带衰退记忆的方程整体解的长时间动力学行为
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摘要
在这篇论文中,通过运用无穷维动力系统关于吸引子理论的最新研究成果并且结合一些能量估计技巧,我们研究了两类方程:具有衰退记忆的非经典扩散方程和具有衰退记忆的半线性热方程.我们对其弱解和强解的长时间动力学行为进行了深入的讨论,并且证明了以上方程对应动力系统的全局吸引子或一致吸引子的存在性.
首先,我们研究了在自治外力项作用下具有衰退记忆的非经典扩散方程u_t—△u_t—△u—∫_0~∞k(s)△u(t—s)ds=f(u)+g(x).当非线性项f(u)满足临界指数增长,以及外力项g仅属于空间H~(-1)(Ω)或L~2(Ω)时,通过应用半群分解技术和紧性转移定理来克服证明过程中存在的许多实质性困难,最终获得了方程的全局吸引子在弱拓扑空间和强拓扑空间的存在性结果(见定理3.2.9和定理3.3.8).继而,我们研究了在非自治外力项作用下具有衰退记忆的非经典扩散方程u_t—△ut—△u—∫_0~∞k(s)△u(t—s)ds=f(u)+g(x,t).当非线性项f(u)满足临界指数增长,以及符号空间仅为L_b~2(R;L~2(Ω))或L_b~2(R;H_0~1(Ω))(而非平移紧)时,通过分解解过程,来进行解的渐近正则性估计,从而得到了方程在弱拓扑空间和强拓扑空间对应的过程族的紧的一致(w.r.t。g∈H(g_0))吸引子的存在性(见定理4.2.15和定理4.3.5).
最后,我们研究了在自治外力项作用下具有衰退记忆的半线性热方程u_t—△u—∫_0~∞k(s)△u(t—s)ds+f(u)=g(x).在非线性项满足超临界指数增长条件下,通过应用抽象半群理论,进行先验估计,获得了解的存在唯一性,在此基础上利用收缩函数来验证解半群的渐近紧性,最终我们证明了方程的全局吸引子在弱拓扑空间L~2(Ω)×L_μ~2(R~+;H_0~1(Ω))和强拓扑空间H_0~1(Ω)×L_μ~2(R~+;D(A))中的存在性(见定理5.2.9和定理5.3.7).
我们所研究的方程满足的条件很弱,关于非线性项通常是满足临界指数增长或超临界指数增长、而且在自治情形下外力项仅属于正则性较低的空间,在非自治情形下外力项仅为平移有界(符号空间不紧),所以得到的结果极大地改进和推广了已有的一些结果.本文使用的主要工具为抽象半群理论,紧性转移定理及收缩函数.
In this doctoral dissertation,applying the recent theoretical results about attractors and combining with some estimates of energy functional,we mainly consider the two classes of equations as follows:the non-classical diffusion equations with fading memory and the semilinear heat equations with fading memory.Long-time behavior of weak solution and strong solution is discussed and the existence of global attractors or uniform attractors respectively for either autonomous or non-autonomous case is gained.
At first,we research the non-classical diffusion equations with fading memory u_t-△u_t-△u-integral from n=0 to∞k(s)△u(t-s)ds=f(u)+g(x) for the autonomous case.Because nonlinear term satisfies critical exponential growth condition and forcing term g only belongs to H~(-1)(Ω) or L~2(Ω),there exist some virtuality difficulties in the process of proof.Conquering above difficulties through decomposition technique of semigroup and compactness transition theorem, furthermore,we prove the existence of global attractors in weak topological space and in strong topological space(see Theorem 3.2.9 and Theorem 3.3.8). After that,we discuss the non-classical diffusion equations with fading memory u_t-△u_t-△u-integral from n=0 to∞k(s)△u(t-s)ds= f(u)+g(x,t) for the non-autonomous case.When the nonlinear term satisfies critical exponential growth condition and the time-dependant forcing term is translation bounded(that is,only belongs to n_b~2(R;L~2(Ω)) or L_b~2(R;H_0~1(Ω))) instead of translation compact,after decomposing the solution process we test and verify the asymptotic regularity of solutions.Based on the result we show the existence of compactly uniform attractors together their structure in both weak and strong topological spaces(see Theorem 4.2.15 and Theorem 4.3.5).
In the end,we consider the semilinear heat equations with fading memory u_t-△u-integral from n=0 to∞k(s)△u(t-s)ds+f(u)=g(x).When the nonlinearity adheres to polynomial growth of arbitrary order,applying abstract semigroup theory,we make the priori estimate and obtain the existence and uniqueness of solutions. And then the asymptotic compactness of solution semigroup is gained by utilizing contract function theory.According to these results as indicated above, we show the existence of global attractors in both L~2(Ω)×L_μ~2(R~+;H_0~1(Ω)) and H_0~1(Ω)×L_μ~2(R~+;D(A))(see Theorem 5.2.9 and Theorem 5.3.7).
Since the equations we study satisfy the weaker assumptions as follows:the nonlinearity adheres to critical exponential growth or polynomial growth of arbitrary order;and for the autonomous case,the forcing term only belong the lower regularity space,or for the non-autonomous case,the forcing term is only translation bounded,these results we gain improve and extend some known results extremely.The principal tools in my paper are:abstract semigroup theory,compactness transition theorem and contract function.
首先,我们研究了在自治外力项作用下具有衰退记忆的非经典扩散方程u_t—△u_t—△u—∫_0~∞k(s)△u(t—s)ds=f(u)+g(x).当非线性项f(u)满足临界指数增长,以及外力项g仅属于空间H~(-1)(Ω)或L~2(Ω)时,通过应用半群分解技术和紧性转移定理来克服证明过程中存在的许多实质性困难,最终获得了方程的全局吸引子在弱拓扑空间和强拓扑空间的存在性结果(见定理3.2.9和定理3.3.8).继而,我们研究了在非自治外力项作用下具有衰退记忆的非经典扩散方程u_t—△ut—△u—∫_0~∞k(s)△u(t—s)ds=f(u)+g(x,t).当非线性项f(u)满足临界指数增长,以及符号空间仅为L_b~2(R;L~2(Ω))或L_b~2(R;H_0~1(Ω))(而非平移紧)时,通过分解解过程,来进行解的渐近正则性估计,从而得到了方程在弱拓扑空间和强拓扑空间对应的过程族的紧的一致(w.r.t。g∈H(g_0))吸引子的存在性(见定理4.2.15和定理4.3.5).
最后,我们研究了在自治外力项作用下具有衰退记忆的半线性热方程u_t—△u—∫_0~∞k(s)△u(t—s)ds+f(u)=g(x).在非线性项满足超临界指数增长条件下,通过应用抽象半群理论,进行先验估计,获得了解的存在唯一性,在此基础上利用收缩函数来验证解半群的渐近紧性,最终我们证明了方程的全局吸引子在弱拓扑空间L~2(Ω)×L_μ~2(R~+;H_0~1(Ω))和强拓扑空间H_0~1(Ω)×L_μ~2(R~+;D(A))中的存在性(见定理5.2.9和定理5.3.7).
我们所研究的方程满足的条件很弱,关于非线性项通常是满足临界指数增长或超临界指数增长、而且在自治情形下外力项仅属于正则性较低的空间,在非自治情形下外力项仅为平移有界(符号空间不紧),所以得到的结果极大地改进和推广了已有的一些结果.本文使用的主要工具为抽象半群理论,紧性转移定理及收缩函数.
In this doctoral dissertation,applying the recent theoretical results about attractors and combining with some estimates of energy functional,we mainly consider the two classes of equations as follows:the non-classical diffusion equations with fading memory and the semilinear heat equations with fading memory.Long-time behavior of weak solution and strong solution is discussed and the existence of global attractors or uniform attractors respectively for either autonomous or non-autonomous case is gained.
At first,we research the non-classical diffusion equations with fading memory u_t-△u_t-△u-integral from n=0 to∞k(s)△u(t-s)ds=f(u)+g(x) for the autonomous case.Because nonlinear term satisfies critical exponential growth condition and forcing term g only belongs to H~(-1)(Ω) or L~2(Ω),there exist some virtuality difficulties in the process of proof.Conquering above difficulties through decomposition technique of semigroup and compactness transition theorem, furthermore,we prove the existence of global attractors in weak topological space and in strong topological space(see Theorem 3.2.9 and Theorem 3.3.8). After that,we discuss the non-classical diffusion equations with fading memory u_t-△u_t-△u-integral from n=0 to∞k(s)△u(t-s)ds= f(u)+g(x,t) for the non-autonomous case.When the nonlinear term satisfies critical exponential growth condition and the time-dependant forcing term is translation bounded(that is,only belongs to n_b~2(R;L~2(Ω)) or L_b~2(R;H_0~1(Ω))) instead of translation compact,after decomposing the solution process we test and verify the asymptotic regularity of solutions.Based on the result we show the existence of compactly uniform attractors together their structure in both weak and strong topological spaces(see Theorem 4.2.15 and Theorem 4.3.5).
In the end,we consider the semilinear heat equations with fading memory u_t-△u-integral from n=0 to∞k(s)△u(t-s)ds+f(u)=g(x).When the nonlinearity adheres to polynomial growth of arbitrary order,applying abstract semigroup theory,we make the priori estimate and obtain the existence and uniqueness of solutions. And then the asymptotic compactness of solution semigroup is gained by utilizing contract function theory.According to these results as indicated above, we show the existence of global attractors in both L~2(Ω)×L_μ~2(R~+;H_0~1(Ω)) and H_0~1(Ω)×L_μ~2(R~+;D(A))(see Theorem 5.2.9 and Theorem 5.3.7).
Since the equations we study satisfy the weaker assumptions as follows:the nonlinearity adheres to critical exponential growth or polynomial growth of arbitrary order;and for the autonomous case,the forcing term only belong the lower regularity space,or for the non-autonomous case,the forcing term is only translation bounded,these results we gain improve and extend some known results extremely.The principal tools in my paper are:abstract semigroup theory,compactness transition theorem and contract function.
引文
[1]J.Arrieta,A.N.Carvalho and J.K.Hale,A damped hyperbolic equations with critical exponents,Comm.Partial Differentail Equations,17(1992),841-866.
[2]E.C.Aifantis,On the problem of diffusion in solids,Acta Mech.,37(1980),265-296.
[3]A.R.Bishop,K.Fesser,P.S.Lonmdahl and S.E.Trullinger,Influence of solutions in the state on chaos in the driven damped sine-Gordon system,Physica D,7(1983),259-279.
[4]S.Borini,V.Pata,Uniform attractors for a strongly damped wave equations with linear memory,Asymptot.Anal.,20(1999),263-277.
[5]A.V.Barbin and M.I.Vishik,Attractors of evolution equations,North-Holland,Amsterdam,1992.
[6]J.M.Ball,Global attractors for damped semilinear wave equations,Discrete Contin.Dyn.Syst.,10(2004),31-52.
[7]H.Br(?)zis,Op(?)rateurs Maximaux Monotones Et Semi-groups De Contractions Dans Les Espaces De Hilbert,North-Holland Math.Stud.5,North-Holland,Amsterdam,1973.
[8]D.N.Cheban and J.Duan,Almost periodic motions and global attractors of the non-autonomous Navier-Stokes equations,Journal of Dynamical Differential Equations,16(2004),1-34.
[9]B.D.Coleman and M.E.Gurtin,Equipresence and constitutive equations for rigid heat conductors,Z.Angew.Math.Phys.,18(1967),199-208.
[10]V.V.Chepyzhov,S.Gatti,M.Grasselli,A.Miranville,V.Pata,Trajectory and global attractors for evolution equations with memory,Applied Mathematics Letters,19(2006),87-96.
[11]D.N.Cheban,P.E.Kloeden and B.Schmalfuβ,The relationship between pullback,forward and global attractors of nonautonomous dynamical systems,Nonlinear Dyn.Syst.Theory,2(2002),9-28.
[12]T.Caraballo,J.A.Langa and J.C.Robinson,Attractors for differential equations with variable delays,J.Math.Anal.Appl.,260(2001),421-438.
[13]V.V.Chepyzhov,A.Miranville,On trajectory and global attractors for semilinear heat equations with fading memory,Indiana University Mathematics Journal,55(2006),119-167.
[14]B.D.Coleman and W.Noll,Foundations of linear viscoelasticity,Rev.Mod.Phys.,33(1961),239-249.
[15]V.V.Chepyzhov and M.I.Vishik,Attractors for equations of mathematical physics,Amer.Math.Soc.Colloq.Publ.,Vol.49,Amer.Math.Soc,Providence,RI,2002.
[16]V.V.Chepyzhov and M.I.Vishik,Attractors of nonautonomous dynamical systems and their dimension,J.Math.Pures Appl.,73(1994),279-333.
[17]V.V.Chepyzhov and M.I.Vishik,Non-autonomous evolutionary equations with translation-compact symbols and their attractors,C.R.Acad.Sci Paris s(?)r.I Math.,321(1995),153-158.
[18]D.N.Cheban,Global attractors of non-autonomous dissipative dynamical systems,Interdisciplinary Mathematical Sciences,World scientific,New Jersy,USA,Vol.1,2004.
[19]杜先云,戴正德,耗散KDV型方程Cauchy问题的整体吸引子,数学物理学报,20(3)(2000),289-295.
[20]C.M.Dafermos,On abstract Volterra equations with applications to linear viscoelacticity,J.Differential Equations,7(1970),554-569.
[21]C.M.Dafermos,Semiflows generated by compact and uniform processes and some generalizations,Math.Systems Theory,8(1974),142-149.
[22]C.M.Dafermos,Asymptotic stability in viscoelasticity,Arch.Rational Mech.Anal.,37(1970),297-308.
[23]M.Fabrizio and B.Lazzari,On the existence and asymptotic stability of solutions for linear viscoelastic solids,Arch.Rational Mech.Anal.,116(1991),139-152.
[24]M.Fabrizio and A.Morro,Mathematical Problem in Linear Viscoelasticity,SIAM Studies in Applied Mathematics,Philadelphia,1992.
[25]E.Feireisl,Attractors for semilinear damped wave equations on R~3,Nonlinear Analysis,TMA.Vol.23,2(1994),187-195.
[26]J.M.Ghidaglia and A.Marzochhi,Longtime behaviour of strongly damped wave equations,global attractors and their dimension,SIAM J.Math.Anal.,22(1991),879-895.
[27]C.Gatti,A.Miranville,V.Pata,S.V.Zelik,Attractors for semilinear equations of viscoelasticity with very low dissipation,R.Mountain J.Math.,38(2008),1117-1138.
[28]C.Giorgi,M.G.Naso and V.Pata,Exponential stability in linear heat conduction with memory:a semigroup approach,Comm.Appl.Anal.,5(2001),121-133.
[29]M.E.Gurtin and A.Pipkin,A gengral theory of heat conduction with finite wave speed,Arch.Rational Mech.Anal.,31(1968),113-126.
[30]C.Giorgi and V.Pata,Asymptotic behavior of a nonlinear hyperbolic heat equation with memory,Nonlin.Diff.Eq.Appl.,8(2001),157-171.
[31]C.Giorgi and V.Pata,Stability of abstract linear thermoelastic systems with memory,Math.Models Methods Appl.Sci.,11(2001),627-644.
[32]M.Grasselli and V.Pata,A reaction-diffussion equation with memory,Discrete Contin.Dynam.Systems.,15(2006),1079-1088.
[33]C.Giorgi,V.Pata and A.Marzocchi,Asymptotic behavior of a semilinear problem in heat conduction with memory,Nonlin.Diff.Eq.Appl.,5(1998),333-354.
[34]M.Grasselli,V.Pata and G.Prouse,Longtime behavior of a viscoelastic timosheko beam,Discrete Contin.Dynam.Systems.,Vol.10,1,2(2004),337-348.
[35]C.Giorgi,J.E.M.Rivera and V.Pata,Global attractors for a semilinear hyperbolic equations in viscoelasticity,J.Math.Anal.Appl.,260(2001),83-99.
[36]J.M.Ghidaglia and R.Temam,Attractors for damped nonlinear hyperbolic equations,J.Math.Pures Appl.,66(1987),273-319.
[37]C.Giorgi and F.M.Vegni,On the dissipative semigroup associated to the semilinear Mindlin-Timoshenko beam with memory,Quad.Sere.Mat.Brescia,16(2002).
[38]郭柏灵,无穷维动力系统,国防工业出版社,2000.
[39]J.K.Hale,Asymptotic behavior of dissipative systems,Amer.Math.Soc.,Providence,RJ,1988.
[40]A.Haraux,Two remarks on dissipative hyperbolic problems,S(?)minarie du Coll(?)ge de Prance,J.-L.Lions Ed.,Pitman Boston,1985.
[41]J.L.Lions and E.Magenes,Non-homogeneous boundary value problems and applications,Springer-Verlag,Berlin,1972.
[42]Y.C.Liu and F.Wang,A class of multidimensional nonlinear Sobolev-Galpern equation,Acta Math.Appl.Sinica,17(1994),569-577.
[43]李德生,王宗信,王志林,一类非经典扩散方程整体解的存在唯一性及长时间行为,应用数学学报,21(2)(2005).
[44]S.S.Lu,H.Q.Wu and C.K.Zhong,Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,Discrete Contin.Dyn.Syst.,13(2005),701-719.
[45]Z.Liu and S.Zheng,On the exponential stability of linear viscoelasticity and thermoviscoelasticity,Quart.Appl.Math.,54(1996),21-31.
[46]O.A.Ladyzhenskaya,Attractors for semigroups and evolution equations,Leizioni Lincei,Cambridge Univ.Press,Cambridge,New York,1991.
[47]J.L.Lions,Quelques methods de resolution des problems aux limites non lineaires,Dunod,Paris,1969.
[48]卢松松,兰州大学博士论文,2005.
[49]I.Moise,R.Rosa and X.Wang,Attractors for noncompact nonautonomous systems via energy equations,Discrete Contin..Dynam.Systems.,10(2004),473-496.
[50]R.K.Miller,G.R.Sell,Topological dynamical and its relation to integral and non-autonomous system,Intern.Symposium,I,Acad.Press,New York,(1976),223-248.
[51]Q.F.Ma,S.H.Wang and Z.K.Zhong,Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,Indiana University Math.J,51(6)(2002),1541-1559.
[52]Q.Z.Ma and C.K.Zhong,Existence of strong global attractors for hyperbolic equation with linear memory,Applied Mathematics and Computation.157(2004),745-758.
[53]S.Ma and C.K.Zhong,The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces,Discrete Contin.Dynam.Systems.,18(2007),53-70.
[54]马闪,兰州大学博士论文,2007.
[55]J.Meixner,On the linear theory of heat conduction,Arch.Rational Mech.Anal.,39(1970),108.
[56]R.K.Miller,Almost periodic differential equations as dynamical systems with applications to the existence of almost periodic solutions,J.Differential Equations,1(1965),337-345.
[57]J.Nunziato,On heat conduction in materials with memory,Quart.Appl.Math.,29(1971),187-204.
[58]J.G.Peter and M.E.Gurtin,On the theory of heat condition involving two temperatures,Z.Ange.Math.Phys.,19(1968),614-627.
[59]V.Pata and A.Zucchi,Attractors for a damped hyperbolic equation with linear memory,Adv.Math.Sci.Appl.,11(2)(2001),505-529.
[60]H.Poincar(?),Sur less equations de ia dynamique et problem des troic corps,Acta.Math.B(1890),1-270.
[61]M.Renardy,W.J.Hrusa and J.A.Nobel,Mathematical problems in visoelasticity,Pitman Monographs and Surveys in Pure and Applied Mathematics,Vol.35,Wiley,New York,1987.
[62]J.E.M.Rivera,M.G.Naso and F.M.Vegni,Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory,J.Math.Anal.Appl.,286(2003),692-704.
[63]J.C.Robinson,Infinite-dimensional dynamical systems,An introduction to dissipative parabolic PDEs and the theory of global attractors,Cambridge University Press,2001.
[64]C.Y.Sun,D.M.Cao and J.Q.Duan,Non-autonomous wave dynamics with memory-Asymptotic regularity and uniform attractor,Discrete Contin.Dyn.Syst.,Series B,9(2008),743-761.
[65]C.Y.Sun,D.M.Cao and J.Q.Duan,Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,Nonlinearity,19(2006),2645-2665.
[66]C.Y.Sun,D.M.Cao and J.Q.Duan,Uniform attractors for non-autonomous wave equations with nonlinear damping,SIAM Journal on Applied Dynamical Systems,6(2007),293-318.
[67]C.Y.Sun,S.Y.Wang and C.K.Zhong,Global attractors for a nonclassical diffusion equation,Acta Mathematica Sinica,English Series,23(2007),1271-1280.
[68]日本数学会,数学百科辞典,科学出版社,1983.
[69]C.Y.Sun and M.H.Yang,Dynamics of the nonclassical diffusion equtions,Asymptotic Analysis,59(2008),51-81.
[70]孙春友,兰州大学博士论文,2005.
[71]G.R.Sell and Y.You,Dynamics of evolutionary equations,Springer,New York,2002.
[72]G.R.Sell,Non-autonomous differential equations and topological dynamics,Ⅰ,Ⅱ,Trans.Amer.Math.Soc.,127(1967),241-262,263-283.
[73]S.Seo,Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions,Pacific Journal of Mathmatics,Vol.193,1(2000).
[74]R.Temam,Infinite dimensional dynamical systems in mechanics and physics,2nd edn,Berlin,Springer,1997.
[75]T.W.Ting,Certain non-steady flows of second order fluids,Arch.Rational Mech.Anal.,14(1963),1-26.
[76]S.Y.Wang,D.S.Li,C.K.Zhong,On the dynamics of a class of nonclassical parabolic equations,J.Math.Anal.Appl.,317(2006),565-582.
[77]王素云,兰州大学博士论文,2004.
[78]王自强,理性力学基础,科学出版社,2000.
[79]文兰,动力系统简介,数学进展,31(4)(2002),293-294.
[80]Y.L.Xiao,Attractors for a nonclassical diffusion equation,Acta Mathematicae Applicatae Sinica,English Series,18(2002),273-276.
[8]杨美华,兰州大学博士论文,2006.
[82]C.K.Zhong,M.H.Yang and C.Y.Sun,The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reactiondiffusion equations,J.Differential Equations,223(2006),367-399.
[83]S.V.Zelik,Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,Comm.Pure Appl.Anal.,3(2004),921-934.
[2]E.C.Aifantis,On the problem of diffusion in solids,Acta Mech.,37(1980),265-296.
[3]A.R.Bishop,K.Fesser,P.S.Lonmdahl and S.E.Trullinger,Influence of solutions in the state on chaos in the driven damped sine-Gordon system,Physica D,7(1983),259-279.
[4]S.Borini,V.Pata,Uniform attractors for a strongly damped wave equations with linear memory,Asymptot.Anal.,20(1999),263-277.
[5]A.V.Barbin and M.I.Vishik,Attractors of evolution equations,North-Holland,Amsterdam,1992.
[6]J.M.Ball,Global attractors for damped semilinear wave equations,Discrete Contin.Dyn.Syst.,10(2004),31-52.
[7]H.Br(?)zis,Op(?)rateurs Maximaux Monotones Et Semi-groups De Contractions Dans Les Espaces De Hilbert,North-Holland Math.Stud.5,North-Holland,Amsterdam,1973.
[8]D.N.Cheban and J.Duan,Almost periodic motions and global attractors of the non-autonomous Navier-Stokes equations,Journal of Dynamical Differential Equations,16(2004),1-34.
[9]B.D.Coleman and M.E.Gurtin,Equipresence and constitutive equations for rigid heat conductors,Z.Angew.Math.Phys.,18(1967),199-208.
[10]V.V.Chepyzhov,S.Gatti,M.Grasselli,A.Miranville,V.Pata,Trajectory and global attractors for evolution equations with memory,Applied Mathematics Letters,19(2006),87-96.
[11]D.N.Cheban,P.E.Kloeden and B.Schmalfuβ,The relationship between pullback,forward and global attractors of nonautonomous dynamical systems,Nonlinear Dyn.Syst.Theory,2(2002),9-28.
[12]T.Caraballo,J.A.Langa and J.C.Robinson,Attractors for differential equations with variable delays,J.Math.Anal.Appl.,260(2001),421-438.
[13]V.V.Chepyzhov,A.Miranville,On trajectory and global attractors for semilinear heat equations with fading memory,Indiana University Mathematics Journal,55(2006),119-167.
[14]B.D.Coleman and W.Noll,Foundations of linear viscoelasticity,Rev.Mod.Phys.,33(1961),239-249.
[15]V.V.Chepyzhov and M.I.Vishik,Attractors for equations of mathematical physics,Amer.Math.Soc.Colloq.Publ.,Vol.49,Amer.Math.Soc,Providence,RI,2002.
[16]V.V.Chepyzhov and M.I.Vishik,Attractors of nonautonomous dynamical systems and their dimension,J.Math.Pures Appl.,73(1994),279-333.
[17]V.V.Chepyzhov and M.I.Vishik,Non-autonomous evolutionary equations with translation-compact symbols and their attractors,C.R.Acad.Sci Paris s(?)r.I Math.,321(1995),153-158.
[18]D.N.Cheban,Global attractors of non-autonomous dissipative dynamical systems,Interdisciplinary Mathematical Sciences,World scientific,New Jersy,USA,Vol.1,2004.
[19]杜先云,戴正德,耗散KDV型方程Cauchy问题的整体吸引子,数学物理学报,20(3)(2000),289-295.
[20]C.M.Dafermos,On abstract Volterra equations with applications to linear viscoelacticity,J.Differential Equations,7(1970),554-569.
[21]C.M.Dafermos,Semiflows generated by compact and uniform processes and some generalizations,Math.Systems Theory,8(1974),142-149.
[22]C.M.Dafermos,Asymptotic stability in viscoelasticity,Arch.Rational Mech.Anal.,37(1970),297-308.
[23]M.Fabrizio and B.Lazzari,On the existence and asymptotic stability of solutions for linear viscoelastic solids,Arch.Rational Mech.Anal.,116(1991),139-152.
[24]M.Fabrizio and A.Morro,Mathematical Problem in Linear Viscoelasticity,SIAM Studies in Applied Mathematics,Philadelphia,1992.
[25]E.Feireisl,Attractors for semilinear damped wave equations on R~3,Nonlinear Analysis,TMA.Vol.23,2(1994),187-195.
[26]J.M.Ghidaglia and A.Marzochhi,Longtime behaviour of strongly damped wave equations,global attractors and their dimension,SIAM J.Math.Anal.,22(1991),879-895.
[27]C.Gatti,A.Miranville,V.Pata,S.V.Zelik,Attractors for semilinear equations of viscoelasticity with very low dissipation,R.Mountain J.Math.,38(2008),1117-1138.
[28]C.Giorgi,M.G.Naso and V.Pata,Exponential stability in linear heat conduction with memory:a semigroup approach,Comm.Appl.Anal.,5(2001),121-133.
[29]M.E.Gurtin and A.Pipkin,A gengral theory of heat conduction with finite wave speed,Arch.Rational Mech.Anal.,31(1968),113-126.
[30]C.Giorgi and V.Pata,Asymptotic behavior of a nonlinear hyperbolic heat equation with memory,Nonlin.Diff.Eq.Appl.,8(2001),157-171.
[31]C.Giorgi and V.Pata,Stability of abstract linear thermoelastic systems with memory,Math.Models Methods Appl.Sci.,11(2001),627-644.
[32]M.Grasselli and V.Pata,A reaction-diffussion equation with memory,Discrete Contin.Dynam.Systems.,15(2006),1079-1088.
[33]C.Giorgi,V.Pata and A.Marzocchi,Asymptotic behavior of a semilinear problem in heat conduction with memory,Nonlin.Diff.Eq.Appl.,5(1998),333-354.
[34]M.Grasselli,V.Pata and G.Prouse,Longtime behavior of a viscoelastic timosheko beam,Discrete Contin.Dynam.Systems.,Vol.10,1,2(2004),337-348.
[35]C.Giorgi,J.E.M.Rivera and V.Pata,Global attractors for a semilinear hyperbolic equations in viscoelasticity,J.Math.Anal.Appl.,260(2001),83-99.
[36]J.M.Ghidaglia and R.Temam,Attractors for damped nonlinear hyperbolic equations,J.Math.Pures Appl.,66(1987),273-319.
[37]C.Giorgi and F.M.Vegni,On the dissipative semigroup associated to the semilinear Mindlin-Timoshenko beam with memory,Quad.Sere.Mat.Brescia,16(2002).
[38]郭柏灵,无穷维动力系统,国防工业出版社,2000.
[39]J.K.Hale,Asymptotic behavior of dissipative systems,Amer.Math.Soc.,Providence,RJ,1988.
[40]A.Haraux,Two remarks on dissipative hyperbolic problems,S(?)minarie du Coll(?)ge de Prance,J.-L.Lions Ed.,Pitman Boston,1985.
[41]J.L.Lions and E.Magenes,Non-homogeneous boundary value problems and applications,Springer-Verlag,Berlin,1972.
[42]Y.C.Liu and F.Wang,A class of multidimensional nonlinear Sobolev-Galpern equation,Acta Math.Appl.Sinica,17(1994),569-577.
[43]李德生,王宗信,王志林,一类非经典扩散方程整体解的存在唯一性及长时间行为,应用数学学报,21(2)(2005).
[44]S.S.Lu,H.Q.Wu and C.K.Zhong,Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,Discrete Contin.Dyn.Syst.,13(2005),701-719.
[45]Z.Liu and S.Zheng,On the exponential stability of linear viscoelasticity and thermoviscoelasticity,Quart.Appl.Math.,54(1996),21-31.
[46]O.A.Ladyzhenskaya,Attractors for semigroups and evolution equations,Leizioni Lincei,Cambridge Univ.Press,Cambridge,New York,1991.
[47]J.L.Lions,Quelques methods de resolution des problems aux limites non lineaires,Dunod,Paris,1969.
[48]卢松松,兰州大学博士论文,2005.
[49]I.Moise,R.Rosa and X.Wang,Attractors for noncompact nonautonomous systems via energy equations,Discrete Contin..Dynam.Systems.,10(2004),473-496.
[50]R.K.Miller,G.R.Sell,Topological dynamical and its relation to integral and non-autonomous system,Intern.Symposium,I,Acad.Press,New York,(1976),223-248.
[51]Q.F.Ma,S.H.Wang and Z.K.Zhong,Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,Indiana University Math.J,51(6)(2002),1541-1559.
[52]Q.Z.Ma and C.K.Zhong,Existence of strong global attractors for hyperbolic equation with linear memory,Applied Mathematics and Computation.157(2004),745-758.
[53]S.Ma and C.K.Zhong,The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces,Discrete Contin.Dynam.Systems.,18(2007),53-70.
[54]马闪,兰州大学博士论文,2007.
[55]J.Meixner,On the linear theory of heat conduction,Arch.Rational Mech.Anal.,39(1970),108.
[56]R.K.Miller,Almost periodic differential equations as dynamical systems with applications to the existence of almost periodic solutions,J.Differential Equations,1(1965),337-345.
[57]J.Nunziato,On heat conduction in materials with memory,Quart.Appl.Math.,29(1971),187-204.
[58]J.G.Peter and M.E.Gurtin,On the theory of heat condition involving two temperatures,Z.Ange.Math.Phys.,19(1968),614-627.
[59]V.Pata and A.Zucchi,Attractors for a damped hyperbolic equation with linear memory,Adv.Math.Sci.Appl.,11(2)(2001),505-529.
[60]H.Poincar(?),Sur less equations de ia dynamique et problem des troic corps,Acta.Math.B(1890),1-270.
[61]M.Renardy,W.J.Hrusa and J.A.Nobel,Mathematical problems in visoelasticity,Pitman Monographs and Surveys in Pure and Applied Mathematics,Vol.35,Wiley,New York,1987.
[62]J.E.M.Rivera,M.G.Naso and F.M.Vegni,Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory,J.Math.Anal.Appl.,286(2003),692-704.
[63]J.C.Robinson,Infinite-dimensional dynamical systems,An introduction to dissipative parabolic PDEs and the theory of global attractors,Cambridge University Press,2001.
[64]C.Y.Sun,D.M.Cao and J.Q.Duan,Non-autonomous wave dynamics with memory-Asymptotic regularity and uniform attractor,Discrete Contin.Dyn.Syst.,Series B,9(2008),743-761.
[65]C.Y.Sun,D.M.Cao and J.Q.Duan,Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,Nonlinearity,19(2006),2645-2665.
[66]C.Y.Sun,D.M.Cao and J.Q.Duan,Uniform attractors for non-autonomous wave equations with nonlinear damping,SIAM Journal on Applied Dynamical Systems,6(2007),293-318.
[67]C.Y.Sun,S.Y.Wang and C.K.Zhong,Global attractors for a nonclassical diffusion equation,Acta Mathematica Sinica,English Series,23(2007),1271-1280.
[68]日本数学会,数学百科辞典,科学出版社,1983.
[69]C.Y.Sun and M.H.Yang,Dynamics of the nonclassical diffusion equtions,Asymptotic Analysis,59(2008),51-81.
[70]孙春友,兰州大学博士论文,2005.
[71]G.R.Sell and Y.You,Dynamics of evolutionary equations,Springer,New York,2002.
[72]G.R.Sell,Non-autonomous differential equations and topological dynamics,Ⅰ,Ⅱ,Trans.Amer.Math.Soc.,127(1967),241-262,263-283.
[73]S.Seo,Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions,Pacific Journal of Mathmatics,Vol.193,1(2000).
[74]R.Temam,Infinite dimensional dynamical systems in mechanics and physics,2nd edn,Berlin,Springer,1997.
[75]T.W.Ting,Certain non-steady flows of second order fluids,Arch.Rational Mech.Anal.,14(1963),1-26.
[76]S.Y.Wang,D.S.Li,C.K.Zhong,On the dynamics of a class of nonclassical parabolic equations,J.Math.Anal.Appl.,317(2006),565-582.
[77]王素云,兰州大学博士论文,2004.
[78]王自强,理性力学基础,科学出版社,2000.
[79]文兰,动力系统简介,数学进展,31(4)(2002),293-294.
[80]Y.L.Xiao,Attractors for a nonclassical diffusion equation,Acta Mathematicae Applicatae Sinica,English Series,18(2002),273-276.
[8]杨美华,兰州大学博士论文,2006.
[82]C.K.Zhong,M.H.Yang and C.Y.Sun,The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reactiondiffusion equations,J.Differential Equations,223(2006),367-399.
[83]S.V.Zelik,Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,Comm.Pure Appl.Anal.,3(2004),921-934.