非线性发展方程（组）整体解及其渐近性态
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摘要
对于数学以及其他自然科学分支(例如物理,力学,材料科学,化学等)中提出的各类非线性发展方程的整体解的存在唯一性以及其大时间渐近性态的研究具有理论上和实际上的重要意义,并有广泛的应用。长期以来,非线性发展方程因其本身重要的应用背景以及非线性带来的数学困难引起了国内外数学家的广泛的研究兴趣。
本学位论文主要是关于非线性发展方程(组)整体解大时间渐近性态的研究。非线性发展方程(组)的整体解的渐近性态,包括整体解当时间趋于无穷大时是否趋于某个平衡态(Equilibrium),以及对应的无限维动力系统是否存在整体吸引子,是非线性发展方程研究的两个基本问题。通常,人们将所考虑的非线性发展方程(组)看成某个Sobolev函数空间中从某个初始数据出发的轨道,那么我们考虑的问题分别为当时间趋于无穷大时从这个函数空间中给定的初始数据出发的单个轨道是否收敛于某个平衡态以及从这个函数空间或者其某个完备的闭子空间中任意有界集出发的一族轨道是否存在整体吸引子?
对于非线性发展方程(组)对应的无限维动力系统的整体吸引子存在性的研究是本文主要的研究兴趣所在。国际上有大量的工作致力于考察由连续介质力学,物理学,材料科学所提出的无限维动力系统的整体吸引子的存在性问题。例如,热力学及生物数学中提出的反应-扩散方程,材料相变理论中的Cahn-Hilliard方程以及Phase-Field方程组,2维不可压缩Navier-Stokes方程,耗散的波动方程,Ginzburg-Landau方程以及Sine-Gordon方程等。相应有专著S.Zheng[72],Temam[41],Babin& Vishik[5],Hale[22]以及Sell & You[55]等。
本学位论文对由形状记忆合金材料的相变过程而提出的一维非线性热粘弹性方程组进行研究,在证明了方程组整体解存在唯一性的基础上,分别得到了具有Ginzburg-Landau形式且满足铰链支座边界条件的一维非线性热粘弹性方程组所对应的无限维动力系统的整体吸引子的存在性以及具有常值温度边界条件的一维非线性热粘弹性方程组整体弱解当时间趋向于无穷大时对一稳态解的收敛性。这些结果为此前文献中所未见。
具体的,本文的主要内容如下:
第一章绪论,简要回顾问题的背景,研究现状及我们证明的思想和方法。介绍了本文考虑的问题的特点,数学上的困难以及本文工作的创新之处。最后,简要列举了必要的一些基本定理和常用不等式。
第二章,考虑具有Ginzburg-Landau形式的,且满足铰链支座边界条件的一维非线性热粘弹性方程组,克服了非线性项以及高阶导数项带来的一系列的数学困难,我们不但证明了方程组整体解的存在唯一性,而且进一步得到了其对应无限维动力系统在我们所定义的完备闭子空间上整体吸引子的存在性。
第三章,考虑了具有常值温度边界条件的一维非线性热粘弹性方程组,我们解决了由于温度函数满足非齐次Dirichlet边界条件而导致能量估计中含有的边界项所带来的数学困难,在证明了整体弱解存在唯一性的基础上,进一步得到了该整体解当时间趋于无穷大时对某个平衡态的收敛性,成功的将之前文献中所考虑的非线性粘弹性方程的结果推广到非线性热粘弹性方程组的情形。
下面简要列举本论文中所考虑问题的特点,数学困难以及本文工作的主要贡献
(1)第二章中,我们考虑了与Hoffmann&Zochowshi[25]相同的模型,不同的是我们应用了不同的能量估计的技巧得到了解不依赖于时间T的一致先验估计,这对于我们接下来研究解的大时间渐近性态是至关重要的。
(2)在第二章整体解的存在唯一性的证明中,可以看到解的存在空间H为不完备的,而且在H上成立能量守恒等式,也就是说,空间H上不可能存在整体吸引子。为了解决类似的这种问题,Zheng,Shen&Qin([62],[63],[70],[71])引入参数β_i并定义H的子空间H_(β_i),证明了方程组在H_(β_i)上整体吸引子的存在性。受此启发,我们在第二章中同样地引入参数β_i定义H的完备闭子空间H_(β_1,β_2,β_3),证明了方程组在H_(β_1,β_2,β_3)上整体吸引子的存在性,不同的是,我们在H_(β_1,β_2,β_3)定义中引入限制条件θ≥β_1＞0代替θ＞0,克服了限制条件θ＞0使得空间H_(β_i),非闭不完备的缺陷,同时克服了θ≥β_1＞0带来的数学上一系列困难,首次证明了非线性热粘弹性方程组在完备子空间上整体吸引子的存在性。
(3)第三章中我们考虑了绝对温度函数θ满足非齐次Dirichlet边界条件的初边值问题。处理此类问题中一个一直困扰的数学困难是如何去估计分步积分中出现的关于θ边界项,这也是本文的主要贡献之一。
(4)第三章中我们得到的所有的先验估计均不依赖于时间T,当时间趋向于无穷大时,我们不但证明了温度函数θ趋向于正常数θ_T,而且得到了函数u对一平衡态的收敛性,将Pego关于非线性粘弹性方程的结果推广到了我们所考虑的非线性热粘弹性方程组,这为θ在满足其它边界条件所对应的初边值问题所没有的结果。
Nonlinear evolution equations,i.e.,partial differential equations with time t as one of the independent variables,arise not only from many fields of mathematics, but also from other branches of science such as physics,mechanics and material science. The complexity and challenges in the theoretical study of nonlinear evolution equations have attracted a lot of interests from many mathematicians for a long time.
The present thesis is devoted to the study of the asymptotic behavior as time tends to infinity of global solution of nonlinear evolution equations.The asymptotic behavior of global solution of nonlinear evolution equations,which include convergence to a certain equilibrium as time goes to infinity and the study for the related infinite-dimensional dynamical system,has become two of the main concerns in the field of nonlinear evolution equation since 1980s.We usually view the solution of the nonlinear evolution equations as an orbit in a certain Sobolev space,starting from the initial datum u_0.Now we are concerned with the asymptotic behavior of the single orbit starting from an arbitrary,but fixed initial datum and the existence of global attractor of a family of orbits starting from initial data varying in any bounded set of a Sobolev space.
The study of global attractor for the infinite-dimensional dynamical system arises from continuum physics,continuum mechanics,material sciences has become a hot topic in research since the 1980s.There is a lot of work has been done in this direction,such as the reaction-diffusion equations from chemical dynamics and biological sciences,Cahn-Hilliard equations and Phase-Field equations from material science,the incompressible Navier-Stokes equations in space dimension 2, the dissipative wave equations etc(see the books Zheng[72],Temam[41],Babin and Vishik[5],Hale[22],Sell and You[55]and references therein).
This thesis is devoted to the study of one-dimensional nonlinear thermoviscoelastic systems arise from the study of phase transitions in shape memory alloys. Based on the global existence and uniqueness of the solutions,we further obtain the existence of global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions and the convergence to equilibrium for the nonlinear thermoviscoelastic systems with constant temperature boundary conditions respectively.All the results obtained in this thesis have never been found in the previous literature.
The thesis is organized as follows:
Chapter 1 is a preliminary chapter in which we not only recall the history in the literature,but also illustrate the main idea of the proof of the existence of global attractor.We discuss the new features and associated mathematical difficulties of the problems under consideration.Some basic materials and frequently used inequalities are also presented.
Chapter 2 is concerned with the Ginzburg-Landau thermoviscoelastic system with hinged boundary conditions.Overcome the mathematical difficulties due to the nonlinearity and high order derivative,we obtain the existence and uniqueness of the global solution,the asymptotic behavior of the solution as time tends to infinity and the compactness of the orbit.Furthermore,we investigate dynamics of the system and prove the existence of global attractor.
Chapter 3 is concerned with the nonlinear thermoviscoelastic system with constant temperature boundary conditions,we show the global existence and uniqueness of the weak solution and the convergence to a steady state as time tends to infinity, which can be considered to be an extension of Pego[42]result on isothermal case to the non-isothermal case.
We briefly point out the new features,mathematical difficulties of the problems considered in this thesis and our main contributions.
First,in chapter 2 we consider the same model with Hoffmann and Zochowchi [25].Different with[25],we derive delicate uniform a prior estimates independent of T in the proof,which is essential for the study of the asymptotic behavior of the solution as time tends to infinity.
Second,the existing setting of the global solution in chapter 2 is Sobolev space H which is not complete.On the other hand,there exists an energy conservation in H,which means there can be no global attractor for initial data varying in the whole space.In order to solve these problems,Zheng,Shen & Qin([62],[63],[70], [71]) introduced subspaces defined by some parametersβ_i,i.e.,H_(β_i),and proved the existence of global attractor in H_(β_i).Motivated by these results,we consider the dynamics in closed subspace H_(β_1,β_2,β_3) in chapter 2.The difference is that here we use the constraintθ≥β_1＞0 instead ofθ＞0,which brought H_(β_1,β_2,β_3) to be closed and complete.Overcome the mathematical difficulties arise from the constraintθ≥β_1＞0,we prove the existence of global attractor in complete subspaces H_(β_1,β_2,β_3) for the first time.
Third,one of the main contributions in chapter 3 is to handle the boundary termθ_x|x=0,1 appears in the integration by parts due to the non-homogeneous Dirichlet boundary conditionθ|x=0,1=T_0＞0,which seems to have been a major obstacle for people working on such problems.
Fourth,all the estimates obtained in chapter 3 does not depend on T,we prove that as time goes to infinity,θwill converge toθ_T uniformly,the strain u will almost pointwise converge to an equilibrium function u_∞.Such results are still open for the other initial boundary problems for this system.
本学位论文主要是关于非线性发展方程(组)整体解大时间渐近性态的研究。非线性发展方程(组)的整体解的渐近性态,包括整体解当时间趋于无穷大时是否趋于某个平衡态(Equilibrium),以及对应的无限维动力系统是否存在整体吸引子,是非线性发展方程研究的两个基本问题。通常,人们将所考虑的非线性发展方程(组)看成某个Sobolev函数空间中从某个初始数据出发的轨道,那么我们考虑的问题分别为当时间趋于无穷大时从这个函数空间中给定的初始数据出发的单个轨道是否收敛于某个平衡态以及从这个函数空间或者其某个完备的闭子空间中任意有界集出发的一族轨道是否存在整体吸引子?
对于非线性发展方程(组)对应的无限维动力系统的整体吸引子存在性的研究是本文主要的研究兴趣所在。国际上有大量的工作致力于考察由连续介质力学,物理学,材料科学所提出的无限维动力系统的整体吸引子的存在性问题。例如,热力学及生物数学中提出的反应-扩散方程,材料相变理论中的Cahn-Hilliard方程以及Phase-Field方程组,2维不可压缩Navier-Stokes方程,耗散的波动方程,Ginzburg-Landau方程以及Sine-Gordon方程等。相应有专著S.Zheng[72],Temam[41],Babin& Vishik[5],Hale[22]以及Sell & You[55]等。
本学位论文对由形状记忆合金材料的相变过程而提出的一维非线性热粘弹性方程组进行研究,在证明了方程组整体解存在唯一性的基础上,分别得到了具有Ginzburg-Landau形式且满足铰链支座边界条件的一维非线性热粘弹性方程组所对应的无限维动力系统的整体吸引子的存在性以及具有常值温度边界条件的一维非线性热粘弹性方程组整体弱解当时间趋向于无穷大时对一稳态解的收敛性。这些结果为此前文献中所未见。
具体的,本文的主要内容如下:
第一章绪论,简要回顾问题的背景,研究现状及我们证明的思想和方法。介绍了本文考虑的问题的特点,数学上的困难以及本文工作的创新之处。最后,简要列举了必要的一些基本定理和常用不等式。
第二章,考虑具有Ginzburg-Landau形式的,且满足铰链支座边界条件的一维非线性热粘弹性方程组,克服了非线性项以及高阶导数项带来的一系列的数学困难,我们不但证明了方程组整体解的存在唯一性,而且进一步得到了其对应无限维动力系统在我们所定义的完备闭子空间上整体吸引子的存在性。
第三章,考虑了具有常值温度边界条件的一维非线性热粘弹性方程组,我们解决了由于温度函数满足非齐次Dirichlet边界条件而导致能量估计中含有的边界项所带来的数学困难,在证明了整体弱解存在唯一性的基础上,进一步得到了该整体解当时间趋于无穷大时对某个平衡态的收敛性,成功的将之前文献中所考虑的非线性粘弹性方程的结果推广到非线性热粘弹性方程组的情形。
下面简要列举本论文中所考虑问题的特点,数学困难以及本文工作的主要贡献
(1)第二章中,我们考虑了与Hoffmann&Zochowshi[25]相同的模型,不同的是我们应用了不同的能量估计的技巧得到了解不依赖于时间T的一致先验估计,这对于我们接下来研究解的大时间渐近性态是至关重要的。
(2)在第二章整体解的存在唯一性的证明中,可以看到解的存在空间H为不完备的,而且在H上成立能量守恒等式,也就是说,空间H上不可能存在整体吸引子。为了解决类似的这种问题,Zheng,Shen&Qin([62],[63],[70],[71])引入参数β_i并定义H的子空间H_(β_i),证明了方程组在H_(β_i)上整体吸引子的存在性。受此启发,我们在第二章中同样地引入参数β_i定义H的完备闭子空间H_(β_1,β_2,β_3),证明了方程组在H_(β_1,β_2,β_3)上整体吸引子的存在性,不同的是,我们在H_(β_1,β_2,β_3)定义中引入限制条件θ≥β_1＞0代替θ＞0,克服了限制条件θ＞0使得空间H_(β_i),非闭不完备的缺陷,同时克服了θ≥β_1＞0带来的数学上一系列困难,首次证明了非线性热粘弹性方程组在完备子空间上整体吸引子的存在性。
(3)第三章中我们考虑了绝对温度函数θ满足非齐次Dirichlet边界条件的初边值问题。处理此类问题中一个一直困扰的数学困难是如何去估计分步积分中出现的关于θ边界项,这也是本文的主要贡献之一。
(4)第三章中我们得到的所有的先验估计均不依赖于时间T,当时间趋向于无穷大时,我们不但证明了温度函数θ趋向于正常数θ_T,而且得到了函数u对一平衡态的收敛性,将Pego关于非线性粘弹性方程的结果推广到了我们所考虑的非线性热粘弹性方程组,这为θ在满足其它边界条件所对应的初边值问题所没有的结果。
Nonlinear evolution equations,i.e.,partial differential equations with time t as one of the independent variables,arise not only from many fields of mathematics, but also from other branches of science such as physics,mechanics and material science. The complexity and challenges in the theoretical study of nonlinear evolution equations have attracted a lot of interests from many mathematicians for a long time.
The present thesis is devoted to the study of the asymptotic behavior as time tends to infinity of global solution of nonlinear evolution equations.The asymptotic behavior of global solution of nonlinear evolution equations,which include convergence to a certain equilibrium as time goes to infinity and the study for the related infinite-dimensional dynamical system,has become two of the main concerns in the field of nonlinear evolution equation since 1980s.We usually view the solution of the nonlinear evolution equations as an orbit in a certain Sobolev space,starting from the initial datum u_0.Now we are concerned with the asymptotic behavior of the single orbit starting from an arbitrary,but fixed initial datum and the existence of global attractor of a family of orbits starting from initial data varying in any bounded set of a Sobolev space.
The study of global attractor for the infinite-dimensional dynamical system arises from continuum physics,continuum mechanics,material sciences has become a hot topic in research since the 1980s.There is a lot of work has been done in this direction,such as the reaction-diffusion equations from chemical dynamics and biological sciences,Cahn-Hilliard equations and Phase-Field equations from material science,the incompressible Navier-Stokes equations in space dimension 2, the dissipative wave equations etc(see the books Zheng[72],Temam[41],Babin and Vishik[5],Hale[22],Sell and You[55]and references therein).
This thesis is devoted to the study of one-dimensional nonlinear thermoviscoelastic systems arise from the study of phase transitions in shape memory alloys. Based on the global existence and uniqueness of the solutions,we further obtain the existence of global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions and the convergence to equilibrium for the nonlinear thermoviscoelastic systems with constant temperature boundary conditions respectively.All the results obtained in this thesis have never been found in the previous literature.
The thesis is organized as follows:
Chapter 1 is a preliminary chapter in which we not only recall the history in the literature,but also illustrate the main idea of the proof of the existence of global attractor.We discuss the new features and associated mathematical difficulties of the problems under consideration.Some basic materials and frequently used inequalities are also presented.
Chapter 2 is concerned with the Ginzburg-Landau thermoviscoelastic system with hinged boundary conditions.Overcome the mathematical difficulties due to the nonlinearity and high order derivative,we obtain the existence and uniqueness of the global solution,the asymptotic behavior of the solution as time tends to infinity and the compactness of the orbit.Furthermore,we investigate dynamics of the system and prove the existence of global attractor.
Chapter 3 is concerned with the nonlinear thermoviscoelastic system with constant temperature boundary conditions,we show the global existence and uniqueness of the weak solution and the convergence to a steady state as time tends to infinity, which can be considered to be an extension of Pego[42]result on isothermal case to the non-isothermal case.
We briefly point out the new features,mathematical difficulties of the problems considered in this thesis and our main contributions.
First,in chapter 2 we consider the same model with Hoffmann and Zochowchi [25].Different with[25],we derive delicate uniform a prior estimates independent of T in the proof,which is essential for the study of the asymptotic behavior of the solution as time tends to infinity.
Second,the existing setting of the global solution in chapter 2 is Sobolev space H which is not complete.On the other hand,there exists an energy conservation in H,which means there can be no global attractor for initial data varying in the whole space.In order to solve these problems,Zheng,Shen & Qin([62],[63],[70], [71]) introduced subspaces defined by some parametersβ_i,i.e.,H_(β_i),and proved the existence of global attractor in H_(β_i).Motivated by these results,we consider the dynamics in closed subspace H_(β_1,β_2,β_3) in chapter 2.The difference is that here we use the constraintθ≥β_1＞0 instead ofθ＞0,which brought H_(β_1,β_2,β_3) to be closed and complete.Overcome the mathematical difficulties arise from the constraintθ≥β_1＞0,we prove the existence of global attractor in complete subspaces H_(β_1,β_2,β_3) for the first time.
Third,one of the main contributions in chapter 3 is to handle the boundary termθ_x|x=0,1 appears in the integration by parts due to the non-homogeneous Dirichlet boundary conditionθ|x=0,1=T_0＞0,which seems to have been a major obstacle for people working on such problems.
Fourth,all the estimates obtained in chapter 3 does not depend on T,we prove that as time goes to infinity,θwill converge toθ_T uniformly,the strain u will almost pointwise converge to an equilibrium function u_∞.Such results are still open for the other initial boundary problems for this system.
引文
[1]G.Andrews.On the existence of solutions to the equation U_(tt)=u_(xxt)+σ(u_x)_x[J].J.Differential Equations,1980,35:200-231.
[2]R.Adams.Sobolev Spaces[M].New York:Academic Press,1975.
[3]G.Andrews,J.M.Ball.Asymptotic behavior and changes of phase in one-dimensional nonlinear viscoelasticity[J].J.Differential Equations,1982,44:306-341.
[4]H.W.Alt,K.H.Hoffman,M.Niezgodka,J.Sprekels.A numerical study of stuctural phase transitions in shape memory alloys[J].Inst.Math.Univ.Augsburg preprint,1985,90.
[5]A.V.Babin,M.I.Vishik.Attractors of evolution equations[M].Amsterdam:North-Holland,1992.
[6]J.M.Ball.Initial-boundary value problems for an extensible beam[J].J.Math.Anal.Appl,1973,42:61-90.
[7]J.M.Ball.Stability theory for an extensible beam[J]J.Differential Equations,1973,14:399-418.
[8]A.Benaddi,B.Rao.Energy decay rate of wave equations with indefinite damping[J].J.Differential Equations,2000,161:337-357.
[9]M.Brokate,J.Sprekels.Hysteresis and phase transitions[M].Appl.Math.Sci.New York:Springer,1996:121.
[10]Z.Chen,K.H.Hoffmann.On a one-dimensional thermoviscoelastic model for structural phase transitions in shape memory alloys[J].J.Differential Equations,1994,112:325-350.
[11]G.Chen,S.A.Fulling,F.J.Narcowich,S.Sun.Exponential decay of energy of evolution equations with locally distributed damping[J].SIAM J.Appl.Math.,1991,51:266-301.
[12]C.M.Dafermos.Global smooth solutions to the initial boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity[J].SIAM J.Math.Anal.,1982,13:397-408.
[13]C.M.Dafermos,L.Hsiao.Global smooth thermomechanical process in one-dimensional nonlinear thermoviscoelasticity [J]. Nonlinear Anal.T.M.A., 1982, 6: 435-454.
[14] C.Elliott, S.Zheng. On the Cahn-Hilliard equation [J]. Arch. Rational Meth.Anal.,1986, 96: 339-357.
[15] A.Eden, A.J.Milani. Exponential attractors for extensible beam equation [J]. Nonlin-earity, 1993, 6: 457-479.
[16] F.Falk. Ginzberg-Landau theory of static domain walls in shape memory alloys [J].Physica B, 1983, 51: 177-185.
[17] F.Falk. Ginzberg-Landau theory of solitary waves in shape memory alloys [J]. Physica B, 1984, 54: 159-167.
[18] E.Feireisl, E.Zuazua. Global attractors for semilinear wave equations with locally distributed nonlinear damping and critial exponent [J]. Comm.PDEs, 1993, 18: 1539-1555.
[19] E.Feireisl. Global attractor for semilinear damped wave equations with supercritical exponent [J]. J.Differential Equations, 1995, 116:431-447.
[20] P.Freitas. On some eigenvalue problems related to the wave equation with indefinite damping [J]. J.Differential Equations, 1996, 127: 320-335.
[21] P.Freitas, E.Zuazua. Stability results for the wave equation with indefinite damping [J]. J.Differential Equations, 1996, 132: 338-353.
[22] J.K.Hale. Asymptotic behavior of dissipative systems, AMS Math.Surveys and Monographs, 25[M]. Rhode Island, Providence: AMS, 1988.
[23] A.Haranx, M.A. Jendoubi. Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity[J]. Calc.Var., 1999, 9: 95-124.
[24] A.Haranx, M.A.Jendoubi. Decay estimates to equilibrium for some evolution equations wiht an analytic nonlinearity[J]. Asymptotic Analysis, 2001, 26:21-36.
[25] K.H. Hoffmann, A.Zochowski. Existence of solutions to some non-linear thermoelastic systems vith viscosity [J]. Math.Meth.Appl.Sci., 1992, 15: 187-204.
[26] L.Hsiao, H.Jian. Asymptotic behavior of solutions to the system of one-dimensional nonlinear thermoviscoelaaticity [J]. Chinese.Ann.Math., 1998, B19(2): 143-152.
[27] L.Hsiao, T.Luo. Large-time behavior of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity. [J]. Quart.Appl.Math., 1998, 18: 201-219.
[28] F.Huang. Characteristic conditions for exponential stability of linear dynamical sys- tems in Hilbert spaces [J]. Ann.Diff.Eqs, 1985, 1: 43-56.
[29] S.-Z.Huang, Gradient Inequalities, with applications to asymptotic behavior and stability of gradient-like systems, Mathematical Surveys and Monographs, 126[M].Rhode Island: AMS, 2006.
[30] S.Jiang. Global large solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity [J]. Quart.Appl.Math., 1993, 51: 731-744.
[31] S.Jiang, R.Racke. Evolution Equations in Thermoelasticity, π Monographs Surveys Pure Appl.Math. 112[M]. Boca Raton: Chapman Hall/CRC, 2000.
[32] K.Liu, Z.Liu, R.Rao. Exponential stability of an abstract nondissipative linear system [J]. SIAM J.Control Optim., 2001, 40: 149-165.
[33] Z.Liu, S.Zheng. Semigroups Associated with Dissipative Systems, Pitman Research Notes in Mathematics series No 398 [M] Boca Raton, London, New York, Washington,D.C: Chapman Hall/ CRC, 1999.
[34] H.Matano. Convergence of solutions of one-dimensional semilinear parabolic equations [J]. J.Math.Kyoto Univ., 1978, 18(2): 221-227.
[35] J.E.Munoz Rivera, R.Racke. Timoshenko systems with indefinite damping [J].J.Math.Anal.Appl, 2008, 341, 1068-1083.
[36] J.E.Munoz Rivera, R.Racke. Exponential stability for wave equations with nondissipative damping [J]. Nonlinear Analysis, 2008, 68: 2531-2551.
[37] G.Menz. Exponential stability of wave equations with potential and indefinite damping [J]. J.Differential Equations, 2007, 242: 171-191.
[38] M.Niezgodka, J.Sprekels. Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys [J]. Math.Meth.Appl.Sci, 1988, 10:197-223.
[39] M.Niezgodka, S.Zheng, J.Sprekels. Global solutions to a model of structural phase transitions in shape memory alloys[J]. J.Math.Anal.Appl, 1988, 130: 39-54.
[40] L.Nirenberg. Topic in nonlinear functional analysis [M]. New York: Courant Institute of Mathematical Science, 1974.
[41] R.Temam. Infinite-dimensional dynamical system in mechanics and physics, Applied Mathematics and Science, Vol. 68 [M]. New York: Springer, 1998.
[42] R.L.Pego. Phase transitions in one-dimensional nonlinear viscoelasticity: admissibil- ity and stability [J]. Arch.Rat.Mech.Anal., 1987, 97: 353-394.
[43] J.Pr(?)ss. On the spectrum of Co-semigroups [J]. Trans.AMS, 1984, 284: 847-857.
[44] J.Pr(?)ss, R.Racke, S.Zheng. Maximal regularity and asmptotic behavior of solutions for the Cahn-Hilliard Equations with dynamic boundary condition [J].Ann.Math.Pura Appl., 2006, 185(4): 627-648.
[45] Y.Qin. Asymptotic behaviour for global smooth solution to a one-dimensional nonlinear thermoviscoelastic system [J]. J.Partial Differential Equations, 1999, 12: 111-134.
[46] Y.Qin. Global existence and asymptotic behavior of the solution to the system in one-dimensional nonlinear thermoviscoelasticity [J]. Quart.Appl.Math., 2001,59: 113-142.
[47] Y.Qin. Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity [J]. IMA Joural of Applied Mathematics, 2005, 1-18.
[48] Y.Qin, J.Fang. Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density [J]. Nonlinear.Anal., 2006, 65: 892-917.
[49] Y.Qin, H.Liu, C.Song. Global attractor for a nonlinear thermoviscoelastic system in shape memory alloys [J]. Proc.Roy.Soc.Edinburgh A, 2008, 138A: 1103-1135.
[50] R.Racke. Lectures on nonlinear evolution equations. Initial value problems. Aspects of Mathematics E19 [M]. Braunschweig/Wiesbaden, 1992.
[51] R.Racke, S.Zheng. Global existence and asymptotic behavior in nonlinear thermoviscoelasticity [J]. J.Differential Equations, 1997, 134: 46-67.
[52] R.Racke, S.Zheng. The Cahn-Hilliard equation with dynamical boundary conditions [J]. Adv.Diflf.Eqs., 2003,8(1): 83-110.
[53] R.Racke, Y.Wang. Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound [J]. J.Ana.Appl., 2005, 24:117-135.
[54] R.Racke, C.Shang. Global Attractors for Nonlinear Beam Equations [J]. Konstanzer Schiriften Math.Inf., 2009, 249.
[55] G.Sell, Y.You, Dynamics of evolutionary equations[M]. New York: Springer, 2001.
[56] C.Shang. Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions [J]. Asymptotic Analysis, 2007,55: 229-251.
[57] C.Shang. Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions [J]. J.Math.Anal.Appl., 2008, 343: 1-21.
[58] C.Shang. Nonlinear Beam Equation with indefinite damping, submitted.
[59] L.Simon. Asymptotics for a class of nonlinear evolution equation with applicationa to geometric problems [J]. Ann.of Math., 1983, 118: 525-571.
[60] W.Shen , S.Zheng. On the coupled Cahn-Hilliard equations [J]. Comm.PDE, 1993,18: 701-727.
[61] W.Shen, S.Zheng, P.Zhu. Global existence and aspmptotic behavior of weak solutions to nonlinear thermoviscoelastic system with clamped boundary conditions [J] Quart.Appl.Math., 1999, 57: 93-116.
[62] W.Shen, S.Zheng. Maximal attractor for the coupled Cahn-Hilliard Equations [J].Nonlinear Anal., 2002, 49: 21-34.
[63] W. Shen, S. Zheng. Maximal attractor for the phase-field equations of Penrose-Fife type[J]. Applied Mathematics Letters, 2002 15: 1019-1023.
[64] J.Sprekels, S.Zheng. Global solutions to the equations of a Ginzburg-Landau theory for stuctural phase transition in shape memory alloys [J]. Physica D, 1989, 39: 59-76.
[65] J.Sprekels, S.Zheng. Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys[J]. Physica D, 1998, 121:252-262.
[66] J.Sprekels, S.Zheng, P.Zhu. Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memeory alloys[J]. SIAM J.Math.Anal, 1998, 29: 69-84.
[67] H.Wu, S.Zheng. Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions [J]. J.Differential Equations, 2004, 204: 511-531.
[68] H.Wu, S.Zheng. Asymptotic behavior of solutions to the Cahn-Hilliard equation with dynamic boundary conditions. In: N.Kenmochi et al. Proceedings of international conference on: Nonlinear partial differential equations and their applications,GAKUTO international Series[J]. Math.Sci.and Appl., 2004, 20. Toyko: Gakkotosho Co.Ltd., 382-390.
[69] S. Zheng. Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems,Pitman Series Monograph and Surveys in Pure and Applied Mathematics, Vol. 76[M].London:Longman Group Limited, 1995.
[70] S. Zheng, Y. Qin. Maximal attractor for the system of one-dimensional polytropic viscous ideal gas[J].Quart.Appl.Math.,2001 59:579-599.
[71]S.Zheng,Y.Qin.Universal attractors for the Navier-Stokes equations of compressible and heat-conductive fluid in bounded annular domains in R~n[J].Arch.Rational.Meth.Anal.,2001,160:153-179.
[72]S.Zheng.Nonlinear Evolution Equations,Pitman Monographs and Surverys in Pure and Applied Mathematics,133[M].BocaRaton,Florida:Chapman Hall/CRC,2004.
[2]R.Adams.Sobolev Spaces[M].New York:Academic Press,1975.
[3]G.Andrews,J.M.Ball.Asymptotic behavior and changes of phase in one-dimensional nonlinear viscoelasticity[J].J.Differential Equations,1982,44:306-341.
[4]H.W.Alt,K.H.Hoffman,M.Niezgodka,J.Sprekels.A numerical study of stuctural phase transitions in shape memory alloys[J].Inst.Math.Univ.Augsburg preprint,1985,90.
[5]A.V.Babin,M.I.Vishik.Attractors of evolution equations[M].Amsterdam:North-Holland,1992.
[6]J.M.Ball.Initial-boundary value problems for an extensible beam[J].J.Math.Anal.Appl,1973,42:61-90.
[7]J.M.Ball.Stability theory for an extensible beam[J]J.Differential Equations,1973,14:399-418.
[8]A.Benaddi,B.Rao.Energy decay rate of wave equations with indefinite damping[J].J.Differential Equations,2000,161:337-357.
[9]M.Brokate,J.Sprekels.Hysteresis and phase transitions[M].Appl.Math.Sci.New York:Springer,1996:121.
[10]Z.Chen,K.H.Hoffmann.On a one-dimensional thermoviscoelastic model for structural phase transitions in shape memory alloys[J].J.Differential Equations,1994,112:325-350.
[11]G.Chen,S.A.Fulling,F.J.Narcowich,S.Sun.Exponential decay of energy of evolution equations with locally distributed damping[J].SIAM J.Appl.Math.,1991,51:266-301.
[12]C.M.Dafermos.Global smooth solutions to the initial boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity[J].SIAM J.Math.Anal.,1982,13:397-408.
[13]C.M.Dafermos,L.Hsiao.Global smooth thermomechanical process in one-dimensional nonlinear thermoviscoelasticity [J]. Nonlinear Anal.T.M.A., 1982, 6: 435-454.
[14] C.Elliott, S.Zheng. On the Cahn-Hilliard equation [J]. Arch. Rational Meth.Anal.,1986, 96: 339-357.
[15] A.Eden, A.J.Milani. Exponential attractors for extensible beam equation [J]. Nonlin-earity, 1993, 6: 457-479.
[16] F.Falk. Ginzberg-Landau theory of static domain walls in shape memory alloys [J].Physica B, 1983, 51: 177-185.
[17] F.Falk. Ginzberg-Landau theory of solitary waves in shape memory alloys [J]. Physica B, 1984, 54: 159-167.
[18] E.Feireisl, E.Zuazua. Global attractors for semilinear wave equations with locally distributed nonlinear damping and critial exponent [J]. Comm.PDEs, 1993, 18: 1539-1555.
[19] E.Feireisl. Global attractor for semilinear damped wave equations with supercritical exponent [J]. J.Differential Equations, 1995, 116:431-447.
[20] P.Freitas. On some eigenvalue problems related to the wave equation with indefinite damping [J]. J.Differential Equations, 1996, 127: 320-335.
[21] P.Freitas, E.Zuazua. Stability results for the wave equation with indefinite damping [J]. J.Differential Equations, 1996, 132: 338-353.
[22] J.K.Hale. Asymptotic behavior of dissipative systems, AMS Math.Surveys and Monographs, 25[M]. Rhode Island, Providence: AMS, 1988.
[23] A.Haranx, M.A. Jendoubi. Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity[J]. Calc.Var., 1999, 9: 95-124.
[24] A.Haranx, M.A.Jendoubi. Decay estimates to equilibrium for some evolution equations wiht an analytic nonlinearity[J]. Asymptotic Analysis, 2001, 26:21-36.
[25] K.H. Hoffmann, A.Zochowski. Existence of solutions to some non-linear thermoelastic systems vith viscosity [J]. Math.Meth.Appl.Sci., 1992, 15: 187-204.
[26] L.Hsiao, H.Jian. Asymptotic behavior of solutions to the system of one-dimensional nonlinear thermoviscoelaaticity [J]. Chinese.Ann.Math., 1998, B19(2): 143-152.
[27] L.Hsiao, T.Luo. Large-time behavior of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity. [J]. Quart.Appl.Math., 1998, 18: 201-219.
[28] F.Huang. Characteristic conditions for exponential stability of linear dynamical sys- tems in Hilbert spaces [J]. Ann.Diff.Eqs, 1985, 1: 43-56.
[29] S.-Z.Huang, Gradient Inequalities, with applications to asymptotic behavior and stability of gradient-like systems, Mathematical Surveys and Monographs, 126[M].Rhode Island: AMS, 2006.
[30] S.Jiang. Global large solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity [J]. Quart.Appl.Math., 1993, 51: 731-744.
[31] S.Jiang, R.Racke. Evolution Equations in Thermoelasticity, π Monographs Surveys Pure Appl.Math. 112[M]. Boca Raton: Chapman Hall/CRC, 2000.
[32] K.Liu, Z.Liu, R.Rao. Exponential stability of an abstract nondissipative linear system [J]. SIAM J.Control Optim., 2001, 40: 149-165.
[33] Z.Liu, S.Zheng. Semigroups Associated with Dissipative Systems, Pitman Research Notes in Mathematics series No 398 [M] Boca Raton, London, New York, Washington,D.C: Chapman Hall/ CRC, 1999.
[34] H.Matano. Convergence of solutions of one-dimensional semilinear parabolic equations [J]. J.Math.Kyoto Univ., 1978, 18(2): 221-227.
[35] J.E.Munoz Rivera, R.Racke. Timoshenko systems with indefinite damping [J].J.Math.Anal.Appl, 2008, 341, 1068-1083.
[36] J.E.Munoz Rivera, R.Racke. Exponential stability for wave equations with nondissipative damping [J]. Nonlinear Analysis, 2008, 68: 2531-2551.
[37] G.Menz. Exponential stability of wave equations with potential and indefinite damping [J]. J.Differential Equations, 2007, 242: 171-191.
[38] M.Niezgodka, J.Sprekels. Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys [J]. Math.Meth.Appl.Sci, 1988, 10:197-223.
[39] M.Niezgodka, S.Zheng, J.Sprekels. Global solutions to a model of structural phase transitions in shape memory alloys[J]. J.Math.Anal.Appl, 1988, 130: 39-54.
[40] L.Nirenberg. Topic in nonlinear functional analysis [M]. New York: Courant Institute of Mathematical Science, 1974.
[41] R.Temam. Infinite-dimensional dynamical system in mechanics and physics, Applied Mathematics and Science, Vol. 68 [M]. New York: Springer, 1998.
[42] R.L.Pego. Phase transitions in one-dimensional nonlinear viscoelasticity: admissibil- ity and stability [J]. Arch.Rat.Mech.Anal., 1987, 97: 353-394.
[43] J.Pr(?)ss. On the spectrum of Co-semigroups [J]. Trans.AMS, 1984, 284: 847-857.
[44] J.Pr(?)ss, R.Racke, S.Zheng. Maximal regularity and asmptotic behavior of solutions for the Cahn-Hilliard Equations with dynamic boundary condition [J].Ann.Math.Pura Appl., 2006, 185(4): 627-648.
[45] Y.Qin. Asymptotic behaviour for global smooth solution to a one-dimensional nonlinear thermoviscoelastic system [J]. J.Partial Differential Equations, 1999, 12: 111-134.
[46] Y.Qin. Global existence and asymptotic behavior of the solution to the system in one-dimensional nonlinear thermoviscoelasticity [J]. Quart.Appl.Math., 2001,59: 113-142.
[47] Y.Qin. Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity [J]. IMA Joural of Applied Mathematics, 2005, 1-18.
[48] Y.Qin, J.Fang. Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density [J]. Nonlinear.Anal., 2006, 65: 892-917.
[49] Y.Qin, H.Liu, C.Song. Global attractor for a nonlinear thermoviscoelastic system in shape memory alloys [J]. Proc.Roy.Soc.Edinburgh A, 2008, 138A: 1103-1135.
[50] R.Racke. Lectures on nonlinear evolution equations. Initial value problems. Aspects of Mathematics E19 [M]. Braunschweig/Wiesbaden, 1992.
[51] R.Racke, S.Zheng. Global existence and asymptotic behavior in nonlinear thermoviscoelasticity [J]. J.Differential Equations, 1997, 134: 46-67.
[52] R.Racke, S.Zheng. The Cahn-Hilliard equation with dynamical boundary conditions [J]. Adv.Diflf.Eqs., 2003,8(1): 83-110.
[53] R.Racke, Y.Wang. Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound [J]. J.Ana.Appl., 2005, 24:117-135.
[54] R.Racke, C.Shang. Global Attractors for Nonlinear Beam Equations [J]. Konstanzer Schiriften Math.Inf., 2009, 249.
[55] G.Sell, Y.You, Dynamics of evolutionary equations[M]. New York: Springer, 2001.
[56] C.Shang. Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions [J]. Asymptotic Analysis, 2007,55: 229-251.
[57] C.Shang. Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions [J]. J.Math.Anal.Appl., 2008, 343: 1-21.
[58] C.Shang. Nonlinear Beam Equation with indefinite damping, submitted.
[59] L.Simon. Asymptotics for a class of nonlinear evolution equation with applicationa to geometric problems [J]. Ann.of Math., 1983, 118: 525-571.
[60] W.Shen , S.Zheng. On the coupled Cahn-Hilliard equations [J]. Comm.PDE, 1993,18: 701-727.
[61] W.Shen, S.Zheng, P.Zhu. Global existence and aspmptotic behavior of weak solutions to nonlinear thermoviscoelastic system with clamped boundary conditions [J] Quart.Appl.Math., 1999, 57: 93-116.
[62] W.Shen, S.Zheng. Maximal attractor for the coupled Cahn-Hilliard Equations [J].Nonlinear Anal., 2002, 49: 21-34.
[63] W. Shen, S. Zheng. Maximal attractor for the phase-field equations of Penrose-Fife type[J]. Applied Mathematics Letters, 2002 15: 1019-1023.
[64] J.Sprekels, S.Zheng. Global solutions to the equations of a Ginzburg-Landau theory for stuctural phase transition in shape memory alloys [J]. Physica D, 1989, 39: 59-76.
[65] J.Sprekels, S.Zheng. Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys[J]. Physica D, 1998, 121:252-262.
[66] J.Sprekels, S.Zheng, P.Zhu. Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memeory alloys[J]. SIAM J.Math.Anal, 1998, 29: 69-84.
[67] H.Wu, S.Zheng. Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions [J]. J.Differential Equations, 2004, 204: 511-531.
[68] H.Wu, S.Zheng. Asymptotic behavior of solutions to the Cahn-Hilliard equation with dynamic boundary conditions. In: N.Kenmochi et al. Proceedings of international conference on: Nonlinear partial differential equations and their applications,GAKUTO international Series[J]. Math.Sci.and Appl., 2004, 20. Toyko: Gakkotosho Co.Ltd., 382-390.
[69] S. Zheng. Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems,Pitman Series Monograph and Surveys in Pure and Applied Mathematics, Vol. 76[M].London:Longman Group Limited, 1995.
[70] S. Zheng, Y. Qin. Maximal attractor for the system of one-dimensional polytropic viscous ideal gas[J].Quart.Appl.Math.,2001 59:579-599.
[71]S.Zheng,Y.Qin.Universal attractors for the Navier-Stokes equations of compressible and heat-conductive fluid in bounded annular domains in R~n[J].Arch.Rational.Meth.Anal.,2001,160:153-179.
[72]S.Zheng.Nonlinear Evolution Equations,Pitman Monographs and Surverys in Pure and Applied Mathematics,133[M].BocaRaton,Florida:Chapman Hall/CRC,2004.