初边值问题的适定性与解的Blowup
问题。
在本文的第一部分，我们讨论出自粘弹性力学的具强阻尼、非线性应变、非线性阻
尼和非线性力源项的多维拟线性发展方程初边值问题的整体解的存在性、解的渐近性、
稳定性及解的Blowup问题。
我们利用位势井方法，单调性方法研究了在小初始能量情况下，上述初边值问题的
整体弱解的存在性和衰减性，用Galerkin方法和紧致性方法研究了在大初值情况下上述
初这值问题的整体弱解的存在性，分别利用补偿能量方法和能量方法研究了上述初边值
问题的整体解的不存在性问题。我们证明，上述方程中诸非线性项的增长阶（假定它们
均具有不超过多项式的增长阶）指数、初始能量的状态和相应的定解问题的整体解存在
与解在有限时刻发生Blowup现象之间存在着密切的联系，得到了增长指数、初始能量
的状态与整体解的存在与不存在之间的类似于门槛的结果。
我们利用H_o~k-Galerkin方法，进一步研究了具强阻尼、非线性应变和非线性外力项
的多维非线性发展方程的初边值问题。证明了在小初始能量情况下，上述初边值问题的
整体古典解的存在性、稳定性和衰减性，得到了一系列新的结果。
在本文的第二部分，我们讨论出自浅水波理论及等离子物理的“坏的”Boussinesq型
方程初边值问题的局部解的存在性及解的Blowup问题，利用一种新的数学思想、即将
抽象初值问题的解视为抽象空间中由初始点出发的“流”。通过建立一系列等距同构的
Hilbert空间，利用这些空间的拓扑不变性、利用Galenkin逼近和解的逐次延拓，证明
了在相当宽松的条件下，上述问题存在局部广义解。
我们分别利用Jensen不等式、常微分方程的比较原理、能量方法和Fourier变换方
法，证明了在一定条件下，“坏的”Boussinesq型方程的上述初边值问题的局部解必在
有限时刻发生Blowup现象。
Abstract
The paper studies the existence, non-existence, asymptotic property and stability of global solutions of the initial boundary value problems to two classes of nonlinear model equations in mathematical physics.
The paper is divided into two parts. In the first part, it discusses the existence of global weak solutions of the initial boundary value problems to a class of multidimensional quasi linear evolution equations with strong damping, nonlinear strain, nonlinear damping and source terms, which come from Viscoelasticity Mechanics. Making use of the potential well method as well as the monotonicity method, the paper arrives at the global existence and asymptotic property of weak solutions to the problems under the assumptions that the initial energy is properly small. Under the circumstances of large initial data, taking advantage of the Galerkin method a well as the compactness method, the paper gets the global weak solutions of the problems. And using the compensating energy method and the energy method respectively, the paper studies the non-existence of global weak solutions to the problems. These results implies that there exist close relations or thresholds among the growth indexes of the nonlinear terms, assuming that their growth indexes are no more than that of polynomials respectively, the value of initial energy, and the existence and non-existence of global weak solutions of the above-mentioned problems. Furthermore, the paper studies the initial boundary value problems to a class of multidimensional nonlinear evolution equations with strong damping, nonlinear strain and nonlinear external force terms. Under the assumptions that initial data is properly small, by virtue of the H -Galerkin method, the paper obtains the
global existence, asymptotic property and stability of the classical solutions to the problems
and gains a series of new results.
In the second part, the paper studies the local existence and the blowup of solutions of the initial boundary value problems to the ad?Boussinesq type equations, which arise from the theory of shallow waves and the Plasma Physics. Based on a new mathematical idea, i.e., viewing the solutions of the problems as streams starting from their initial points, by establishing a series of isometricly isomorphic Hilbert spaces, by the topological invariance of these spaces, and by exploiting the Galerkin approximation and the continuation of solutions step by step, the paper proved that under rather mild conditions for the nonlinear terms and the initial data, the initial boundary value problems of the ad?Boussinesq type equation have local generalized solutions. And by using the Jensen inequality, the comparison principle of ordinaiy differential equations, the energy method and the Fourier transform method respectively, the paper proved that under certain conditions the above-mentioned solutions will blow up in finite time.
问题。
在本文的第一部分，我们讨论出自粘弹性力学的具强阻尼、非线性应变、非线性阻
尼和非线性力源项的多维拟线性发展方程初边值问题的整体解的存在性、解的渐近性、
稳定性及解的Blowup问题。
我们利用位势井方法，单调性方法研究了在小初始能量情况下，上述初边值问题的
整体弱解的存在性和衰减性，用Galerkin方法和紧致性方法研究了在大初值情况下上述
初这值问题的整体弱解的存在性，分别利用补偿能量方法和能量方法研究了上述初边值
问题的整体解的不存在性问题。我们证明，上述方程中诸非线性项的增长阶（假定它们
均具有不超过多项式的增长阶）指数、初始能量的状态和相应的定解问题的整体解存在
与解在有限时刻发生Blowup现象之间存在着密切的联系，得到了增长指数、初始能量
的状态与整体解的存在与不存在之间的类似于门槛的结果。
我们利用H_o~k-Galerkin方法，进一步研究了具强阻尼、非线性应变和非线性外力项
的多维非线性发展方程的初边值问题。证明了在小初始能量情况下，上述初边值问题的
整体古典解的存在性、稳定性和衰减性，得到了一系列新的结果。
在本文的第二部分，我们讨论出自浅水波理论及等离子物理的“坏的”Boussinesq型
方程初边值问题的局部解的存在性及解的Blowup问题，利用一种新的数学思想、即将
抽象初值问题的解视为抽象空间中由初始点出发的“流”。通过建立一系列等距同构的
Hilbert空间，利用这些空间的拓扑不变性、利用Galenkin逼近和解的逐次延拓，证明
了在相当宽松的条件下，上述问题存在局部广义解。
我们分别利用Jensen不等式、常微分方程的比较原理、能量方法和Fourier变换方
法，证明了在一定条件下，“坏的”Boussinesq型方程的上述初边值问题的局部解必在
有限时刻发生Blowup现象。
Abstract
The paper studies the existence, non-existence, asymptotic property and stability of global solutions of the initial boundary value problems to two classes of nonlinear model equations in mathematical physics.
The paper is divided into two parts. In the first part, it discusses the existence of global weak solutions of the initial boundary value problems to a class of multidimensional quasi linear evolution equations with strong damping, nonlinear strain, nonlinear damping and source terms, which come from Viscoelasticity Mechanics. Making use of the potential well method as well as the monotonicity method, the paper arrives at the global existence and asymptotic property of weak solutions to the problems under the assumptions that the initial energy is properly small. Under the circumstances of large initial data, taking advantage of the Galerkin method a well as the compactness method, the paper gets the global weak solutions of the problems. And using the compensating energy method and the energy method respectively, the paper studies the non-existence of global weak solutions to the problems. These results implies that there exist close relations or thresholds among the growth indexes of the nonlinear terms, assuming that their growth indexes are no more than that of polynomials respectively, the value of initial energy, and the existence and non-existence of global weak solutions of the above-mentioned problems. Furthermore, the paper studies the initial boundary value problems to a class of multidimensional nonlinear evolution equations with strong damping, nonlinear strain and nonlinear external force terms. Under the assumptions that initial data is properly small, by virtue of the H -Galerkin method, the paper obtains the
global existence, asymptotic property and stability of the classical solutions to the problems
and gains a series of new results.
In the second part, the paper studies the local existence and the blowup of solutions of the initial boundary value problems to the ad?Boussinesq type equations, which arise from the theory of shallow waves and the Plasma Physics. Based on a new mathematical idea, i.e., viewing the solutions of the problems as streams starting from their initial points, by establishing a series of isometricly isomorphic Hilbert spaces, by the topological invariance of these spaces, and by exploiting the Galerkin approximation and the continuation of solutions step by step, the paper proved that under rather mild conditions for the nonlinear terms and the initial data, the initial boundary value problems of the ad?Boussinesq type equation have local generalized solutions. And by using the Jensen inequality, the comparison principle of ordinaiy differential equations, the energy method and the Fourier transform method respectively, the paper proved that under certain conditions the above-mentioned solutions will blow up in finite time.
引文
[1] Greenberg, J. M., MacCamy, R. C. and Mizel, J. J., On the existence, uniqueness and stability of the equationσ(u_x)u_(xx)+λu_(xxt)=P_ou_(ss). J Math. Mech., 1968,17,707--728．
[2] Greenberg, J. M.. On the existence, uniqueness and stability of the equation P_oX_(xx)=E(X_x)X_(xx)+λX_(xxt).Math. Anal.Appl, 1969,25,575--591．
[3] Dafermos, C. M., The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity. J. Diff. Eqns,1969, 6,71-86．
[4] Greenberg, J. M. and MacCamy, R. C., On the exponential stability of solutions of E(u_x)u_(xx)+λu_(xtx)=pu_(tt). J. Math. Anal. Appl., 1970,31,406-417．
[5] Davis, P., A quasi-linear hyperbolic and related third-order equation. J. Math. Anal. Appl., 1975,51,596--606．
[6] Andrews, G., On the existence of solutions to the equation u_(tt)-u_(xxt)=σ(u_x)x. J. Diff Eqns. 1980,35,200--231． [7] Andrews, G. and Ball, J. M., Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eqns,1982,44,306--341．
[8] Yamada, Y., Some remarks on equation Y_u-σ(Y_x)Y_(xx)-Y_(xtx)=f. Osaka. J. Math.. 1980,17． 303--323．
[9] Engler, H., Strong solutions for strongly damped quasilinear wave equations. Contemp. Math., 1987,64,219--237．
[10] 刘亚成,刘大成,方程u_(tt)=u_(xx1) +σ(u_x)_x的初边值问题、周期边界问题与初值问题.数学年刊, 1988,9(A),459--470．
[11] Kuttler,K. and Hicks,D., Initial-boundary value problems for the equation u_(tt)=σ(u_x)_x+(X(u_x)u_(xt))_x+f. Quart. Appl. Math.. XLVl. 1988,393-407．
[12] Clement, J., Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math., 1974,226． 745--752．
[13] Clements, J.. On the existence and uniqueness of solutions of the equation u_(tt)-/X×σ(u_(x1) )-△_Nu_s=f.Canad.Math. Bull. ,1975． 18, 181-187．
[14] Ebihara, Y, On Some nonlinear evolution equations with the strong dissipation. J. Diff. Eqns,1978,30,l49--l64．
[15] Ebihara, Y., Some evolution equations with the quasilinear strong dissipation. J. Math. Pures et Appl., 1979,58,229-245．
[16] Webb, G. R, Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canada J. Math., 1980,32,631--643．
[17] Pecher, H., On global regular solutions of third-order partial differential equations. J. Math. Anal. Appl., 1980,73,278-299．
[18] Friedman, A. and Necas, J., Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math., 1988,135,29-55.
[19] Ang, D. D. and Dinh, A. P. N., On the strongly damped wave equation u_u-△u-△u_(?)+f(u)=0. SIAM J. Math. Anal., 1988,19,1409-1418.
[20] 刘亚成，刘大成，一类三维非线性拟双曲方程的初边值问题．数学年刊，1988，9(A)，213-223．
[21] Kawashima, S. and Shibata Y., Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys., 1992,148,189-208.
[22] Kobaynshi, T., Pecher, H. and Shibata, Y., On a global in time existence theorem of small solutions to a nonlinear wave equation with viscosity. Math. Ann., 1993,296,215-234.
[23] Nakao, M., Energy decay for the quasilinear wave equation with viscosity. Math. Z, 1995,219,289-299.
[24] Whitham，G．B．著，线性与非线性波．北京：科学出版社，1984．
[25] 朱位秋，弹性杆中的非线性波．固体力学学报，1980，2，247-253．
[26] 庄蔚，杨桂通，孤波在非线性弹性杆中的传播应用数学和力学，1986，7．
[27] 张善元，庄蔚，非线性弹性杆中的应变孤波．力学学报，1988，20：1，58-66．
[28] 杨志坚，陈国旺，一类四阶拟线性波动方程初边值问题的整体解．数学物理学报，1995，15(增刊)1-9．
[29] Zhijian, Y. and Changming, S., Blowup of solutions for a class of quasilinear evolution equations. Nonlinear Analysis, T M. A., 1997,8,2017-2032.
[30] Zhijian, Y., Initial-boundary value problem and Cauchy problem for a quasilinear evolution equation. Acta Math. Sci., 1999,19(supp),52-61.
[31] Christensen, R. M., Theory of viscoelasticity. An Introduction, 2nd ed. New York: Academic Press, 1982
[32] Lee, E. H. and Kanter, L., Wave propagation in finite rods of viscoelastic material. J. Appl. Phys., 1953,24,115-1122.
[33] 崔尚斌，半线性卷积双曲型积分微分方程的初边值问题．应用数学学报，1988,11，271-286．
[34] Hai, D. D., On a wave equation with nonlinear strong damping. Nonlinear Analysis, T. M A., 1990,15,1005-1015.
[35] Hal, D. D., On a quasilinear wave equation with nonlinear damping, Proc. R. Soc. Edinb., 1988 110A,227-239.
[36] Prestel, M. A., Forced oscillations for the solutions of nonlinear hyperbolic equation. Nonlinear Analysis, T. M. A., 1982,6,209-216.
[37] Biler, P., Decay of solutions of semilinear strongly damped generalized wave equation. Math. Meth Appl. Sci., 1991,14 (16),427-444.
[38] Naumkin, P. and Shishmarev, I., Nonlinear nonlocal equations in the theory of waves. Transl. of Math. Monographs, Vol.133, Amer. Math Soc., Providence, RI, 1994.
[39] Nakao, M. and Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. Math. Z, 1993,214,325-342.
[40] Georgiev, V., Existence of a solution of the wave equation with nonlinear Damping and source terms. J. Diff. Eqns., 1994,109,295-308.
[41] Ikehata, R, Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Analysis, T. M. A., 1996,27,1165-1175.
[42] Ono, K., On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. Math. Meth. Appl. Sci., 1997,20,151-177.
[43] Ang, D. D. and Dinh, A. P. N., Strong solutions ofa quasilinear wave equation with nonlinear damping. SIAM J. Math. Anal., 1988,19,337-347.
[44] 周毓麟，符鸿源，广义Sine-Gordon型非线性高阶双曲方程组．数学学报，1983，26，234-249．
[45] 孙和生，半线性退化发展方程的混合初边值问题．数学年刊，1987，8(A)，32-43．
[46] Sobolev, S., Some application of functional analysis to mathematical physics. Leninggrad, 1950.
[47] Mizohata, S., The theory, of partial differential equations. London: Cambridge Univ. Press, 1973.
[48] Boussinesq, J., Theorie des ondes et des remous qui se propagent le long dun canalrectangulaire horizontal, en comuniquant au liquid contene dans ce canal des vitessessunsiblement parielles de la surface au fond. J. Math. Pures Appl. Ser., 1872,2(17),55-108.
[49] Makhankov, V. G., Dynamics of classical solitons. Phys. Rev. Lett., 1978,35(c),1-128.
[50] Berryman, J., Stability of solitary waves in shallow water. Phys. Fluids 1976,19,771-777.
[51] Bona, J. L., On the stability theory waves. Proc. Roy. Soc. Lond., 1975, A 344, 363-374.
[52] Bona. J. L. and Smith, R., A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc., 1976,79,167-182.
[53] Linares, F., Global existence of small solutions for a generalized Boussinesq equation. J. Diff. Eqns., 1993,106,257-293.
[54] Zabusky, N. J. and Kruskal, M. D., Interaction of "solitons" in a collisionless plasma and recurrence of initial states. Phys. Rev Lett., 1965.15,240-243.
[55] Zakharov, V. E., On slochastization of one-dimensional chains of nonlinear oscitlators. Soviet Phys. JETP., 1974.38.108-110.
[56] Deift, P., Tomei, C. and Trubowitz, E., Inverse scattering and the Boussinesq equation. Commun. Pure Appl. Math., 1982,35,567-628.
[57] Kalantarov, V. and Ladyzhenskaya, O., The occurrence of collapse for quasilinear equation of parabolic and hyperbolic types. J Sov. Math., 1978,10,53-70.
[58] Sachs. R. L., On the blowup of certain solutions of the "good" Boussinesq equation. Appl. Anal., 1990,34,145-152.
[59] Bona. J. L. and Sachs, R. L., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys., 1988,118.15-29.
[60] 郭柏灵，复Schrodinger场和Boussinesq型自洽场相互作用下一类方程组的整体解．数学学报．1983，26，295-306．
[61] 郭柏灵，非线性演化方程．上海：上海科技教育出版社，1995．
[62] Kalantorov, V. K. and Ladyzhenskaya. A., Occurrence of blow-up for quasilinear partial differential equations of parabolic and hyperbolic types. Zap. Nauk. Som. Lomi., 1977,69,77-102.
[63] Levine, H. A. and Sleeman, B. D., A note on the nonexistence of global solutions of initial boundary value problems for the Boussinesq equation u_(?)=3u_(?)=+u_(?)-12(u~2)_(?). J. Math. Anal. Appl., 1985,107,206-210.
[64] Varlamov, V., Asymptotics as t→+∞ of a solution to the periodic Cauchy problem for the damped Boussinesq equation. Math. Meth. Appl. Sci., 1997,20,805-812.
[65] Yang Zhijian, Existence and non-existence of global solutions to a generalized Modification of the Improved Boussinesq equation. Math. Meth. Appl. Sci., 1998,21,1467-1477.
[66] Varlamov, V., On the Cauchy problem for the damped Boussinesq equation. Dif. Int. Eqs., 1996,9(3),619-634.
[67] Hirota, R., Exact n-soliton solutions of the wave equation of long wave in shallow water and in nonlinear lattices. J. Math. Phys., 1973,14,810-814.
[68] Clarkson P. A. and Kruskal, M. D., New similarity reductions of the Boussinesq equation. J. Math. Phys., 1989,30(10),2201-2213.
[69] Clarkson, P., New exact solutions of the Boussinesq equations. Euro. J. Appl. Maths., 1990,1,279-300.
[70] Ladyzenskaya, O. A., Solonnikov, V. A. and Ural ceva, N. N., Linear and quasilinear equation of parabolic type. Amer. Math. Soc. Transl. of Math. Monographs, Vol 23.
[71] Palais, B., Blowup for nonlinear equations using a comparison principle in Fourier space Comm. Pure Appl. Math, 1988,41,165-196.
[72]杨志坚，一类非线性发展方程初边值问题解的Blowup．高校应用数学学报，1993，8(4)，351-359．
[73]杨志坚，陈国旺，关于一类拟线性拟抛物方程解的Blowup问题．高校应用数学学报，1994，9(3)，228-235．
[74]杨志坚，陈国旺，一类广义Boussinesq方程解的Blowup．数学物理学报，1996，16(1)，31-40．
[75] Budd, C., Dold, B. and Stuart, A., Blowup in partial differential equation with conserved first integral. SLAMJ. Appl. Math., 1993,53(3),718-742.
[76] Schaaf, R., Stationary solutions of chemataxis systems. Trans, Amer. Math. Soc., 1985,29(2)531-556.
[77] Furter, J. and Grinfeld, M., Local vs. non-Local interactions in population dynamics. J. Math. Biol., 1989,27:65-80.
[78] Walter. W., Differential and integral inequality. New York: Springer-Verlag, 1973.
[79]李岳生，基本不等式与微分方程解的唯一性．吉林大学自然科学学报，1960，(1)7-22．
[80]叶其孝，李正元著，反应扩散方程引论．北京：科学出版社，1994．
[81] Miller, R. K. and Michel, A. N., Ordinary differential equation. New York: Academic Press, 1982.
[82] Ladyzhenskaya, O. A., The boundary value problems of mathematical physics. New York: Springer Verlag, 1985.
[83] Maz ja, V. G., Sobolev Spaces. Berlin: Springer-Verlag, 1985.
[84] Hutson, V. and Pym, J. S., Application of functional analysis and operator theory. London: Academic Press INC., 1980.
[85] Chen Guowang and Yang Zhijian, Initial valuc problem for a class of nonlinear pseudo-hyperbolic equations. Acta Math. Appl. Sinica, 1993,19(2),166-173.
[86] Chen Guowang, Yang Zhijian and Zhao Zhancai, Initial value problems and first boundary problems for a class of quasilinear wave equations. Acta Math. Appl. Sinica, 1993,9(4),289-301.
[87] Chen Fengxin, Guo Boling and Wang Ping, Long time behavior of strongly damped nonlinear wave equations. J. Diff. Eqns, 1998,147,231-241.
[88] Chen Guowang, Xing Jiasheng and Yang Zhijian, Cauchy problem for generalized IMBq equation with several variables. Nonlinear Analysis, T. M. A., 1996,26(7),1255-1270.
[89] Guo Boling and Yuan Guangwei, On the suitable weak solutions for the Cauchy problem of the Boussinesq equations. Nonlinear Analysis, T. M. A., 1996,26(8),1367-1386.
[90]郭柏灵，袁光伟，具有L~P数据的Boussinesq方程的初边值问题．数学年刊，1996，17：A(5)，595-606．
[91]杨志坚，陈国旺，具有阻尼项的非线性波动方程的初值问题．应用数学学报，2000，23(1)，45-54．
[92]杨志坚，陈国旺，Boussinesq型方程的周期边界问题与初值问题的解的存在性．应用数学学报，2000，23，(2)，
[93]陈国旺，一类多维高阶非线性耦合发展方程组的第一边值问题．应用数学学报，1988，11(1)，79-94．
[94]陈国旺，非线性高阶抛物双曲型耦合方程组的第一边界问题．数学年刊，1988，9A(2)，139-150．
[95]刘亚成，万维明，吕淑娟，神经传播型方程初边值问题解的Blowup．应用数学学报，1997，20(2)，289-293．
[96] Chen Guowang and Wang Shubin, Existence and nonexistence of global solutions for the generalized IMBq equation. Nonlinear analysis, T. M. A., 1999,36(8),961-980.
[97] Adams, R. A., Sobolev spaces. New York: Academic Press, 1975.
[2] Greenberg, J. M.. On the existence, uniqueness and stability of the equation P_oX_(xx)=E(X_x)X_(xx)+λX_(xxt).Math. Anal.Appl, 1969,25,575--591．
[3] Dafermos, C. M., The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity. J. Diff. Eqns,1969, 6,71-86．
[4] Greenberg, J. M. and MacCamy, R. C., On the exponential stability of solutions of E(u_x)u_(xx)+λu_(xtx)=pu_(tt). J. Math. Anal. Appl., 1970,31,406-417．
[5] Davis, P., A quasi-linear hyperbolic and related third-order equation. J. Math. Anal. Appl., 1975,51,596--606．
[6] Andrews, G., On the existence of solutions to the equation u_(tt)-u_(xxt)=σ(u_x)x. J. Diff Eqns. 1980,35,200--231． [7] Andrews, G. and Ball, J. M., Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eqns,1982,44,306--341．
[8] Yamada, Y., Some remarks on equation Y_u-σ(Y_x)Y_(xx)-Y_(xtx)=f. Osaka. J. Math.. 1980,17． 303--323．
[9] Engler, H., Strong solutions for strongly damped quasilinear wave equations. Contemp. Math., 1987,64,219--237．
[10] 刘亚成,刘大成,方程u_(tt)=u_(xx1) +σ(u_x)_x的初边值问题、周期边界问题与初值问题.数学年刊, 1988,9(A),459--470．
[11] Kuttler,K. and Hicks,D., Initial-boundary value problems for the equation u_(tt)=σ(u_x)_x+(X(u_x)u_(xt))_x+f. Quart. Appl. Math.. XLVl. 1988,393-407．
[12] Clement, J., Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math., 1974,226． 745--752．
[13] Clements, J.. On the existence and uniqueness of solutions of the equation u_(tt)-/X×σ(u_(x1) )-△_Nu_s=f.Canad.Math. Bull. ,1975． 18, 181-187．
[14] Ebihara, Y, On Some nonlinear evolution equations with the strong dissipation. J. Diff. Eqns,1978,30,l49--l64．
[15] Ebihara, Y., Some evolution equations with the quasilinear strong dissipation. J. Math. Pures et Appl., 1979,58,229-245．
[16] Webb, G. R, Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canada J. Math., 1980,32,631--643．
[17] Pecher, H., On global regular solutions of third-order partial differential equations. J. Math. Anal. Appl., 1980,73,278-299．
[18] Friedman, A. and Necas, J., Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math., 1988,135,29-55.
[19] Ang, D. D. and Dinh, A. P. N., On the strongly damped wave equation u_u-△u-△u_(?)+f(u)=0. SIAM J. Math. Anal., 1988,19,1409-1418.
[20] 刘亚成，刘大成，一类三维非线性拟双曲方程的初边值问题．数学年刊，1988，9(A)，213-223．
[21] Kawashima, S. and Shibata Y., Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys., 1992,148,189-208.
[22] Kobaynshi, T., Pecher, H. and Shibata, Y., On a global in time existence theorem of small solutions to a nonlinear wave equation with viscosity. Math. Ann., 1993,296,215-234.
[23] Nakao, M., Energy decay for the quasilinear wave equation with viscosity. Math. Z, 1995,219,289-299.
[24] Whitham，G．B．著，线性与非线性波．北京：科学出版社，1984．
[25] 朱位秋，弹性杆中的非线性波．固体力学学报，1980，2，247-253．
[26] 庄蔚，杨桂通，孤波在非线性弹性杆中的传播应用数学和力学，1986，7．
[27] 张善元，庄蔚，非线性弹性杆中的应变孤波．力学学报，1988，20：1，58-66．
[28] 杨志坚，陈国旺，一类四阶拟线性波动方程初边值问题的整体解．数学物理学报，1995，15(增刊)1-9．
[29] Zhijian, Y. and Changming, S., Blowup of solutions for a class of quasilinear evolution equations. Nonlinear Analysis, T M. A., 1997,8,2017-2032.
[30] Zhijian, Y., Initial-boundary value problem and Cauchy problem for a quasilinear evolution equation. Acta Math. Sci., 1999,19(supp),52-61.
[31] Christensen, R. M., Theory of viscoelasticity. An Introduction, 2nd ed. New York: Academic Press, 1982
[32] Lee, E. H. and Kanter, L., Wave propagation in finite rods of viscoelastic material. J. Appl. Phys., 1953,24,115-1122.
[33] 崔尚斌，半线性卷积双曲型积分微分方程的初边值问题．应用数学学报，1988,11，271-286．
[34] Hai, D. D., On a wave equation with nonlinear strong damping. Nonlinear Analysis, T. M A., 1990,15,1005-1015.
[35] Hal, D. D., On a quasilinear wave equation with nonlinear damping, Proc. R. Soc. Edinb., 1988 110A,227-239.
[36] Prestel, M. A., Forced oscillations for the solutions of nonlinear hyperbolic equation. Nonlinear Analysis, T. M. A., 1982,6,209-216.
[37] Biler, P., Decay of solutions of semilinear strongly damped generalized wave equation. Math. Meth Appl. Sci., 1991,14 (16),427-444.
[38] Naumkin, P. and Shishmarev, I., Nonlinear nonlocal equations in the theory of waves. Transl. of Math. Monographs, Vol.133, Amer. Math Soc., Providence, RI, 1994.
[39] Nakao, M. and Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. Math. Z, 1993,214,325-342.
[40] Georgiev, V., Existence of a solution of the wave equation with nonlinear Damping and source terms. J. Diff. Eqns., 1994,109,295-308.
[41] Ikehata, R, Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Analysis, T. M. A., 1996,27,1165-1175.
[42] Ono, K., On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. Math. Meth. Appl. Sci., 1997,20,151-177.
[43] Ang, D. D. and Dinh, A. P. N., Strong solutions ofa quasilinear wave equation with nonlinear damping. SIAM J. Math. Anal., 1988,19,337-347.
[44] 周毓麟，符鸿源，广义Sine-Gordon型非线性高阶双曲方程组．数学学报，1983，26，234-249．
[45] 孙和生，半线性退化发展方程的混合初边值问题．数学年刊，1987，8(A)，32-43．
[46] Sobolev, S., Some application of functional analysis to mathematical physics. Leninggrad, 1950.
[47] Mizohata, S., The theory, of partial differential equations. London: Cambridge Univ. Press, 1973.
[48] Boussinesq, J., Theorie des ondes et des remous qui se propagent le long dun canalrectangulaire horizontal, en comuniquant au liquid contene dans ce canal des vitessessunsiblement parielles de la surface au fond. J. Math. Pures Appl. Ser., 1872,2(17),55-108.
[49] Makhankov, V. G., Dynamics of classical solitons. Phys. Rev. Lett., 1978,35(c),1-128.
[50] Berryman, J., Stability of solitary waves in shallow water. Phys. Fluids 1976,19,771-777.
[51] Bona, J. L., On the stability theory waves. Proc. Roy. Soc. Lond., 1975, A 344, 363-374.
[52] Bona. J. L. and Smith, R., A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc., 1976,79,167-182.
[53] Linares, F., Global existence of small solutions for a generalized Boussinesq equation. J. Diff. Eqns., 1993,106,257-293.
[54] Zabusky, N. J. and Kruskal, M. D., Interaction of "solitons" in a collisionless plasma and recurrence of initial states. Phys. Rev Lett., 1965.15,240-243.
[55] Zakharov, V. E., On slochastization of one-dimensional chains of nonlinear oscitlators. Soviet Phys. JETP., 1974.38.108-110.
[56] Deift, P., Tomei, C. and Trubowitz, E., Inverse scattering and the Boussinesq equation. Commun. Pure Appl. Math., 1982,35,567-628.
[57] Kalantarov, V. and Ladyzhenskaya, O., The occurrence of collapse for quasilinear equation of parabolic and hyperbolic types. J Sov. Math., 1978,10,53-70.
[58] Sachs. R. L., On the blowup of certain solutions of the "good" Boussinesq equation. Appl. Anal., 1990,34,145-152.
[59] Bona. J. L. and Sachs, R. L., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys., 1988,118.15-29.
[60] 郭柏灵，复Schrodinger场和Boussinesq型自洽场相互作用下一类方程组的整体解．数学学报．1983，26，295-306．
[61] 郭柏灵，非线性演化方程．上海：上海科技教育出版社，1995．
[62] Kalantorov, V. K. and Ladyzhenskaya. A., Occurrence of blow-up for quasilinear partial differential equations of parabolic and hyperbolic types. Zap. Nauk. Som. Lomi., 1977,69,77-102.
[63] Levine, H. A. and Sleeman, B. D., A note on the nonexistence of global solutions of initial boundary value problems for the Boussinesq equation u_(?)=3u_(?)=+u_(?)-12(u~2)_(?). J. Math. Anal. Appl., 1985,107,206-210.
[64] Varlamov, V., Asymptotics as t→+∞ of a solution to the periodic Cauchy problem for the damped Boussinesq equation. Math. Meth. Appl. Sci., 1997,20,805-812.
[65] Yang Zhijian, Existence and non-existence of global solutions to a generalized Modification of the Improved Boussinesq equation. Math. Meth. Appl. Sci., 1998,21,1467-1477.
[66] Varlamov, V., On the Cauchy problem for the damped Boussinesq equation. Dif. Int. Eqs., 1996,9(3),619-634.
[67] Hirota, R., Exact n-soliton solutions of the wave equation of long wave in shallow water and in nonlinear lattices. J. Math. Phys., 1973,14,810-814.
[68] Clarkson P. A. and Kruskal, M. D., New similarity reductions of the Boussinesq equation. J. Math. Phys., 1989,30(10),2201-2213.
[69] Clarkson, P., New exact solutions of the Boussinesq equations. Euro. J. Appl. Maths., 1990,1,279-300.
[70] Ladyzenskaya, O. A., Solonnikov, V. A. and Ural ceva, N. N., Linear and quasilinear equation of parabolic type. Amer. Math. Soc. Transl. of Math. Monographs, Vol 23.
[71] Palais, B., Blowup for nonlinear equations using a comparison principle in Fourier space Comm. Pure Appl. Math, 1988,41,165-196.
[72]杨志坚，一类非线性发展方程初边值问题解的Blowup．高校应用数学学报，1993，8(4)，351-359．
[73]杨志坚，陈国旺，关于一类拟线性拟抛物方程解的Blowup问题．高校应用数学学报，1994，9(3)，228-235．
[74]杨志坚，陈国旺，一类广义Boussinesq方程解的Blowup．数学物理学报，1996，16(1)，31-40．
[75] Budd, C., Dold, B. and Stuart, A., Blowup in partial differential equation with conserved first integral. SLAMJ. Appl. Math., 1993,53(3),718-742.
[76] Schaaf, R., Stationary solutions of chemataxis systems. Trans, Amer. Math. Soc., 1985,29(2)531-556.
[77] Furter, J. and Grinfeld, M., Local vs. non-Local interactions in population dynamics. J. Math. Biol., 1989,27:65-80.
[78] Walter. W., Differential and integral inequality. New York: Springer-Verlag, 1973.
[79]李岳生，基本不等式与微分方程解的唯一性．吉林大学自然科学学报，1960，(1)7-22．
[80]叶其孝，李正元著，反应扩散方程引论．北京：科学出版社，1994．
[81] Miller, R. K. and Michel, A. N., Ordinary differential equation. New York: Academic Press, 1982.
[82] Ladyzhenskaya, O. A., The boundary value problems of mathematical physics. New York: Springer Verlag, 1985.
[83] Maz ja, V. G., Sobolev Spaces. Berlin: Springer-Verlag, 1985.
[84] Hutson, V. and Pym, J. S., Application of functional analysis and operator theory. London: Academic Press INC., 1980.
[85] Chen Guowang and Yang Zhijian, Initial valuc problem for a class of nonlinear pseudo-hyperbolic equations. Acta Math. Appl. Sinica, 1993,19(2),166-173.
[86] Chen Guowang, Yang Zhijian and Zhao Zhancai, Initial value problems and first boundary problems for a class of quasilinear wave equations. Acta Math. Appl. Sinica, 1993,9(4),289-301.
[87] Chen Fengxin, Guo Boling and Wang Ping, Long time behavior of strongly damped nonlinear wave equations. J. Diff. Eqns, 1998,147,231-241.
[88] Chen Guowang, Xing Jiasheng and Yang Zhijian, Cauchy problem for generalized IMBq equation with several variables. Nonlinear Analysis, T. M. A., 1996,26(7),1255-1270.
[89] Guo Boling and Yuan Guangwei, On the suitable weak solutions for the Cauchy problem of the Boussinesq equations. Nonlinear Analysis, T. M. A., 1996,26(8),1367-1386.
[90]郭柏灵，袁光伟，具有L~P数据的Boussinesq方程的初边值问题．数学年刊，1996，17：A(5)，595-606．
[91]杨志坚，陈国旺，具有阻尼项的非线性波动方程的初值问题．应用数学学报，2000，23(1)，45-54．
[92]杨志坚，陈国旺，Boussinesq型方程的周期边界问题与初值问题的解的存在性．应用数学学报，2000，23，(2)，
[93]陈国旺，一类多维高阶非线性耦合发展方程组的第一边值问题．应用数学学报，1988，11(1)，79-94．
[94]陈国旺，非线性高阶抛物双曲型耦合方程组的第一边界问题．数学年刊，1988，9A(2)，139-150．
[95]刘亚成，万维明，吕淑娟，神经传播型方程初边值问题解的Blowup．应用数学学报，1997，20(2)，289-293．
[96] Chen Guowang and Wang Shubin, Existence and nonexistence of global solutions for the generalized IMBq equation. Nonlinear analysis, T. M. A., 1999,36(8),961-980.
[97] Adams, R. A., Sobolev spaces. New York: Academic Press, 1975.