三维Ginzburg Landau方程的动力学行为
|
摘要
该文由两部分构成。首先，在第一部分，考虑Ginzburg-Landau方程在三维空间的整体吸引子的存在性，整体吸引子的正则性，指数吸引子和三无界区域R~3上的整体吸引子。第二部分，考虑广义Ginzburg-Landau方程在三维空间的整体吸引子的存在性和时间周期解的存在性。
该文由七章构成。在第一章，介绍Ginzburg-Landau方程的物理背景，研究状况及本文的工作内容。在第二章，考虑Ginzburg-Landau方程在三维空间的整体吸引子的存在性，首先考虑Ginzburg-Landau方程的局部解的存在性，对于一给定的扰动项N(u)，证明N(u)是收缩的且是局部Lipschitz连续的。因此，可获得局部解的存在性。然后，使用先验估计的方法获得整体解的存在性。同时也获得了紧的有界吸收集的存在性，从而得到整体吸引子的存在性，最后讨论了它的Hausdorff维数和分形维数。第三章，在三维空间中研究Ginzburg-Landau方程整体光滑解的存在性和整体吸引子的正则性，首先利用系列精细的估计证明整体光滑解的存在性。进而研究在H~m中的吸引子的存在性。考虑H~1中的解半群{S~((1))(t)}t≥0，采用分解其解算子S~1(t)=S_1~1(t)+S_2~1(t)的方法，证明了S_1~1(t)u_0比S~1(t)u_0更加正则，而当t→∞时，‖S_2~1(t)u_0‖_1趋于零，其中u_0∈H~2。进而证明H~1中的吸引子A_1等于H~m(2σ+1≥m≥2)中的吸引子A_m。第四章，研究Ginzburg-Landau方程在三维空间的指数吸引子的存在性。首先证明解算子S(t)是Lipschitz连续的，然后证明离散解算子S_*=S(t_*)满足挤压性，从而得到指数吸引子M的存在性。它的分形维数是有限的。第五章，讨论Ginzburg-Landau方程在三维无界区域R~3上的整体吸引子。通过引入加权空间，以克服古典Sobolev空间在无界区域嵌入的非紧性。在加权空间进行先验估计，获得解算子S(t)在加权空间紧的有界吸收集，从而在加权空间得到整体吸引子的存在性。第六章，它是论文的第二部分，研究带导数项的广义Ginzburg-Landau方程在三维空间的整体吸引子的存在性。主要困难在于解在空间H~1，H~2的先验估计。我们充分考虑方程的特点，将高阶非线性项化为一非负定二次型使得解在H~1中的吸收集的存在性得以证明。通过能量估计的
巧妙组合以及多种形式的不等式进行精细的估计以获得H“的吸收集的
存在性.从而得到整体吸引子的存在性，讨论了它的Hausdorff维数和分形
维数.第七章，讨论广义Ginzburg一Landau方程在三维空间的时间周期解的
存在性。首先应用Galerkin方法和Larey一schaude:不动点定理证明近似解的
存在性，然后进行近似解的高阶导数的先验估计(关于空间变量和时间
变量)。最后，使用标准的紧性讨论的方法得到这个系统的时间周期解的
存在性。
This dissertation consists of two part. In first part we consider the existence of the global attractor the complex Ginzburg-Landau equation in three dimen-sions space, the regularity of the global attractor, the exponential attractors and the existence of the global attractor in whole R~(3). Then, in secondary part, we consider the existence of the global attractor and the time periodic solution for generlized complex Ginzburg-Landau equation in three dimensions space.
This dissertation consists of seven chapters. In chapter 1, we briefly in-troduce the background in physics and developments in the Ginzburg-Landau equation. In which the main work of the dissertation is described. In chapter 2, we study the existence of the global attractor the complex Ginzburg-Landau equation in three dimensions space. First, we consider existence of local solu-tion. For a given perturbation N(u), we prove N(u) is contractive and locally Lipschitz continuous. Therefore, we obtain existence of local solution. Then we obtain the existence of global solution by the using the method of priori estimate. At the same time, we give the existence of bounded absorbing set. Moreover, we obtain the existence of global attractor. Finally, we study the dimensionality of Hausdorff and fractal of attractor. In chapter 3, we study the existence of global smooth solution and regularity of the global attractor for the complex Ginzburg-Landau equation in three dimensions space. First, applying a, series fine
estimate, we prove existence of global smooth solution. Moreover, we obtain the existence of the attractor in H~(m). Considering solution semigroup in H~(1), we decomposing solution operator We prove that is more regular than the solution tends to zero as t goes to infinity, uniformly for . Furthermore, attractor A_(1) in H~(1) is equal to attractor A_(m) in H~(m)(2σ + 1 ≥ m ≥ 2). In chapter 4, we study the existence of the exponential attractor of the complex Ginzburg-Landau equation in three dimen-sions space. We first show that the solution operator S(t) is Lipschitz continuous, then prove the discrete solution operator S_(*) = 5(t_(*)) satisfy the squeezing property, finally, we get the existence of the exponential attractor M. whose fractal dimensionality is finite. In chapter 5, we study the existence of the global attrac-
tor for the complex Ginzburg-Landau equation in 3-D unbounded domain. By introducing weighted space and using the method of priori estimatehe, uniformly compactness are achieved for S(t) in weighted space to overcome the noncom-pactness of the classical Sobolev embedding in unbounded domain. In chapter 6, it's the secondary part of this dissertation, we study the existence of the global attractor for the generalized complex Ginzburg-Landau equation with derivative term in three dimensions space. Considering enough the property of the equa-tion, we prove existence of absorbing set of the solution in H~(1) by the method of changeing the higher order nonlinear term as nonnegative guadratic form and prove existence of absorbing set of the solution in H~(2) by the method of linear combining of energy inequality and multiform classical inequality. Moreover, we obtain the existence of global attractor. Finally, we study the dimensionality of Hausdorff and fractal of attractor. In chapter 7, we discuss the existence of time periodic solution of the generalized complex Ginzburg-Landau equation in three dimensions space. First, we apply the method of Galerkin and the fixed theorem of the Larey-Schauder to prove the existence of the approximate solution. Next, we give the priori estimates of the higher order derivatives (with respect to spa-tial variable and time variable) of the approximate solution. Finally, we use the method of standard compactness arguments to get the existence of time periodic solution of this system.
该文由七章构成。在第一章，介绍Ginzburg-Landau方程的物理背景，研究状况及本文的工作内容。在第二章，考虑Ginzburg-Landau方程在三维空间的整体吸引子的存在性，首先考虑Ginzburg-Landau方程的局部解的存在性，对于一给定的扰动项N(u)，证明N(u)是收缩的且是局部Lipschitz连续的。因此，可获得局部解的存在性。然后，使用先验估计的方法获得整体解的存在性。同时也获得了紧的有界吸收集的存在性，从而得到整体吸引子的存在性，最后讨论了它的Hausdorff维数和分形维数。第三章，在三维空间中研究Ginzburg-Landau方程整体光滑解的存在性和整体吸引子的正则性，首先利用系列精细的估计证明整体光滑解的存在性。进而研究在H~m中的吸引子的存在性。考虑H~1中的解半群{S~((1))(t)}t≥0，采用分解其解算子S~1(t)=S_1~1(t)+S_2~1(t)的方法，证明了S_1~1(t)u_0比S~1(t)u_0更加正则，而当t→∞时，‖S_2~1(t)u_0‖_1趋于零，其中u_0∈H~2。进而证明H~1中的吸引子A_1等于H~m(2σ+1≥m≥2)中的吸引子A_m。第四章，研究Ginzburg-Landau方程在三维空间的指数吸引子的存在性。首先证明解算子S(t)是Lipschitz连续的，然后证明离散解算子S_*=S(t_*)满足挤压性，从而得到指数吸引子M的存在性。它的分形维数是有限的。第五章，讨论Ginzburg-Landau方程在三维无界区域R~3上的整体吸引子。通过引入加权空间，以克服古典Sobolev空间在无界区域嵌入的非紧性。在加权空间进行先验估计，获得解算子S(t)在加权空间紧的有界吸收集，从而在加权空间得到整体吸引子的存在性。第六章，它是论文的第二部分，研究带导数项的广义Ginzburg-Landau方程在三维空间的整体吸引子的存在性。主要困难在于解在空间H~1，H~2的先验估计。我们充分考虑方程的特点，将高阶非线性项化为一非负定二次型使得解在H~1中的吸收集的存在性得以证明。通过能量估计的
巧妙组合以及多种形式的不等式进行精细的估计以获得H“的吸收集的
存在性.从而得到整体吸引子的存在性，讨论了它的Hausdorff维数和分形
维数.第七章，讨论广义Ginzburg一Landau方程在三维空间的时间周期解的
存在性。首先应用Galerkin方法和Larey一schaude:不动点定理证明近似解的
存在性，然后进行近似解的高阶导数的先验估计(关于空间变量和时间
变量)。最后，使用标准的紧性讨论的方法得到这个系统的时间周期解的
存在性。
This dissertation consists of two part. In first part we consider the existence of the global attractor the complex Ginzburg-Landau equation in three dimen-sions space, the regularity of the global attractor, the exponential attractors and the existence of the global attractor in whole R~(3). Then, in secondary part, we consider the existence of the global attractor and the time periodic solution for generlized complex Ginzburg-Landau equation in three dimensions space.
This dissertation consists of seven chapters. In chapter 1, we briefly in-troduce the background in physics and developments in the Ginzburg-Landau equation. In which the main work of the dissertation is described. In chapter 2, we study the existence of the global attractor the complex Ginzburg-Landau equation in three dimensions space. First, we consider existence of local solu-tion. For a given perturbation N(u), we prove N(u) is contractive and locally Lipschitz continuous. Therefore, we obtain existence of local solution. Then we obtain the existence of global solution by the using the method of priori estimate. At the same time, we give the existence of bounded absorbing set. Moreover, we obtain the existence of global attractor. Finally, we study the dimensionality of Hausdorff and fractal of attractor. In chapter 3, we study the existence of global smooth solution and regularity of the global attractor for the complex Ginzburg-Landau equation in three dimensions space. First, applying a, series fine
estimate, we prove existence of global smooth solution. Moreover, we obtain the existence of the attractor in H~(m). Considering solution semigroup in H~(1), we decomposing solution operator We prove that is more regular than the solution tends to zero as t goes to infinity, uniformly for . Furthermore, attractor A_(1) in H~(1) is equal to attractor A_(m) in H~(m)(2σ + 1 ≥ m ≥ 2). In chapter 4, we study the existence of the exponential attractor of the complex Ginzburg-Landau equation in three dimen-sions space. We first show that the solution operator S(t) is Lipschitz continuous, then prove the discrete solution operator S_(*) = 5(t_(*)) satisfy the squeezing property, finally, we get the existence of the exponential attractor M. whose fractal dimensionality is finite. In chapter 5, we study the existence of the global attrac-
tor for the complex Ginzburg-Landau equation in 3-D unbounded domain. By introducing weighted space and using the method of priori estimatehe, uniformly compactness are achieved for S(t) in weighted space to overcome the noncom-pactness of the classical Sobolev embedding in unbounded domain. In chapter 6, it's the secondary part of this dissertation, we study the existence of the global attractor for the generalized complex Ginzburg-Landau equation with derivative term in three dimensions space. Considering enough the property of the equa-tion, we prove existence of absorbing set of the solution in H~(1) by the method of changeing the higher order nonlinear term as nonnegative guadratic form and prove existence of absorbing set of the solution in H~(2) by the method of linear combining of energy inequality and multiform classical inequality. Moreover, we obtain the existence of global attractor. Finally, we study the dimensionality of Hausdorff and fractal of attractor. In chapter 7, we discuss the existence of time periodic solution of the generalized complex Ginzburg-Landau equation in three dimensions space. First, we apply the method of Galerkin and the fixed theorem of the Larey-Schauder to prove the existence of the approximate solution. Next, we give the priori estimates of the higher order derivatives (with respect to spa-tial variable and time variable) of the approximate solution. Finally, we use the method of standard compactness arguments to get the existence of time periodic solution of this system.
引文
[1] V. I. Ginzburg and L. D. Landau, On theory of supperconductivity, [J]. Zh.Eksp.Theor. Fiz. 20 (1950) , 1064-1082．
[2] E. M. Lifishitz and L. P. Pitaevstii, [M]. Statistical physics, Vol. 2 Oxford; Pergamon, 1980．
[3] A. C. Newell and J. C. Wkitehead, Finite Bandwidth, Finite Amplitude Convection, [J]. Fluid Mech. 38 (1969) , 279-303．
[4] L. A. Segel, Distant Side-Walls Case slow Amplitude Modulation of Cellular Convection, [J]. Fluid Mech. 38 (1969) , 203-224．
[5] S. Kogelman and R. C. Diprima, Stability of spatially periodic supercritical flow in hydrodynamics, [J]. Phys. Fluid Vol. 13 (1970) 1-23．
[6] A. C. Newell and J. C. Wkitehead, Review of the Finite Bandwidth concept, in Proceedings International Union of Theoretical and Applied Mechanics, [C]. Symposium on Instablity of Continuus System (1969) , H.Leipholz ed., Springer-Verlin 1971, 279-303．
[7] R. C. DiPrima, W. Eckhaus and L. A. Segel, Non-linear Wave-Number Interaction in Near-Critical Two-Dimensional Flows, [J]. Fluid Mech. 49 (1971) , 705-744．
[8] Y. Kuramoto, Diffusion induced chaos in reaction system, [J]. Progr. Theor, Phys,. Suppl. Vol.64 (1978) 364-
[9] A. C. Newell and J. V. Moloney, [M]. Nonlinear Optics, Addison-Wesley Publishing Company. 1992．
[10] J. M. Ghidaglia and B. Heron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, [J]. Physica D 28 (1987) , 282-304．
[11] C. Doering, J. D. Gibbon, D. holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation. [J]. Nonlinearity, Vol. 1, (1988) , 279-309．
[12] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial equation, [J]. Physica D, 41 (1990) , 232-252．
[13] J. D. Rodriguez and L. Sirovich, Low-dimensional dynamics for the complex Ginzburg-Landau equation, [J]. Physica D 43 (1990) 77-86．
[14] Michele Bartuccelli, Petre Constantin, Charles R.Doering, John D.Gibbon abd Magnus Gisselfalt, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, [J]. Physica D 44 (1990) , 421-444．
[15] J. Ginibre, G. velo, The cauchy problem in local spaces for the complex Ginzburg-Landau equation, I.Compactness methods, [J]. Physica D 95 (1996) , 191-228．
[16] L. Sirovich, J. D. Rodriguez, Two boundary value problem dor the Ginzburg-Landau equation, [J]. Physica D 43 (1990) ,63-76．
[17] C. Bu, On the Cauchy problem for the 1 + 2 complex Ginzburg-Landau equation, [J]. Austral math. Soc .Ser. B, 36(1994) , 313-324．
[18] Charles R. Doering, John D.Gibbon and C.David Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, [J]. Physica D 71 (1994) , 285-318．
[19] C.David Levermore and Marcel Oliver, The complex Ginzburg-Landau equation as a model problem. [A]. Lectures in applied Mathematics Volume 31, 1996． Americam Mathematical Society.
[20] Michele Bartuccelli, John D.Gibbon abd Marcel Oliver, Length scale in solutions of the complex Ginzburg-Landau equation, [J]. Physica D 89 (1996) , 267-286．
[21] Alexander Mielke and Guido Schneider, Attractors for modulation equations on unbounded domains-existencce and comparision, [J]. Nonlinearity 8 (1995) , 743-768．
[22] Alexander Mielke, The complex Ginzburg-Landau equation on large and unbounded do-mains:sharper bounds and attractors, [J]. Nonlinearity 10 (1997) , 199-222．
[23] Y. Yang, Global spatially-periodic solutions to the Ginzburg-Landau equation, [J]. Proc. R.Soc. Edinburgh, S. Vol. 110 (1988) 263-273．
[24] J. A. Glazier and P. Kolodner, Interaction of nonlinear pluses in convection in binary fluids, [J]. Phys. Rev. Lett. Vol. 63 (1989) 2801-2804．
[25] H. R. Brand, R. J. Deissler. Interaction of localized for Subcritical Bifurcations. [J]. Phys. Rev. Lett., 1989, 63:2801-2804．
[26] Jinqiao Duan, Philip Holmes and Edriss S Titi, Global existence theory for a Generalized Ginzburg-Landau equation, [J]. Nonlinearity 5 (1992) 1303-1314．
[27] Jinqiao Duan and Philip Holmes, On the Cauchy problem of a Generalized Ginzburg-Landau equation, [J]. Nonlinear Analysis, Theory, Methods & applications, Vol.22． No.8． pp1033-1040,1994．
[28] Guo Boling and Gao Hongjun, Finite dimensional behavior of generalized Ginzburg-Landau equation, [J]. Progress in natural science, 4 (1994) ,423-434．
[29] A.Doelman, On the nonlinear evolution of patterns (modulation equations and theia solution), [D]. Ph.D. Thesis, University of Utrech, the Netherland, 1990．
[30] Jinqiao Duan, Edriss S Titi and Philip Holmes, Regularity ,approximation and asymptotic dynamics for a Generalized Ginzburg-Landau equation, [J]. Nonlinearity 6 (1993) 915-933．
[31] Gao Hongjun and Guo Boling, On the number of determining nodes for the generalized Ginzburg-Landau equation, [J]. J. Partial Diff. Eqs. 10 (1997) , 97-106．
[32] Gao Hongjun and Guo Boling, Number of determining nodes generalized Ginzburg-Landau equation, [J]. Progress in natural science Vol. 6 No. 1 February 1996．
[33] Hidetsugu Sakguchi and Boris A.Malomed, Instablities and splitting of pulses in coupled Ginzburg-Landau equation, [J]. Physica D 154 (2001) 229-239．
[34] B. Guo and X.wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions, [J]. Phys. 89D, (1995) , 83-99．
[35] Guo Boling and Bixiang Wang, Exponential attractors for the generalized Ginzburg-Landau equation, [J]. Acta Mathemactica sinica, English Series July 2000 Vol. 16 No. 3 p. 513-526．
[36] Cao Zhengchao, Guo Boling and Wang Bixiang Global existence theory for the two-dimensional derivative Ginzburg-Landau equations, [J]. Chinese Science Bulletin Vol. 43 No.5 March (1998) .
[37] Rong yuan, Periodic Solution, Almost Periodic Solution, and Regularity of Attractors for the Generalization on Ginzburg-Landau equations, [D]. Ph.D. Thesis, 北京，中国工程物理研究院.
[38] Hongjun Gao and Jinqiao Duan, On the initial-value problem for the generalized two-dimensional Ginzburg-Landau eqyation, [J]. Journal of Mathematical Analysis and applications 216 (1997) , 536-548．
[39] Hongjin Gao, Guoguang Lin and Jinqiao Duan, Maximal attractor for the Generalized 2D Ginzburg-Landau equation.
[40] Yongshong Li and Guo Boling, Global existence of solution to the derivative 2D Ginzburg-Landau
eqyation, [J]. Journal of mathematical analysis and application 249, 412-432．
[41] Yongshong Li and Guo Boling, Global attractor for the derivative 2D Ginzburg-Landau equation in an unbounded domain.
[42] R. Temam, Infinite Dimensional Dynamical systems in mechanics and Physics, [M]. 1988, Springer-Verlag.
[43] P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbluence. [J]. Physica D 30 (1988) , 284-296．
[44] P.Constantin, C. Foias and R. Temam, Attractor respresenting turbulent flows, [M]. Mem. Amer.Math. Soc., 53:314 (1985) .
[45] J. k. Hale, Asmptotic behavior of disspative systems, [M]. Math. Surveys and Monographys, Vol. 25 Mem. Amer.Math. Soc. Providence, 1988．
[46] A.V.Babin and M.I.Vishik, Attractor of partial differential evlutions equations and estimate of the dimension, [M]. Russian Math, surveys, Vol.38 (1983) 151-213．
[47] D. Henry, Geometric Theory of Semilinear Parabolic Equation,[M]. Spring-Verlag, Berlin, 1981．
[48] A. Pazy,Semigroups of linear Operators and applications to Partial Differential equation, [M]. Spring-Verlag, Berlin, 1983．
[49] I. Segal, Nonlinear semigroups, [M].Ann. math. , 78, 339-364, 1963．
[50] A. Eden C. Foias B. Nicolaenko and Z. S. She, Exponential attractors and their relevance to fluid dynamics systems, [J]. Physica D 63 (1993) , 350-360．
[51] A. Eden C. Foias B. Nicolaenko and R. Temam, Exponential attractors for Dissipative Evolution. [M]. Reserch in applied Mathematics, Vol.37, John Wiley, New York,1994．
[52] A. V. Babin and Vishik. Attractors of partial differential evolution equations in a unbounded domain. [J]. Pro. Roy. Soc. Edinburgh(Series A), 1990, 116:221-243．
[53] Frederic Abergel, Existence and finite dimensionality of the global attractor for evlutions on unbounded domains, [J]. Journal of Differential Equations 83, 85-108,(1990) .
[54] P.Constantin, A construction of inertial manifolds fors, [J]. Contemporary Math. Vol. 99 (1989) 27-62．
[55] P.Constantin and C. Foias, Navier-Stoker equations, [M]. Univ. of Chicago Press, Chicago, 1988．
[56] C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, [J]. J. Math, pures et appl. 58 1979, p 339-368．
[57] J. M. Ghidaglia and R. Temam, Attractor for damped Nonlinear hyperbolic quations, [J]. J. Math. pures et appl. 66 1987, p. 273-319．
[58] A. Debussche and R. Temam, Convergent families of approximate inertial manifolds, [J]. J. Math. pures appl. 73 1994, p. 489-522．
[59] A. Eden A. J. Milani and B. Nicolaenko, Finite Dimensional Exponential Attractors for semilinear Wave Equations with Damping, [J]. Journal of Mathematical Analysis and applications 169 (1992) , 408-419．
[60] C. Foias and R. Temma, Gevrey class regularity for the solutions of the Navier-Stokes equations, [J]. Function anal. 1989, 87, p. 359-369．
[61] M. I. Vishik, Asymptotic behaviour of solution of evolutionary equations,[M]. Cambridge University Press.
[62] A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, [J]. Math.
pures et appl. 62 1983, p. 441-491．
[63] A. V. Babin, The Attractor of a Navier-Stokes System in an Unbounded channel-like Domains, [J]. Journal of Dynamics and Differential Equations, Vol. 4 No. 4 1992,555-584．
[64] B. Nicolaenko B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equation:nonlinear stability and attractor, [J]. Physica 16 D (1985) , 155-183．
[65] F.Bethuel, Breis and F.Helein, Ginzburg-Landau vortices, [M]. Birkhauser, Boston,1994．
[66] Fang Hua Lin, Some Dynamical properties of Ginzburg-Landau vortices, [J]. Comm. Pure and Applied Math., Vol. XLIX, 1996, 323-359．
[67] Fang Hua Lin, A remark on the previous paper" some dynamical properties of Ginzburg-Landau vortices", [J]. Comm. Pure and Applied Math., Vol. XLIX, 1996, 361-364．
[68] Dai Zhengde and Yang Hanchun, Inertial Fractal set and Fractal structure of Attracyors for the Ginzburg-Landau equation, [J]. Acta Mathemactica Applicatae sinica, Oct.,1998,Vol.14 No.4．
[69] Keite Promislow, Time analyticity and Gevrey regularity for solutions of a class dissipative partial differential equations, [J]. Nonlinear Analysis, Theory & application Vol.16, No. 11, pp. 959-980, 1991．
[70] Ricardo Rose, The global attractors for the 2D Navier-Stokes flow on some unbounded domains, [J]. Nonlinear Analysis, Theory & application Vol. 32 No. 1, pp.71-85, 1998．
[71] A. Carpio, Existence of global solutions to some nonlinear dissipative wave equations, [J]. Math. pures et appl. 73 1994, p. 471-488．
[72] Eduard Feireisl, Bounded,Locally compact global attractors for semilinear damped wave equations on RN, [J]. Differential and Integral Equations Vol. 9 No. 5 1996, 1147-1156．
[73] Zhou Yulin and Guonboling, Global solutions and their large-time behavior of Cauchy problem for equations of deep water type, [J]. Partial diff. Eqs. 9 (1996) , 1-41．
[74] Guo Boling and Miao Changxing, Well-posedness of cauchy problem for coupled system of long-short wave equations, [J]. Partial Diff. Eqs. 11 (1998) , 83-96．
[75] Guo Boling and Wang Bixiang, Attractor for long-short wave equations, [J]. Partial Diff. Eqs. 11 (1998) , 361-383．
[76] Guo Boling and Wang Bixiang, The global solution and its long time behavior for a class of generalized LS equation, [J]. Progress in natural science Vol. 6 No. 5 October 1996．
[77] I. Flahaut, Attractors for the disspative Zakharov system, [J]. Nonlinear Analysis, Theory & application Vol. 16 No. 7/8, pp.599-633,1991．
[78] O. Goubet and I. Moise, Attractors for the disspative Zakharov system, [J]. Nonlinear Analysis, Theory & application Vol. 31 No. 7, pp. 823-847, 1998． (regularity)
[79] Guo Boling and Li Yongsheng, Attractor for the dissipative generalized Klein-Gordon-Schrodinger equation, [J]. Partial Diff. Eqs. 11 (1998) , 260-272．
[80] Guo Boling and Yang Linge The global solution and asymptotic behaviors for one class of system of nonlinear evolution equations, [J]. Partial Diff. Eqs. 10 (1997) , 232-246．
[81] V. V. Chepyzhov, Attractor for damped Non-autonomous dynamical system and their dimension, [J]. Math, pures et appl. 73 1994, p. 279-333．
[82] Shen Lingjun and Zhang Linghai, On the solution of the coupled nonlinear parabolic equayion, [J].Partial diff. Eqs. 9 (1996) 71-83．
[83] W. E. Fitzgibbon and M. E. Parrott, Convergence of Singularly Perturbed Hodgkin-Huxley system,
[J]. Nonlinear Analysis, Theory & application Vol. 22 No. 3, pp.363-379,1994．
[84] Jean-Michel Ghidaglia, Weakly damped forced Korteweg-de Vries equations behavior as a finite dimensional dynamical system in the long time, [J]. Journal of Differential Equations 74, 369-390, (1988) .
[85] Jean-Michel Ghidaglia, A note on the strong convergence towards attractors of damped forced KDV equations, [J]. Partial diff. Eqs. 105 (1994) 356-359．
[86] E.Feireisl Ph. Laurencot and P. Simondon, Global attractor for Degenerate Parabolic equation on unbounded domains, [J]. Journal of Differential Equations 129, 239-261,(1996) .
[87] Anatoli Babin and Basil, Exponential attractors of reation-diffusion systems in an unbounded domain, [J]. Journal of Dynamics and Differential Equation, Vol. 7 No. 4 1995．
[88] George R. Sell, Global Attractors for the There-Dimensional Navier-Stokes Equation, [J]. Journal of Dynamics and Differential Equations Vol. 8 No. 1 1996．
[89] Josef Malek and Jindrich Necas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, [J]. Journal of Differential Equations 127, 498-518 (1996) .
[90] Fengxin Chen Boling Guo and Ping Wang, Long time behavior of strongly damped nonlinear wave equations, [J]. Journal of Differential Equations 147, 231-241 (1998) ,
[91] Paul Massatt, Limiting behavior for strongly damped nonlinear wave equations. [J]. Journal of Differential Equations 48 334-349,(1983) .
[92] Tosio Kato, Blow up of solution of some nonlinear hyperbolic equations, [J], Comm.pure app. Math. 33 4 (1980) .
[93] Eduard Feireisl, Long-time behavior and convergence for semilinear wave equations on RN, [J]. Journal of Dynamics and Differential Equations Vol. 9 No. 1 1997．
[94] A.Eden C.Foias and V. Kalantarov, A remark on two contractions of exponential attractora for α-contractions, [J]. Journal of Dynamics and Differential Equations Vol. 10 No. 10． 1998．
[95] Fandreu J. M. Mazon F. Simondon and J. toledo, Attractor for a degenerate nonlinear diffusion problem with nonlinear boundary condition, [J]. Journal of Dynamics and Differential Equations Vol. 10 No. 3． 1998．
[96] Reinhard Redlinger, Existence of the Global Attractor for a Strongly Coupled Parabolic System Arising in Population Dynamics, [J]. Journal of Differential Equations 118 (1995) , 219-252．
[97] K. Ito and Y. Yan, Viscous Scalar Conservation Law with Nonlinear Flux Feedback and Global Attractors, [J]. Journal of Mathematical Analysis and applications 227 (1998) , 271-299．
[98] Jan W.Cholewa an Tomasz Dlotko, Global attractor for sectorial evolutionary equation, [J]. Journal of Differential Equations 125 (1996) , 27-39．
[99] Le Dung, Global attractors and steady state solutions for a class of reaction-diffusion systems, [J]. Journal of Differential Equations 147 (1998) , 1-29．
[100] Jiahong Wu, The inviscid limit of the complex Ginzburg-Landau eqyation, [J]. Journal of Differential Equations 142 (1998) , 413-433．
[101] Kening Lu and Xing-Bin Pan, Ginzburg-Landau equation with DeGennes boundary condition, [J]. Journal of Differential Equations 129 (1996) , 136-165．
[102] Howard A. Levine and Sang Ro Park Global existence and Global Nonexistence of Solutions of the Cauchy Problem for a Nonlinearly Damped Wave Equation, [J]. Journal of Mathematical Analysis and applications 228 (1998) , 181-205．
[103] Jack K. Hale and Jurgen Scheurle, Smoothness of boundedsolutions of nonlinear evolution equations, [J]. Journal of Differential Equations 56, 142-163．
[104] Li Kaitai anf Zhang Quande Exisstence and nonexistence of global solutions for the equation of dislocation of crystals. [J]. Journal of Differential Equations 146 (1998) , 5-21．
[105] David W. Bange, An exience theorem for periodic solution of a nonlinear parabolic boundary value problem, [J]. Journal of Differential Equations 24 (1977) , 426-436．
[106] Fred B. Weissler, Local existence and nonexistence for semilinear parabolic equation in Lp, [J]. Indiana university Journal Vol. 29 No. 1 (1980) .
[107] Jack K. Hall and Genevieve Raugel, Upper semicontinuity of the attractor for the singularity perturbed Hodgkin-Huxley equation, [J]. Journal of Differential Equations 73 (1988) ,197-214．
[108] C. Galusinski, Existence and continuity of uniform exponential attractor of the singularity perturbed Hodgkin-Huxley system, [J]. Journal of Differential Equations 144 (1998) , 99-169．
[109] W. E. Frrzgibbon and M. E. Parrott and Y. You, Finite dimensionality and upper semicontinuity of the global attractor of singularity perturbed Hodgkin-Huxley system, [J]. Journal of Differential Equations 129 (1996) , 193-237．
[110] Huai-Yu Jian Xiao-Ping Wang and Din-Yu Hsieh, The Global Attractor of Dissipative Nonlinear Evolution System, [J]. Journal of Mathematical Analysis and applications 238 (1999) , 124-142．
[111] Desheng Li and Chenkui Zhong, Global Attractor for Cahn-Hilliard System with Fast Growing Nonlinearity, [J]. Journal of Mathematical Analysis and applications 149 (1998) , 191-210．
[112] Tomasz Dlotko, Global Attractor for Cahn-Hilliard in H2 and H3, [J]. Journal of Differential Equations 113 (1994) , 381-393．
[113] Ioana Moise Ricardo Rose and Xiaoming Wang, Attractor for non-compact semigroups via energy equations, [J]. Nonlinearity 11 (1998) ,1369-1393．
[114] Zhi-Min Chen, Global Solutions of the Navier-Stokes equations in Thin There-Dimensional Domains, [J]. Journal of Mathematical Analysis and applications 233 (1999) , 681-697．
[115] Cheng He, Weighted Estimates for Nonstationary Navier-Stokes equations, [J]. Journal of Differential Equations. 148 (1998) , 422-444．
[116] Hi Jun Choe, Boundary Regularity of Weak Solutions of the Navier-Stokes equations, [J]. Journal of Differential Equations 149 (1998) , 211-247．
[117] Chang-Shou Lin, Interpolation inequalities with weights, [J]. Comm. in partial differential equations 11 (14) (1986) , 1515-1538．
[118] I.Witt, Existence and continuity of the attractor for a singularly perturbed Hodgkin-Huxley system, [J]. Journal of Dynamics Differential Equations Vol. 7 No. 4 1995, 591-639．
[119] Shui-Nee Chow, Kening Lu and George R.Sell, Smoothness of Inertial Manifolds, [J]. Journal of Mathematical Analysis and applications 109 (1992) , 283-312．
[120] Jan W. Cholewa and Tomasz Dlotko, Cauchy Problem with Subcritical Nonlinearity, [J]. Journal of Mathematical Analysis and applications 210 (1997) , 531-548．
[121] Piotr Biler Tadahisa and Wojbor, Fractal Burger Equatiions, [J]. Journal of Differential Equations 148 (1998) , 9-46．
[122] J. Colliander, Wellposendness for Zakharov Systems with Generalized Nonlinearity, [J]. Journal of Differential Equations 148 (1998) , 351-363．
[123] Akihiti Unai, Global C1-solution of time-dependent complex Ginzburg-Landau equations, [J].
Nonlinear Analysis 46 (2001) 329-334.
[124] Philppe Bechouche, Anshar Jünge, Inviscid Limits of Ginzburg-Landau Equation, [J]. Communications in Mathematical Physics 214 (2000), 201-226.
[125] Ling Hsiao, Global Smooth Solutions to the Spatially Periodic Cauchy Problem for Dissipative Nonlinear Evolution Equations, [J]. Journal of Mathematical Analysis and applications 213 (1997), 262-274.
[126] Guillermo Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm, [J]. Nonlinear Analysis 46(2001) 309-327.
[127] Mengxing He, Global Existence and Stability of Solutions for Reaction Diffusion Functional Differential equations, [J]. Journal of Mathematical Analysis and Applications 199, 842-858 (1996).
[128] Linda Sundbye, Global Exientence for the Dirichlet problem for the Viscous Shallow water Equations, [J]. Journal of Mathematical Analysis and Applications 202, 236-258(1996).
[129] Sergiu Klainerman, Global Existence for Nonlinear Wave Equations, [J]. Comm. pure appl. Math. 33 (1980), 43-101.
[130] Carlos E. Kenig, Gustavo Ponce and Luis Vega, Global Solutions for the KDV equation with Unbounded Data, [J]. Journal of Differential Equations 139 (1997), 339-364.
[131] Giovanni P.Galde, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, [M]. Volume 1 Linearized Steady Problem.
[132] J.W.Cholewa & T.Diotko, Global Attractors in Abstract Parabolic Problem, [M]. Cambridge University Press.
[133] 郭柏灵著，[M]．非线性演化方程，上海科技教育出版社，1996．
[134] 郭柏灵著，[M]．无穷维动力系统(上，下册)，国防工业出版社，2000．
[135] 郭柏灵著，[M]．粘性消去法和差分格式的粘性，科学出版社，1993．
[136] 郭柏灵，黄海洋，蒋幕容等著，[M]．金兹堡-郎道方程，科学出版社，2002．
[137] [法]Lions，J．L著，郭柏灵等译，[M]．非线性边值问题的一些解法，1992．
[138] [法]Lions，J．L[意]E．Magenes著，[M]．非齐次边值问题及其应用，1992
[2] E. M. Lifishitz and L. P. Pitaevstii, [M]. Statistical physics, Vol. 2 Oxford; Pergamon, 1980．
[3] A. C. Newell and J. C. Wkitehead, Finite Bandwidth, Finite Amplitude Convection, [J]. Fluid Mech. 38 (1969) , 279-303．
[4] L. A. Segel, Distant Side-Walls Case slow Amplitude Modulation of Cellular Convection, [J]. Fluid Mech. 38 (1969) , 203-224．
[5] S. Kogelman and R. C. Diprima, Stability of spatially periodic supercritical flow in hydrodynamics, [J]. Phys. Fluid Vol. 13 (1970) 1-23．
[6] A. C. Newell and J. C. Wkitehead, Review of the Finite Bandwidth concept, in Proceedings International Union of Theoretical and Applied Mechanics, [C]. Symposium on Instablity of Continuus System (1969) , H.Leipholz ed., Springer-Verlin 1971, 279-303．
[7] R. C. DiPrima, W. Eckhaus and L. A. Segel, Non-linear Wave-Number Interaction in Near-Critical Two-Dimensional Flows, [J]. Fluid Mech. 49 (1971) , 705-744．
[8] Y. Kuramoto, Diffusion induced chaos in reaction system, [J]. Progr. Theor, Phys,. Suppl. Vol.64 (1978) 364-
[9] A. C. Newell and J. V. Moloney, [M]. Nonlinear Optics, Addison-Wesley Publishing Company. 1992．
[10] J. M. Ghidaglia and B. Heron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, [J]. Physica D 28 (1987) , 282-304．
[11] C. Doering, J. D. Gibbon, D. holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation. [J]. Nonlinearity, Vol. 1, (1988) , 279-309．
[12] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial equation, [J]. Physica D, 41 (1990) , 232-252．
[13] J. D. Rodriguez and L. Sirovich, Low-dimensional dynamics for the complex Ginzburg-Landau equation, [J]. Physica D 43 (1990) 77-86．
[14] Michele Bartuccelli, Petre Constantin, Charles R.Doering, John D.Gibbon abd Magnus Gisselfalt, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, [J]. Physica D 44 (1990) , 421-444．
[15] J. Ginibre, G. velo, The cauchy problem in local spaces for the complex Ginzburg-Landau equation, I.Compactness methods, [J]. Physica D 95 (1996) , 191-228．
[16] L. Sirovich, J. D. Rodriguez, Two boundary value problem dor the Ginzburg-Landau equation, [J]. Physica D 43 (1990) ,63-76．
[17] C. Bu, On the Cauchy problem for the 1 + 2 complex Ginzburg-Landau equation, [J]. Austral math. Soc .Ser. B, 36(1994) , 313-324．
[18] Charles R. Doering, John D.Gibbon and C.David Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, [J]. Physica D 71 (1994) , 285-318．
[19] C.David Levermore and Marcel Oliver, The complex Ginzburg-Landau equation as a model problem. [A]. Lectures in applied Mathematics Volume 31, 1996． Americam Mathematical Society.
[20] Michele Bartuccelli, John D.Gibbon abd Marcel Oliver, Length scale in solutions of the complex Ginzburg-Landau equation, [J]. Physica D 89 (1996) , 267-286．
[21] Alexander Mielke and Guido Schneider, Attractors for modulation equations on unbounded domains-existencce and comparision, [J]. Nonlinearity 8 (1995) , 743-768．
[22] Alexander Mielke, The complex Ginzburg-Landau equation on large and unbounded do-mains:sharper bounds and attractors, [J]. Nonlinearity 10 (1997) , 199-222．
[23] Y. Yang, Global spatially-periodic solutions to the Ginzburg-Landau equation, [J]. Proc. R.Soc. Edinburgh, S. Vol. 110 (1988) 263-273．
[24] J. A. Glazier and P. Kolodner, Interaction of nonlinear pluses in convection in binary fluids, [J]. Phys. Rev. Lett. Vol. 63 (1989) 2801-2804．
[25] H. R. Brand, R. J. Deissler. Interaction of localized for Subcritical Bifurcations. [J]. Phys. Rev. Lett., 1989, 63:2801-2804．
[26] Jinqiao Duan, Philip Holmes and Edriss S Titi, Global existence theory for a Generalized Ginzburg-Landau equation, [J]. Nonlinearity 5 (1992) 1303-1314．
[27] Jinqiao Duan and Philip Holmes, On the Cauchy problem of a Generalized Ginzburg-Landau equation, [J]. Nonlinear Analysis, Theory, Methods & applications, Vol.22． No.8． pp1033-1040,1994．
[28] Guo Boling and Gao Hongjun, Finite dimensional behavior of generalized Ginzburg-Landau equation, [J]. Progress in natural science, 4 (1994) ,423-434．
[29] A.Doelman, On the nonlinear evolution of patterns (modulation equations and theia solution), [D]. Ph.D. Thesis, University of Utrech, the Netherland, 1990．
[30] Jinqiao Duan, Edriss S Titi and Philip Holmes, Regularity ,approximation and asymptotic dynamics for a Generalized Ginzburg-Landau equation, [J]. Nonlinearity 6 (1993) 915-933．
[31] Gao Hongjun and Guo Boling, On the number of determining nodes for the generalized Ginzburg-Landau equation, [J]. J. Partial Diff. Eqs. 10 (1997) , 97-106．
[32] Gao Hongjun and Guo Boling, Number of determining nodes generalized Ginzburg-Landau equation, [J]. Progress in natural science Vol. 6 No. 1 February 1996．
[33] Hidetsugu Sakguchi and Boris A.Malomed, Instablities and splitting of pulses in coupled Ginzburg-Landau equation, [J]. Physica D 154 (2001) 229-239．
[34] B. Guo and X.wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions, [J]. Phys. 89D, (1995) , 83-99．
[35] Guo Boling and Bixiang Wang, Exponential attractors for the generalized Ginzburg-Landau equation, [J]. Acta Mathemactica sinica, English Series July 2000 Vol. 16 No. 3 p. 513-526．
[36] Cao Zhengchao, Guo Boling and Wang Bixiang Global existence theory for the two-dimensional derivative Ginzburg-Landau equations, [J]. Chinese Science Bulletin Vol. 43 No.5 March (1998) .
[37] Rong yuan, Periodic Solution, Almost Periodic Solution, and Regularity of Attractors for the Generalization on Ginzburg-Landau equations, [D]. Ph.D. Thesis, 北京，中国工程物理研究院.
[38] Hongjun Gao and Jinqiao Duan, On the initial-value problem for the generalized two-dimensional Ginzburg-Landau eqyation, [J]. Journal of Mathematical Analysis and applications 216 (1997) , 536-548．
[39] Hongjin Gao, Guoguang Lin and Jinqiao Duan, Maximal attractor for the Generalized 2D Ginzburg-Landau equation.
[40] Yongshong Li and Guo Boling, Global existence of solution to the derivative 2D Ginzburg-Landau
eqyation, [J]. Journal of mathematical analysis and application 249, 412-432．
[41] Yongshong Li and Guo Boling, Global attractor for the derivative 2D Ginzburg-Landau equation in an unbounded domain.
[42] R. Temam, Infinite Dimensional Dynamical systems in mechanics and Physics, [M]. 1988, Springer-Verlag.
[43] P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbluence. [J]. Physica D 30 (1988) , 284-296．
[44] P.Constantin, C. Foias and R. Temam, Attractor respresenting turbulent flows, [M]. Mem. Amer.Math. Soc., 53:314 (1985) .
[45] J. k. Hale, Asmptotic behavior of disspative systems, [M]. Math. Surveys and Monographys, Vol. 25 Mem. Amer.Math. Soc. Providence, 1988．
[46] A.V.Babin and M.I.Vishik, Attractor of partial differential evlutions equations and estimate of the dimension, [M]. Russian Math, surveys, Vol.38 (1983) 151-213．
[47] D. Henry, Geometric Theory of Semilinear Parabolic Equation,[M]. Spring-Verlag, Berlin, 1981．
[48] A. Pazy,Semigroups of linear Operators and applications to Partial Differential equation, [M]. Spring-Verlag, Berlin, 1983．
[49] I. Segal, Nonlinear semigroups, [M].Ann. math. , 78, 339-364, 1963．
[50] A. Eden C. Foias B. Nicolaenko and Z. S. She, Exponential attractors and their relevance to fluid dynamics systems, [J]. Physica D 63 (1993) , 350-360．
[51] A. Eden C. Foias B. Nicolaenko and R. Temam, Exponential attractors for Dissipative Evolution. [M]. Reserch in applied Mathematics, Vol.37, John Wiley, New York,1994．
[52] A. V. Babin and Vishik. Attractors of partial differential evolution equations in a unbounded domain. [J]. Pro. Roy. Soc. Edinburgh(Series A), 1990, 116:221-243．
[53] Frederic Abergel, Existence and finite dimensionality of the global attractor for evlutions on unbounded domains, [J]. Journal of Differential Equations 83, 85-108,(1990) .
[54] P.Constantin, A construction of inertial manifolds fors, [J]. Contemporary Math. Vol. 99 (1989) 27-62．
[55] P.Constantin and C. Foias, Navier-Stoker equations, [M]. Univ. of Chicago Press, Chicago, 1988．
[56] C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, [J]. J. Math, pures et appl. 58 1979, p 339-368．
[57] J. M. Ghidaglia and R. Temam, Attractor for damped Nonlinear hyperbolic quations, [J]. J. Math. pures et appl. 66 1987, p. 273-319．
[58] A. Debussche and R. Temam, Convergent families of approximate inertial manifolds, [J]. J. Math. pures appl. 73 1994, p. 489-522．
[59] A. Eden A. J. Milani and B. Nicolaenko, Finite Dimensional Exponential Attractors for semilinear Wave Equations with Damping, [J]. Journal of Mathematical Analysis and applications 169 (1992) , 408-419．
[60] C. Foias and R. Temma, Gevrey class regularity for the solutions of the Navier-Stokes equations, [J]. Function anal. 1989, 87, p. 359-369．
[61] M. I. Vishik, Asymptotic behaviour of solution of evolutionary equations,[M]. Cambridge University Press.
[62] A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, [J]. Math.
pures et appl. 62 1983, p. 441-491．
[63] A. V. Babin, The Attractor of a Navier-Stokes System in an Unbounded channel-like Domains, [J]. Journal of Dynamics and Differential Equations, Vol. 4 No. 4 1992,555-584．
[64] B. Nicolaenko B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equation:nonlinear stability and attractor, [J]. Physica 16 D (1985) , 155-183．
[65] F.Bethuel, Breis and F.Helein, Ginzburg-Landau vortices, [M]. Birkhauser, Boston,1994．
[66] Fang Hua Lin, Some Dynamical properties of Ginzburg-Landau vortices, [J]. Comm. Pure and Applied Math., Vol. XLIX, 1996, 323-359．
[67] Fang Hua Lin, A remark on the previous paper" some dynamical properties of Ginzburg-Landau vortices", [J]. Comm. Pure and Applied Math., Vol. XLIX, 1996, 361-364．
[68] Dai Zhengde and Yang Hanchun, Inertial Fractal set and Fractal structure of Attracyors for the Ginzburg-Landau equation, [J]. Acta Mathemactica Applicatae sinica, Oct.,1998,Vol.14 No.4．
[69] Keite Promislow, Time analyticity and Gevrey regularity for solutions of a class dissipative partial differential equations, [J]. Nonlinear Analysis, Theory & application Vol.16, No. 11, pp. 959-980, 1991．
[70] Ricardo Rose, The global attractors for the 2D Navier-Stokes flow on some unbounded domains, [J]. Nonlinear Analysis, Theory & application Vol. 32 No. 1, pp.71-85, 1998．
[71] A. Carpio, Existence of global solutions to some nonlinear dissipative wave equations, [J]. Math. pures et appl. 73 1994, p. 471-488．
[72] Eduard Feireisl, Bounded,Locally compact global attractors for semilinear damped wave equations on RN, [J]. Differential and Integral Equations Vol. 9 No. 5 1996, 1147-1156．
[73] Zhou Yulin and Guonboling, Global solutions and their large-time behavior of Cauchy problem for equations of deep water type, [J]. Partial diff. Eqs. 9 (1996) , 1-41．
[74] Guo Boling and Miao Changxing, Well-posedness of cauchy problem for coupled system of long-short wave equations, [J]. Partial Diff. Eqs. 11 (1998) , 83-96．
[75] Guo Boling and Wang Bixiang, Attractor for long-short wave equations, [J]. Partial Diff. Eqs. 11 (1998) , 361-383．
[76] Guo Boling and Wang Bixiang, The global solution and its long time behavior for a class of generalized LS equation, [J]. Progress in natural science Vol. 6 No. 5 October 1996．
[77] I. Flahaut, Attractors for the disspative Zakharov system, [J]. Nonlinear Analysis, Theory & application Vol. 16 No. 7/8, pp.599-633,1991．
[78] O. Goubet and I. Moise, Attractors for the disspative Zakharov system, [J]. Nonlinear Analysis, Theory & application Vol. 31 No. 7, pp. 823-847, 1998． (regularity)
[79] Guo Boling and Li Yongsheng, Attractor for the dissipative generalized Klein-Gordon-Schrodinger equation, [J]. Partial Diff. Eqs. 11 (1998) , 260-272．
[80] Guo Boling and Yang Linge The global solution and asymptotic behaviors for one class of system of nonlinear evolution equations, [J]. Partial Diff. Eqs. 10 (1997) , 232-246．
[81] V. V. Chepyzhov, Attractor for damped Non-autonomous dynamical system and their dimension, [J]. Math, pures et appl. 73 1994, p. 279-333．
[82] Shen Lingjun and Zhang Linghai, On the solution of the coupled nonlinear parabolic equayion, [J].Partial diff. Eqs. 9 (1996) 71-83．
[83] W. E. Fitzgibbon and M. E. Parrott, Convergence of Singularly Perturbed Hodgkin-Huxley system,
[J]. Nonlinear Analysis, Theory & application Vol. 22 No. 3, pp.363-379,1994．
[84] Jean-Michel Ghidaglia, Weakly damped forced Korteweg-de Vries equations behavior as a finite dimensional dynamical system in the long time, [J]. Journal of Differential Equations 74, 369-390, (1988) .
[85] Jean-Michel Ghidaglia, A note on the strong convergence towards attractors of damped forced KDV equations, [J]. Partial diff. Eqs. 105 (1994) 356-359．
[86] E.Feireisl Ph. Laurencot and P. Simondon, Global attractor for Degenerate Parabolic equation on unbounded domains, [J]. Journal of Differential Equations 129, 239-261,(1996) .
[87] Anatoli Babin and Basil, Exponential attractors of reation-diffusion systems in an unbounded domain, [J]. Journal of Dynamics and Differential Equation, Vol. 7 No. 4 1995．
[88] George R. Sell, Global Attractors for the There-Dimensional Navier-Stokes Equation, [J]. Journal of Dynamics and Differential Equations Vol. 8 No. 1 1996．
[89] Josef Malek and Jindrich Necas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, [J]. Journal of Differential Equations 127, 498-518 (1996) .
[90] Fengxin Chen Boling Guo and Ping Wang, Long time behavior of strongly damped nonlinear wave equations, [J]. Journal of Differential Equations 147, 231-241 (1998) ,
[91] Paul Massatt, Limiting behavior for strongly damped nonlinear wave equations. [J]. Journal of Differential Equations 48 334-349,(1983) .
[92] Tosio Kato, Blow up of solution of some nonlinear hyperbolic equations, [J], Comm.pure app. Math. 33 4 (1980) .
[93] Eduard Feireisl, Long-time behavior and convergence for semilinear wave equations on RN, [J]. Journal of Dynamics and Differential Equations Vol. 9 No. 1 1997．
[94] A.Eden C.Foias and V. Kalantarov, A remark on two contractions of exponential attractora for α-contractions, [J]. Journal of Dynamics and Differential Equations Vol. 10 No. 10． 1998．
[95] Fandreu J. M. Mazon F. Simondon and J. toledo, Attractor for a degenerate nonlinear diffusion problem with nonlinear boundary condition, [J]. Journal of Dynamics and Differential Equations Vol. 10 No. 3． 1998．
[96] Reinhard Redlinger, Existence of the Global Attractor for a Strongly Coupled Parabolic System Arising in Population Dynamics, [J]. Journal of Differential Equations 118 (1995) , 219-252．
[97] K. Ito and Y. Yan, Viscous Scalar Conservation Law with Nonlinear Flux Feedback and Global Attractors, [J]. Journal of Mathematical Analysis and applications 227 (1998) , 271-299．
[98] Jan W.Cholewa an Tomasz Dlotko, Global attractor for sectorial evolutionary equation, [J]. Journal of Differential Equations 125 (1996) , 27-39．
[99] Le Dung, Global attractors and steady state solutions for a class of reaction-diffusion systems, [J]. Journal of Differential Equations 147 (1998) , 1-29．
[100] Jiahong Wu, The inviscid limit of the complex Ginzburg-Landau eqyation, [J]. Journal of Differential Equations 142 (1998) , 413-433．
[101] Kening Lu and Xing-Bin Pan, Ginzburg-Landau equation with DeGennes boundary condition, [J]. Journal of Differential Equations 129 (1996) , 136-165．
[102] Howard A. Levine and Sang Ro Park Global existence and Global Nonexistence of Solutions of the Cauchy Problem for a Nonlinearly Damped Wave Equation, [J]. Journal of Mathematical Analysis and applications 228 (1998) , 181-205．
[103] Jack K. Hale and Jurgen Scheurle, Smoothness of boundedsolutions of nonlinear evolution equations, [J]. Journal of Differential Equations 56, 142-163．
[104] Li Kaitai anf Zhang Quande Exisstence and nonexistence of global solutions for the equation of dislocation of crystals. [J]. Journal of Differential Equations 146 (1998) , 5-21．
[105] David W. Bange, An exience theorem for periodic solution of a nonlinear parabolic boundary value problem, [J]. Journal of Differential Equations 24 (1977) , 426-436．
[106] Fred B. Weissler, Local existence and nonexistence for semilinear parabolic equation in Lp, [J]. Indiana university Journal Vol. 29 No. 1 (1980) .
[107] Jack K. Hall and Genevieve Raugel, Upper semicontinuity of the attractor for the singularity perturbed Hodgkin-Huxley equation, [J]. Journal of Differential Equations 73 (1988) ,197-214．
[108] C. Galusinski, Existence and continuity of uniform exponential attractor of the singularity perturbed Hodgkin-Huxley system, [J]. Journal of Differential Equations 144 (1998) , 99-169．
[109] W. E. Frrzgibbon and M. E. Parrott and Y. You, Finite dimensionality and upper semicontinuity of the global attractor of singularity perturbed Hodgkin-Huxley system, [J]. Journal of Differential Equations 129 (1996) , 193-237．
[110] Huai-Yu Jian Xiao-Ping Wang and Din-Yu Hsieh, The Global Attractor of Dissipative Nonlinear Evolution System, [J]. Journal of Mathematical Analysis and applications 238 (1999) , 124-142．
[111] Desheng Li and Chenkui Zhong, Global Attractor for Cahn-Hilliard System with Fast Growing Nonlinearity, [J]. Journal of Mathematical Analysis and applications 149 (1998) , 191-210．
[112] Tomasz Dlotko, Global Attractor for Cahn-Hilliard in H2 and H3, [J]. Journal of Differential Equations 113 (1994) , 381-393．
[113] Ioana Moise Ricardo Rose and Xiaoming Wang, Attractor for non-compact semigroups via energy equations, [J]. Nonlinearity 11 (1998) ,1369-1393．
[114] Zhi-Min Chen, Global Solutions of the Navier-Stokes equations in Thin There-Dimensional Domains, [J]. Journal of Mathematical Analysis and applications 233 (1999) , 681-697．
[115] Cheng He, Weighted Estimates for Nonstationary Navier-Stokes equations, [J]. Journal of Differential Equations. 148 (1998) , 422-444．
[116] Hi Jun Choe, Boundary Regularity of Weak Solutions of the Navier-Stokes equations, [J]. Journal of Differential Equations 149 (1998) , 211-247．
[117] Chang-Shou Lin, Interpolation inequalities with weights, [J]. Comm. in partial differential equations 11 (14) (1986) , 1515-1538．
[118] I.Witt, Existence and continuity of the attractor for a singularly perturbed Hodgkin-Huxley system, [J]. Journal of Dynamics Differential Equations Vol. 7 No. 4 1995, 591-639．
[119] Shui-Nee Chow, Kening Lu and George R.Sell, Smoothness of Inertial Manifolds, [J]. Journal of Mathematical Analysis and applications 109 (1992) , 283-312．
[120] Jan W. Cholewa and Tomasz Dlotko, Cauchy Problem with Subcritical Nonlinearity, [J]. Journal of Mathematical Analysis and applications 210 (1997) , 531-548．
[121] Piotr Biler Tadahisa and Wojbor, Fractal Burger Equatiions, [J]. Journal of Differential Equations 148 (1998) , 9-46．
[122] J. Colliander, Wellposendness for Zakharov Systems with Generalized Nonlinearity, [J]. Journal of Differential Equations 148 (1998) , 351-363．
[123] Akihiti Unai, Global C1-solution of time-dependent complex Ginzburg-Landau equations, [J].
Nonlinear Analysis 46 (2001) 329-334.
[124] Philppe Bechouche, Anshar Jünge, Inviscid Limits of Ginzburg-Landau Equation, [J]. Communications in Mathematical Physics 214 (2000), 201-226.
[125] Ling Hsiao, Global Smooth Solutions to the Spatially Periodic Cauchy Problem for Dissipative Nonlinear Evolution Equations, [J]. Journal of Mathematical Analysis and applications 213 (1997), 262-274.
[126] Guillermo Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm, [J]. Nonlinear Analysis 46(2001) 309-327.
[127] Mengxing He, Global Existence and Stability of Solutions for Reaction Diffusion Functional Differential equations, [J]. Journal of Mathematical Analysis and Applications 199, 842-858 (1996).
[128] Linda Sundbye, Global Exientence for the Dirichlet problem for the Viscous Shallow water Equations, [J]. Journal of Mathematical Analysis and Applications 202, 236-258(1996).
[129] Sergiu Klainerman, Global Existence for Nonlinear Wave Equations, [J]. Comm. pure appl. Math. 33 (1980), 43-101.
[130] Carlos E. Kenig, Gustavo Ponce and Luis Vega, Global Solutions for the KDV equation with Unbounded Data, [J]. Journal of Differential Equations 139 (1997), 339-364.
[131] Giovanni P.Galde, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, [M]. Volume 1 Linearized Steady Problem.
[132] J.W.Cholewa & T.Diotko, Global Attractors in Abstract Parabolic Problem, [M]. Cambridge University Press.
[133] 郭柏灵著，[M]．非线性演化方程，上海科技教育出版社，1996．
[134] 郭柏灵著，[M]．无穷维动力系统(上，下册)，国防工业出版社，2000．
[135] 郭柏灵著，[M]．粘性消去法和差分格式的粘性，科学出版社，1993．
[136] 郭柏灵，黄海洋，蒋幕容等著，[M]．金兹堡-郎道方程，科学出版社，2002．
[137] [法]Lions，J．L著，郭柏灵等译，[M]．非线性边值问题的一些解法，1992．
[138] [法]Lions，J．L[意]E．Magenes著，[M]．非齐次边值问题及其应用，1992