热弹方程组及相关模型整体适定性的研究
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摘要
热弹方程是热弹性力学方程组的简称,是根据热弹性体的变形和温度的分布规律建立的数学模型。在本文中,我们主要研究非线性热弹材料模型以及其相关模型包括热粘弹模型、Timoshenko系统等解的整体存在性、渐近性和吸引子的存在性。
本文共分为七章：
第一章是引言和预备知识,主要介绍了所研究问题的相关背景和研究现状以及本文所需要的一些基本概念和引理。
第二章研究了带有第二声的非线性热弹模型,证明了该方程组解的整体存在性和指数稳定性。在证明的过程中,利用常微分方程的技巧,将原来的模型转化为我们熟悉的热弹Ⅰ型方程组,并在适当的假设条件下,充分利用强正定核的性质,得到了较好的结果。
第三章研究了Timoshenko模型在热弹方程中的应用,结合能量扰动方法和多乘子技巧,构造Lyapunov函数,在松弛函数g分别为指数衰减和多项式衰减的条件下,相应地得到了系统解的指数衰减和多项式衰减结果,其中,对于非等波速的情形,首次得到解的能量衰减结果。同时,利用非紧测度的方法,首次得到系统吸引子的存在性。
第四章研究了高维线性热粘弹模型。在本章中,首先利用半群方法得到系统解的整体存在性,接着通过引入部分边界的速度反馈,并构造一般的Lyapunov函数,我们得到了系统解的指数稳定性,并首次得到解的具体衰减率。
第五章研究了一类热粘弹模型。通过引入更一般的边界反馈条件,并对松弛函数作了必要的假设,利用一个重要引理,以及能量扰动方法和多乘子技巧,得到了系统能量的几类衰减结果。另外,对于齐次边界条件的情形,首次得到了系统吸引子的存在性。
第六章讨论了带有混合自由边界的热粘弹方程组,在充分利用H1,H2中的已知结果和一些常用不等式的基础上,采用能量方法,通过精细的估计,得到一系列先验估计,并最终得到解在H4空间中解的整体存在性。
第七章总结了本文的主要工作,并对未来的研究方向作了展望。
Thermoelastic equations describe the elastic and the thermal behavior of elastic, heat conductive media, in particular the reciprocal actions between elastic stresses and temperature differences. The present dissertation is concerned with the global exis-tence and asymptotic behavior of solutions to thermoelastic systems, thermoviscoelas-tic systems and Timoshenko systems. Moreover, the existence of a global attractor is achieved in some case.
This dissertation is divided into seven chapters.
Chapter 1 is preface.
In Chapter 2, we prove the global existence and exponential stability of solutions to nonlinear thermoelastic equations with second sound provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.
In Chapter 3, we consider the stability property for Timoshenko-type systems with past history g (the relaxation kernel). For g decaying polynomially, we prove polyno-mial stability results for the equal wave-speed propagation; for the nonequal wave-speed case, we also establish a decay result under the exponential decay condition on g. Moreover, the existence of a global attractor is achieved.
In Chapter 4, we establish the global existence result for the higher-dimensional linear thermoviscoelastic equations by using a semigroup approach. Using multipler techniques and Lyapunov methods, we prove that the energy in the higher-dimensional linear thermoviscoelasticity decays to zero exponentially by introducing a velocity feed-back on a part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary to increase the loss of energy.
In Chapter 5, we obtain a decay result for higher-dimensional linear thermovis- coelastic equations by introducing a velocity feedback on a part of the boundary and using the multiplier techniques method. Moreover, the existence of a global attractors is abtained.
In Chapter 6, we consider a one-dimensional continuous model of nutron star, which is described by a compressible thermoviscoelastic system with a non-monotone equation of state, due to the effective Skyrme nuclear interaction between particles. We prove that, despite a possible destabilizing influence of the pressure, which is non-monotone and not always positive, the presence of viscosity and a sufficient thermal dissipation can yield the global existence of solutions in H4 with a mixed free boundary problem for our model.
In Chapter 7, we summarize of the results of the dissertation, and predict the work in the future.
本文共分为七章：
第一章是引言和预备知识,主要介绍了所研究问题的相关背景和研究现状以及本文所需要的一些基本概念和引理。
第二章研究了带有第二声的非线性热弹模型,证明了该方程组解的整体存在性和指数稳定性。在证明的过程中,利用常微分方程的技巧,将原来的模型转化为我们熟悉的热弹Ⅰ型方程组,并在适当的假设条件下,充分利用强正定核的性质,得到了较好的结果。
第三章研究了Timoshenko模型在热弹方程中的应用,结合能量扰动方法和多乘子技巧,构造Lyapunov函数,在松弛函数g分别为指数衰减和多项式衰减的条件下,相应地得到了系统解的指数衰减和多项式衰减结果,其中,对于非等波速的情形,首次得到解的能量衰减结果。同时,利用非紧测度的方法,首次得到系统吸引子的存在性。
第四章研究了高维线性热粘弹模型。在本章中,首先利用半群方法得到系统解的整体存在性,接着通过引入部分边界的速度反馈,并构造一般的Lyapunov函数,我们得到了系统解的指数稳定性,并首次得到解的具体衰减率。
第五章研究了一类热粘弹模型。通过引入更一般的边界反馈条件,并对松弛函数作了必要的假设,利用一个重要引理,以及能量扰动方法和多乘子技巧,得到了系统能量的几类衰减结果。另外,对于齐次边界条件的情形,首次得到了系统吸引子的存在性。
第六章讨论了带有混合自由边界的热粘弹方程组,在充分利用H1,H2中的已知结果和一些常用不等式的基础上,采用能量方法,通过精细的估计,得到一系列先验估计,并最终得到解在H4空间中解的整体存在性。
第七章总结了本文的主要工作,并对未来的研究方向作了展望。
Thermoelastic equations describe the elastic and the thermal behavior of elastic, heat conductive media, in particular the reciprocal actions between elastic stresses and temperature differences. The present dissertation is concerned with the global exis-tence and asymptotic behavior of solutions to thermoelastic systems, thermoviscoelas-tic systems and Timoshenko systems. Moreover, the existence of a global attractor is achieved in some case.
This dissertation is divided into seven chapters.
Chapter 1 is preface.
In Chapter 2, we prove the global existence and exponential stability of solutions to nonlinear thermoelastic equations with second sound provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.
In Chapter 3, we consider the stability property for Timoshenko-type systems with past history g (the relaxation kernel). For g decaying polynomially, we prove polyno-mial stability results for the equal wave-speed propagation; for the nonequal wave-speed case, we also establish a decay result under the exponential decay condition on g. Moreover, the existence of a global attractor is achieved.
In Chapter 4, we establish the global existence result for the higher-dimensional linear thermoviscoelastic equations by using a semigroup approach. Using multipler techniques and Lyapunov methods, we prove that the energy in the higher-dimensional linear thermoviscoelasticity decays to zero exponentially by introducing a velocity feed-back on a part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary to increase the loss of energy.
In Chapter 5, we obtain a decay result for higher-dimensional linear thermovis- coelastic equations by introducing a velocity feedback on a part of the boundary and using the multiplier techniques method. Moreover, the existence of a global attractors is abtained.
In Chapter 6, we consider a one-dimensional continuous model of nutron star, which is described by a compressible thermoviscoelastic system with a non-monotone equation of state, due to the effective Skyrme nuclear interaction between particles. We prove that, despite a possible destabilizing influence of the pressure, which is non-monotone and not always positive, the presence of viscosity and a sufficient thermal dissipation can yield the global existence of solutions in H4 with a mixed free boundary problem for our model.
In Chapter 7, we summarize of the results of the dissertation, and predict the work in the future.
引文
[1]Green A. E and Naghdi P. M. A re-examination of the basic postulates of thermomechanics, Proc. Roc. Soc. London Ser. A,1991,432(1885):171-194.
[2]Green A. E and Naghdi P. M. On undamped heat waves in an elastic solid[J]. J. Thermal Stresses,1992,15(2):253-264.
[3]Dafermos C. M. On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity[J]. Arch. Rational Mech. Anal.1968,29(4):241-271.
[4]Henry D, Lopes O and Perissinotto A. On the essential spectrum of a semigroup of thermoe-lasticity[J]. Nonlinear Anal. TMA,1993,21(1):65-75.
[5]Jiang Song, Exponential decay and global existence of spherically symmetric solutions in ther-moelasticity [J]. Chin. Ann. Math.1998,19A(5):629-640.
[6]Munoz Rivera J.E. Energy decay rate in linear thermoelascity[J]. Funkcial Ekvac.1992,35:19-30.
[7]Munoz Rivera J. E. Decomposition of the displacement vector field and decay rates in linear thermoelasticity[J]. SIAM J. Math. Anal.1993,24(2):390-406.
[8]Munoz Rivera J. E. Asymptotic behavior in n-dimensional thermoelasticity [J]. Appl. Math. Lett. 1997,10(5):47-53.
[9]Slemrod M. Global Existence, Uniqueness, and asymptotic stability of classical solutions in one-dimensional thermoelasticity[J]. Archive for Rational Mechanics and Analysis,1981,76(2): 97-133.
[10]Wang Junming and Guo Baozhu, On dynamic behavior of hyperbolic system derived from a thermoelastic equation with memory type[J]. Journal of the Franklin Institute,2007,344(2): 75-96.
[11]Liu Zhuangyi and Zheng Songmu, On the exponential stability of linear viscoelasticity and thermoviscoelasticity[J]. Quarterly Appl. Math.1996,54(1):21-31.
[12]Hansen S. W. Exponential energy decay in a linear thermoelastic rod[J]. J. Math. Anal. Appl. 1992,167(2):429-442.
[13]Shibata Y. Neumann problem for one-dimensional nonlinear thermoelasticity[J]. Partial Differ-ential Equations,1992,27(2):457-480.
[14]Racke R, Shibata Y and Zheng Songmu, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity[J]. Quarterly of Applied Mathematics,1993,51(4):751-763.
[15]Munoz Rivera J. E. and Qin Yuming Global existence and exponential stability of solutions to thermoelastic equations of hyperbolic type[J]. J. Elasticity,2004,75(2):125-145.
[16]Messaoudi S. A. On weak solutions of semilinear thermoelastic equations[J]. Magreb Mathe-matical Review,1992,1(1):31-40.
[17]Messaoudi S. A. A blow up result in a multidimensional semilinear thermoelastic system[J]. Electronic Journal of Diferential Equations,2001,2001(30):1-9.
[18]Messaoudi S. A. Local existence and blow up in thermoelasticity with second sound[J]. CPDE, 2002,26(8):1681-1693.
[19]Racke R. Thermoelasticity with second sound-exponential stability in linear and nonlinear 1-d[J]. Mathematical Methods in Applied Sciences,2002,25(5):409-441.
[20]Racke R. Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound[J]. Quarterly of Applied Mathematics,2003,61(2):315-328.
[21]Wang Weike and Wang Zejun, The pointwise estimates to solutions for one-dimensional linear thermoelastic system with second sound[J]. J. Math. Anal. Appl.2007,326(2):1061-1075.
[22]Yang Lin and Wang Yaguang, Propagation of singularities in Cauchy problems for quasilinear thermoelastic systems in three space variables[J]. J. Math. Anal. Appl.2004,291(2):638-652.
[23]Lazzari B and Nibbi R. On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary[J]. J. Math. Anal. Appl.2008,338(1):317-329.
[24]Quintanilla R. Existence in thermoelasticity without energy dissipation[J]. J Thermal Stresses, 2002,25(2):195-202.
[25]Munoz Rivera J. E and Qin Yuming, Global existence and exponential stability in one-dimensional nonlinear thermoelasticity with thermal memory, Nonlinear Analysis[J].2002, 51(1):11-32.
[26]Chandrasekharaiah D. S. One-dimensional wave propagation in the linear theory of thermoe-lasticity without energy dissipation[J]. J. Thermal Stresses,1996,19(8):695-710.
[27]Green A. E and Naghdi P. M. On thermoelasticity without energy dissipation[J]. J. Elasticity, 1993,31(3):189-208.
[28]Nappa L. Spatial decay estimates for the evolution equations of linear thermoelasticity without energy dissipation[J]. J. Thermal Stresses,1998,21(5):581-592.
[29]Quintanilla R and Racke R. Stability in thermoelasticity of type Ⅲ[J]. Discrete Contin. Dynam-ical Systems, Ser. B,2003,3(3):383-400.
[30]Zhang Xu and Zuazua E. Decay of solutions of the system of thermoelasticity of type Ⅲ[J]. Comm. Contemp. Math.2003,5(1):1-59.
[31]Quintanilla R. Convergence and structural stability in thermoelasticity[J]. Appl. Math. Comput. 2003,135(2-3):287-300.
[32]Quintanilla R. Structural stability and continuous dependence of solutions in thermoelasticity of type Ⅲ[J]. Discrete Contin. Dyn. Syst. Ser. B,2001,1(4):463-470.
[33]Messaoudi S.A and Said-Houari B. Energy decay in a Timoshenko-type system of thermoelas-ticity of type Ⅲ[J]. J. Math. Anal. Appl.2008,438(1):298-307.
[34]Reissig M and Wang Yaguang, Cauchy problems for linear thermoelastic systems of type III in one space variable[J]. Math. Methods Appl. Sci.2005,28(11):1359-1381.
[35]Lebeau G and Zuazua E. Decay rates for the three-dimensional linear system of thermoelastic-ity[J]. Arch. Rational Mech. Anal.1999,148(3):179-231.
[36]Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismaticbars[J]. Philos. Mag.1921,41:744-746.
[37]Kim J. U and Renardy Y. Boundary control of the Timoshenko beam[J]. SIAM J. Control Optim, 1987,25(6):1417-1429.
[38]Munoz Rivera J. E and Racke R. Timoshenko systems with indefinite damping[J]. J. Math. Anal. Appl.2008,341(2):1068-1083.
[39]Shi Donghua and Feng Dexing, Exponential decay of Timoshenko beam with lacally dis-tributed feedback[J]. IMA J. Math. Control Inform.2001,18(3):395-403.
[40]Soufyane A and Wehbe A. Uniform stabilization for the Timoshenko beam by a locally dis-tributed damping[J]. Electron. J. Differ. Equations,2003,2003(29):1-14.
[41]Taylor S. W. A smoothing property of a hyperbolic system and boundary controllability[J]. J. Comput. Appl. Math.2000,114(1):23-40.
[42]Yan Q.'-X. Boundary stabilization of Timoshenko beam[J]. Systems Sci. Math. Sci.2000,13(4): 376-384.
[43]Messaoudi S.A, Pokojovy M and Said-Houari B. Nonlinear damped Timoshenko systems with second sound-Global existence and exponential stability[J]. Math. Methods Appl. Sci.2009, 32(5):505-534.
[44]Mu noz Rivera J.E and Racke R. Mildly dissipative nonlinear Timoshenko systems-Global existence and exponential stability[J]. J. Math. Anal. Appl.2002,276(1):248-278.
[45]Ferandez Sare H. D and Racke R. On the stability of damped Timoshenko systems:Cattaneo versus Fourier's law[J]. Arch. Ration. Mech. Anal.2009,194(1):221-251.
[46]Berrimi S and Messaoudi S.A. Existence and decay of solutions of a viscoelastic equation with a nonlinear source[J]. Nonlinear Anal. TMA.2006,64(10):2314-2331.
[47]Cavalcanti M. M, Domingos Cavalcanti V.N and Martinez P.General decay rate estimates for viscoelastic dissipative systems[J]. Nonlinear Anal.2008,68(1):177-193.
[48]Day W. A. The decay of energy in a viscoelastic body[J]. Mathematika,1980,27(2):268-286.
[49]Fabiano R. H and Ito K. Semigroup theory and numerical approximation for equations arising in linear viscoelascity[J]. SIAM J. Math. Anal.1990,21(2):374-393.
[50]Hsiao Ling and Jian Huaiyu, Asymptotic behaviour of solutions to the system of one-dimensional nonlinear themoviscoelasticity[J]. Chin. Ann. Math. B,1998,19(2):143-152.
[51]Hsiao Ling and Luo Tao, Large time behaviour of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity[J]. Quart. Appl. Math.1998,56(2):201-219.
[52]Jiang Song, Global large solutions to initial boundary value problems in one-dimensional ther-moviscoelasticity[J]. Quart. Appl. Math.1993,51(4):731-744.
[53]Qin Yuming, Asymptotic behaviour for global smooth solution to a one-dimensional nonlinear thermoviscoelastic system[J]. J. Partial Differential Equations,1999,12(2):111-134.
[54]Qin Yuming, Global existence and asymptotic behaviour of the solution to the system in one-dimensional nonlinear thermoviscoelasticity[J]. Quart. Appl. Math.2001,59(1):113-142.
[55]Shen Weixi, Zheng Songmu and Zhu Peicheng. Global existence and asymptotic behaviour of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions [J]. Quart. Appl. Math.1999,57(1):93-116.
[56]Zheng Songmu and Shen Weixi, Global solutions to the Cauchy problem of the equations of one-dimensional thermoviscoelasticity[J]. Journal of Partial Differential Equations,1989,2:26-38.
[57]Brokate M and Sprekels J. Hysteresis and Phase Transitions[M]. Springer-Verlag, Heidelberg, 1996.
[58]Krejci P and Sprekels J. On a system of nonlinear PDEs with temperature-dependent hysteresis in one-dimensional thermoplasticity[J]. J. Math. Anal. Appl.1997,209(1):25-46.
[59]Krejci P and Sprekels J. Weak stabilization of solutions to PDEs with hysteresis in thermo-viscoelastoplasticity, in:R. P. Agawal, F. Meuman[M]. J. Vosmansky (Eds.), EQUADIFF 9-Proceedings Masaryk Univ., Brno 1998,81-96.
[60]Krejci P and Sprekels J. Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity, Appl. Math.1998,43(3):173-205.
[61]Borini Silvia and Pata V. Uniform attractors for a strongly damped wave equation with linear memory[J]. Asymptotic Analysis,1999,20(3-4):263-277.
[62]Giorgi C and Munoz Rivera J. E. Global attractors for a semilinear hyperbolic equation in viscoelasticity[J]. J. Math. Anal. Appl.2001,260(1):83-99.
[63]Hale J.Asymptotic behavior and dynamics in infinite dimensions[M]. in "Nonlinear Differen-tial Equations" (J. Hale and P. Martinez-Amores, Eds. Pitman, Boston,1985.
[64]Qin Yuming. Nonlinear Parabolic-Hyperbolic Systems and Their Attractors[M]. Vol. 184, Operator Theory, Advances in PDEs, Basel, Boston-Berlin, Birkhauser,2008.
[65]Zheng Songmu and Qin Yuming, Maximal attractor for the system of one-dimensional poly-tropic viscous ideal gas[J]. Quart. Appl. Math.2001,59(3):579-599.
[66]Zheng Songmu and Qin Yuming, Universal attractors for the Navier-Stokes equations of com-pressible and heat-conductive fluid in bounded annular domains in Rn[J]. Arch. Rationd Mech. Anal.2001,160(2):153-179.
[67]Martinez P. A new method to obtain decay rate estimates for dissipative systems with localized damping[J]. Rev. Mat. Comput.1999,12(1):419-444.
[68]Tarabek M. A. On the existence of smooth solutions in one-dimensional thermoelasticity with second sound[J]. Quarterly of Applied Mathematics,1992,50(4):727-742.
[69]Messaoudi S. A and Houari B. S. Exponential stability in one-dimensional non-linear thermoe-lasticity with second sound[J]. Math. Mech. Appl. Scil.2005,28(2):205-232.
[70]Messaoudi S. A. Decay of solutions of a nonlinear hyperbolic system describing heat propaga-tion by second sound[J]. Applicable Analysis,2002,81(2):201-209.
[71]Fatori L. H and Munoz Rivera J. E. Energy decay for hyperbolic thermoelastic systems of mem-ory type[J]. Quart. Appl. Math.2001,59(3):441-458.
[72]Coleman B. D, Hrusa W. J and Owen D. R. Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids[J] Archive for Rational Mechan-ics and Analysis,1986,94(3):267-289.
[73]Linz P. Analytical and Numerical Methods for Volterra Equations[M]. SIAM Studies in Applied Mathematics, SIAM, Philadephia, PA,1985.
[74]Staffans O.J. On a nonlinear hyperbolic Volterra equation[J]. SIAM J. Math. Anal.1980,11(5): 793-812.
[75]Raposo C. A, Ferreira J, Santos M. L and Castro N. N. O. Exponential stability for the Timo-shenko system with two weak dampings[J]. Appl. Math. Lett.2005,18(5):535-541.
[76]Liu Zhuangyi and Zheng Songmu, Semigroups associated with dissipative systems[M], Chap-man and Hall/CRC,1999.
[77]Ammar-Khodja F, Benabdallah A, Munoz Rivera J. E and Racke R. Energy decay for Timo-shenko system of memory type[J]. J. Differential Equations,2003,194(1):82-115.
[78]Guesmia A and Messaoudi S. A. On the control of solutions of a viscoelastic equation[J]. Appl. Math. Comput.2008,206(2):589-597.
[79]Messaoudi S. A and Mustafa M.I. A stability result in a memory-type Timoshenko system[J]. Danam. Systems Appl. in press.
[80]Fernandez Sare H. D and Munoz Rivera J. E. Stabiliy of Timoshenko systems with past his-tory[J]. J. Math. Anal. Appl.2008,339(1):482-502.
[81]Alabau-Boussouira F. Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control [J]. Nonlinear Differential Equations Appl.2007,14(4-5):643-669.
[82]Liu Zhuangyi and Rao Bopeng, Energy decay rate of the thermoelastic Bresse system[J]. Z. Math. Phys.2009,60(1):54-69.
[83]Messaoudi S. A and Mustafa M.I. On the stabilization of the Timoshenko system by a weak nonlinear dissipation[J]. Math. Methods Appl. Sci.2009,32(4):454-469.
[84]Dafermos C.M. Asymptotic stability in viscoelasticity[J]. Arch. Ration. Mech. Anal.1970,37(4): 297-308.
[85]Zheng Songmu, Nonlinear Evolution Equations[M]. Pitman Monogr. Survey. Pure Appl. Math, Vol.133, CRC Press, USA,2004.
[86]Messaoudi S.A and Said-Houari B. Uniform decay in a Timoshenko-type system with past his-tory[J]. J. Math. Anal. Appl.2009,360(2):459-475.
[87]Pazy A. Semigroup of linear operators and appplications to partial differential equations[M]. Springer-Verlag, New York,1983.
[88]Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics[M]. Springer-Verlag, New York,1988.
[89]Pata V and Zucchi A. Attractors for a damped hyperbolic equation with linear memory[J]. Adv. Math. Sci. Appl.2001,11(2):505-529.
[90]Dafermos C. M. An abstract Volterra equation with applications to linear viscoelascity[J].J. Differential Equations,1990,7:554-569.
[91]Liu Weijiu, The exponential stabilization of higher-dimensional linear system of thermovis-coelasticity[J]. J. Math. Pures Appl.1998,77(4):355-386.
[92]Liu Weijiu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity[J]. ESAIM:Control Optim. Calc. Var.1998,3:23-48.
[93]Desch W and Miller R. K. Exponential stabilization of Volterra integral equations with singular kernels[J]. J. Integral Equations Applications,1998,1(3):397-433.
[94]Fabiano R. H and Ito K. An approximation framwork for equations in linear viscoelasticity with strongly singular kernels[J]. Quarterly of Applied Mathematics,1994,52(1):65-81.
[95]Liu Kangsheng and Liu Zuangyi. On the type of Co-semigroup associated with the abstract linear viscoelastic system[J]. ZAMP,1996,47(1):1-15.
[96]Leugering G. On boundary feedback stabilization of a viscoelastic membrance[J]. Dynamics and Stability of Systems,1989,4(1):71-79.
[97]Chen G. Energy decay estimates and exact boundary value controllability for the wave equa-tion in a bounded domain[J]. J. Math. Pures Appl.1979,58(3):249-273.
[98]Komornik V and Zuazua E. A direct method for the boundary stabilization of the wave equa-tion[J]. J. Math. Pures Appl.1990,69(1):33-54.
[99]Lagnese J. Boundary stabilization of linear elastodynamic systems, SIAM J. Control Optim. 1983,21(6):968-984.
[100]Leugering G. On boundary feedback stabilization of a viscoelastic beam[J]. Proc. Roy. Soc. Edinburgh Sect. A,1990,114:57-69.
[101]Navarro C. B. Asymptotic stability in linear thermoviscoelasticity[J]. J. Math. Anal. Appl. (1978),65(2):399-431.
[102]Messaoudi S.A and Tatar Nasser-eddine, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem[J]. Math. Methods Appl. Sci.2007,30(6):665-680.
[103]Messaoudi S.A and Tatar Nasser-eddine, Exponential and polynomial decay for a quasilinear viscoelastic problem[J]. Nonlinear Anal. TMA.2008,68(4):785-793.
[104]Messaoudi S. A and Mustafa M.I. On the control of solutions of viscoelastic equations with boundary feedback[J]. Nonlinear Analysis,2009,10(5):3132-3140.
[105]Desch W and Miller R. K. Exponential stabilization of Volterra integrodifferential equations in Hilbert space[J]. J. Differential Equations,1987,70(3):366-389.
[106]Hasse RW. Approaches to nuclear friction[J]. Reports on Progress in Physics,1978,41(7):1027-1101.
[107]Bonche P, Koonin S and Negele J.W. One-dimensional nuclear dynamics in the TDHF approxi-mation[J]. Physical Review C,1976,13(3):1226-1258.
[108]Sureau E. La matiere nucleaire:desetoiles aux noyaux[M]. Hermann:Paris,1998.
[109]Dafermos C.M. Global smooth solutions to the initial boundary value problem for the equa-tions of one-dimensional nonlinear thermoviscoelasticity[J] SIAM J. Math. Anal.1982,13(3): 397-408.
[110]Dafermos C.M and Hsiao L, Global smooth thermomechanical processes in one-dimensional nonlinear thermoelasticity[J]. Nonlinear Analysis Theory Methods and Applications,1982,6(5): 435-454.
[111]Ducomet B. Global existence for a simplified model of nuclear fluid in one dimension[J]. Jour-nal of Mathematical Fluid Mechanics,2000,2(1):1-15.
[112]Ducomet B. Asymptotic behaviour for a non-monotone fluid in one-dimension:the positive temperature case[J] Math. Appl. Sci.2001,24(8):543-559.
[113]Watson S.J. Unique global solvability for initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity[J]. Arch. Rat. Mech. Anal.2000,153(1):1-37.
[114]Racke R and Zheng Songmu, Global existence and asymptotic behaviour in nonlinear themo-viscoelasticity[J]. J. Differential Equations,1997,134(1):46-67.
[115]Qin Yuming, Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity[J]. IMA J. Appl. Math.2005,70(4):509-526.
[116]Qin Yuming, Ma T, Cavalcanti M. M and Andrade D. Exponential stability in H4 for the Navier-Stokes equations of viscous and heat-conductive fluid[J]. Comm. Pure Appl. Anal.2005, 4(3):635-664.
[117]Qin Yuming and Munoz Rivera J. E. Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas[J]. Proc. Roy. Soc. Edinburgh Sec. A,2002,132(3):685-709.
[118]Qin Yuming and Munoz Rivera J. E. Exponential stability and universal attractors for the Navier-Stokes equations of compressible fluids between two horizontal parallel plates in R3[J]. Appl. Numer. Math.2003,47(2):209-235.
[119]Qin Yuming and Fang Jian-an, Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density[J]. Nonlinear Anal. TMA.2006,65(4):892-917.
[120]Qin Yuming. Universal attractor in H4 for the nonlinear one-dimensional compressible Navier-stokes equations[J]. J. Differential Equations,2004,207(1):21-72.
[121]Qin Yuming and Lu Tzon-tzer. Global attractor for a nonlinear viscoelasticity[J]. J. Math. Anal. Appl.2008,341(2):975-997.
[122]Qin Yuming, Liu Hailiang and Song Changming. Global attractor for a nonlinear thermovis-coelastic system in shape memory alloys[J]. Proc. Roy. Soc. Edingburgh Sec. A,2008,138(5): 1103-1135.
[123]Ducomet B. Simplified models of quantum fluids in nuclear physics[J]. Proceedings of Partial Differential Equations and Applications. Math. Bohem.2001,126(2):323-336.
[124]Zheng Songmu and Shen Weixi, Global solutions to the Cauchy problem of quasilinear hyperbolic-parabolic coupled systems[J] Scientia Sinica,1987, A3(11):1133-1149.
[125]Wu Zhonglin and Xu Juanjuan, Global Existence in H2 for a one-dimensional non-monotone fluid[J]. Chinese Quant. J. Math.,2007,22(2):296-304.
[2]Green A. E and Naghdi P. M. On undamped heat waves in an elastic solid[J]. J. Thermal Stresses,1992,15(2):253-264.
[3]Dafermos C. M. On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity[J]. Arch. Rational Mech. Anal.1968,29(4):241-271.
[4]Henry D, Lopes O and Perissinotto A. On the essential spectrum of a semigroup of thermoe-lasticity[J]. Nonlinear Anal. TMA,1993,21(1):65-75.
[5]Jiang Song, Exponential decay and global existence of spherically symmetric solutions in ther-moelasticity [J]. Chin. Ann. Math.1998,19A(5):629-640.
[6]Munoz Rivera J.E. Energy decay rate in linear thermoelascity[J]. Funkcial Ekvac.1992,35:19-30.
[7]Munoz Rivera J. E. Decomposition of the displacement vector field and decay rates in linear thermoelasticity[J]. SIAM J. Math. Anal.1993,24(2):390-406.
[8]Munoz Rivera J. E. Asymptotic behavior in n-dimensional thermoelasticity [J]. Appl. Math. Lett. 1997,10(5):47-53.
[9]Slemrod M. Global Existence, Uniqueness, and asymptotic stability of classical solutions in one-dimensional thermoelasticity[J]. Archive for Rational Mechanics and Analysis,1981,76(2): 97-133.
[10]Wang Junming and Guo Baozhu, On dynamic behavior of hyperbolic system derived from a thermoelastic equation with memory type[J]. Journal of the Franklin Institute,2007,344(2): 75-96.
[11]Liu Zhuangyi and Zheng Songmu, On the exponential stability of linear viscoelasticity and thermoviscoelasticity[J]. Quarterly Appl. Math.1996,54(1):21-31.
[12]Hansen S. W. Exponential energy decay in a linear thermoelastic rod[J]. J. Math. Anal. Appl. 1992,167(2):429-442.
[13]Shibata Y. Neumann problem for one-dimensional nonlinear thermoelasticity[J]. Partial Differ-ential Equations,1992,27(2):457-480.
[14]Racke R, Shibata Y and Zheng Songmu, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity[J]. Quarterly of Applied Mathematics,1993,51(4):751-763.
[15]Munoz Rivera J. E. and Qin Yuming Global existence and exponential stability of solutions to thermoelastic equations of hyperbolic type[J]. J. Elasticity,2004,75(2):125-145.
[16]Messaoudi S. A. On weak solutions of semilinear thermoelastic equations[J]. Magreb Mathe-matical Review,1992,1(1):31-40.
[17]Messaoudi S. A. A blow up result in a multidimensional semilinear thermoelastic system[J]. Electronic Journal of Diferential Equations,2001,2001(30):1-9.
[18]Messaoudi S. A. Local existence and blow up in thermoelasticity with second sound[J]. CPDE, 2002,26(8):1681-1693.
[19]Racke R. Thermoelasticity with second sound-exponential stability in linear and nonlinear 1-d[J]. Mathematical Methods in Applied Sciences,2002,25(5):409-441.
[20]Racke R. Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound[J]. Quarterly of Applied Mathematics,2003,61(2):315-328.
[21]Wang Weike and Wang Zejun, The pointwise estimates to solutions for one-dimensional linear thermoelastic system with second sound[J]. J. Math. Anal. Appl.2007,326(2):1061-1075.
[22]Yang Lin and Wang Yaguang, Propagation of singularities in Cauchy problems for quasilinear thermoelastic systems in three space variables[J]. J. Math. Anal. Appl.2004,291(2):638-652.
[23]Lazzari B and Nibbi R. On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary[J]. J. Math. Anal. Appl.2008,338(1):317-329.
[24]Quintanilla R. Existence in thermoelasticity without energy dissipation[J]. J Thermal Stresses, 2002,25(2):195-202.
[25]Munoz Rivera J. E and Qin Yuming, Global existence and exponential stability in one-dimensional nonlinear thermoelasticity with thermal memory, Nonlinear Analysis[J].2002, 51(1):11-32.
[26]Chandrasekharaiah D. S. One-dimensional wave propagation in the linear theory of thermoe-lasticity without energy dissipation[J]. J. Thermal Stresses,1996,19(8):695-710.
[27]Green A. E and Naghdi P. M. On thermoelasticity without energy dissipation[J]. J. Elasticity, 1993,31(3):189-208.
[28]Nappa L. Spatial decay estimates for the evolution equations of linear thermoelasticity without energy dissipation[J]. J. Thermal Stresses,1998,21(5):581-592.
[29]Quintanilla R and Racke R. Stability in thermoelasticity of type Ⅲ[J]. Discrete Contin. Dynam-ical Systems, Ser. B,2003,3(3):383-400.
[30]Zhang Xu and Zuazua E. Decay of solutions of the system of thermoelasticity of type Ⅲ[J]. Comm. Contemp. Math.2003,5(1):1-59.
[31]Quintanilla R. Convergence and structural stability in thermoelasticity[J]. Appl. Math. Comput. 2003,135(2-3):287-300.
[32]Quintanilla R. Structural stability and continuous dependence of solutions in thermoelasticity of type Ⅲ[J]. Discrete Contin. Dyn. Syst. Ser. B,2001,1(4):463-470.
[33]Messaoudi S.A and Said-Houari B. Energy decay in a Timoshenko-type system of thermoelas-ticity of type Ⅲ[J]. J. Math. Anal. Appl.2008,438(1):298-307.
[34]Reissig M and Wang Yaguang, Cauchy problems for linear thermoelastic systems of type III in one space variable[J]. Math. Methods Appl. Sci.2005,28(11):1359-1381.
[35]Lebeau G and Zuazua E. Decay rates for the three-dimensional linear system of thermoelastic-ity[J]. Arch. Rational Mech. Anal.1999,148(3):179-231.
[36]Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismaticbars[J]. Philos. Mag.1921,41:744-746.
[37]Kim J. U and Renardy Y. Boundary control of the Timoshenko beam[J]. SIAM J. Control Optim, 1987,25(6):1417-1429.
[38]Munoz Rivera J. E and Racke R. Timoshenko systems with indefinite damping[J]. J. Math. Anal. Appl.2008,341(2):1068-1083.
[39]Shi Donghua and Feng Dexing, Exponential decay of Timoshenko beam with lacally dis-tributed feedback[J]. IMA J. Math. Control Inform.2001,18(3):395-403.
[40]Soufyane A and Wehbe A. Uniform stabilization for the Timoshenko beam by a locally dis-tributed damping[J]. Electron. J. Differ. Equations,2003,2003(29):1-14.
[41]Taylor S. W. A smoothing property of a hyperbolic system and boundary controllability[J]. J. Comput. Appl. Math.2000,114(1):23-40.
[42]Yan Q.'-X. Boundary stabilization of Timoshenko beam[J]. Systems Sci. Math. Sci.2000,13(4): 376-384.
[43]Messaoudi S.A, Pokojovy M and Said-Houari B. Nonlinear damped Timoshenko systems with second sound-Global existence and exponential stability[J]. Math. Methods Appl. Sci.2009, 32(5):505-534.
[44]Mu noz Rivera J.E and Racke R. Mildly dissipative nonlinear Timoshenko systems-Global existence and exponential stability[J]. J. Math. Anal. Appl.2002,276(1):248-278.
[45]Ferandez Sare H. D and Racke R. On the stability of damped Timoshenko systems:Cattaneo versus Fourier's law[J]. Arch. Ration. Mech. Anal.2009,194(1):221-251.
[46]Berrimi S and Messaoudi S.A. Existence and decay of solutions of a viscoelastic equation with a nonlinear source[J]. Nonlinear Anal. TMA.2006,64(10):2314-2331.
[47]Cavalcanti M. M, Domingos Cavalcanti V.N and Martinez P.General decay rate estimates for viscoelastic dissipative systems[J]. Nonlinear Anal.2008,68(1):177-193.
[48]Day W. A. The decay of energy in a viscoelastic body[J]. Mathematika,1980,27(2):268-286.
[49]Fabiano R. H and Ito K. Semigroup theory and numerical approximation for equations arising in linear viscoelascity[J]. SIAM J. Math. Anal.1990,21(2):374-393.
[50]Hsiao Ling and Jian Huaiyu, Asymptotic behaviour of solutions to the system of one-dimensional nonlinear themoviscoelasticity[J]. Chin. Ann. Math. B,1998,19(2):143-152.
[51]Hsiao Ling and Luo Tao, Large time behaviour of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity[J]. Quart. Appl. Math.1998,56(2):201-219.
[52]Jiang Song, Global large solutions to initial boundary value problems in one-dimensional ther-moviscoelasticity[J]. Quart. Appl. Math.1993,51(4):731-744.
[53]Qin Yuming, Asymptotic behaviour for global smooth solution to a one-dimensional nonlinear thermoviscoelastic system[J]. J. Partial Differential Equations,1999,12(2):111-134.
[54]Qin Yuming, Global existence and asymptotic behaviour of the solution to the system in one-dimensional nonlinear thermoviscoelasticity[J]. Quart. Appl. Math.2001,59(1):113-142.
[55]Shen Weixi, Zheng Songmu and Zhu Peicheng. Global existence and asymptotic behaviour of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions [J]. Quart. Appl. Math.1999,57(1):93-116.
[56]Zheng Songmu and Shen Weixi, Global solutions to the Cauchy problem of the equations of one-dimensional thermoviscoelasticity[J]. Journal of Partial Differential Equations,1989,2:26-38.
[57]Brokate M and Sprekels J. Hysteresis and Phase Transitions[M]. Springer-Verlag, Heidelberg, 1996.
[58]Krejci P and Sprekels J. On a system of nonlinear PDEs with temperature-dependent hysteresis in one-dimensional thermoplasticity[J]. J. Math. Anal. Appl.1997,209(1):25-46.
[59]Krejci P and Sprekels J. Weak stabilization of solutions to PDEs with hysteresis in thermo-viscoelastoplasticity, in:R. P. Agawal, F. Meuman[M]. J. Vosmansky (Eds.), EQUADIFF 9-Proceedings Masaryk Univ., Brno 1998,81-96.
[60]Krejci P and Sprekels J. Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity, Appl. Math.1998,43(3):173-205.
[61]Borini Silvia and Pata V. Uniform attractors for a strongly damped wave equation with linear memory[J]. Asymptotic Analysis,1999,20(3-4):263-277.
[62]Giorgi C and Munoz Rivera J. E. Global attractors for a semilinear hyperbolic equation in viscoelasticity[J]. J. Math. Anal. Appl.2001,260(1):83-99.
[63]Hale J.Asymptotic behavior and dynamics in infinite dimensions[M]. in "Nonlinear Differen-tial Equations" (J. Hale and P. Martinez-Amores, Eds. Pitman, Boston,1985.
[64]Qin Yuming. Nonlinear Parabolic-Hyperbolic Systems and Their Attractors[M]. Vol. 184, Operator Theory, Advances in PDEs, Basel, Boston-Berlin, Birkhauser,2008.
[65]Zheng Songmu and Qin Yuming, Maximal attractor for the system of one-dimensional poly-tropic viscous ideal gas[J]. Quart. Appl. Math.2001,59(3):579-599.
[66]Zheng Songmu and Qin Yuming, Universal attractors for the Navier-Stokes equations of com-pressible and heat-conductive fluid in bounded annular domains in Rn[J]. Arch. Rationd Mech. Anal.2001,160(2):153-179.
[67]Martinez P. A new method to obtain decay rate estimates for dissipative systems with localized damping[J]. Rev. Mat. Comput.1999,12(1):419-444.
[68]Tarabek M. A. On the existence of smooth solutions in one-dimensional thermoelasticity with second sound[J]. Quarterly of Applied Mathematics,1992,50(4):727-742.
[69]Messaoudi S. A and Houari B. S. Exponential stability in one-dimensional non-linear thermoe-lasticity with second sound[J]. Math. Mech. Appl. Scil.2005,28(2):205-232.
[70]Messaoudi S. A. Decay of solutions of a nonlinear hyperbolic system describing heat propaga-tion by second sound[J]. Applicable Analysis,2002,81(2):201-209.
[71]Fatori L. H and Munoz Rivera J. E. Energy decay for hyperbolic thermoelastic systems of mem-ory type[J]. Quart. Appl. Math.2001,59(3):441-458.
[72]Coleman B. D, Hrusa W. J and Owen D. R. Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids[J] Archive for Rational Mechan-ics and Analysis,1986,94(3):267-289.
[73]Linz P. Analytical and Numerical Methods for Volterra Equations[M]. SIAM Studies in Applied Mathematics, SIAM, Philadephia, PA,1985.
[74]Staffans O.J. On a nonlinear hyperbolic Volterra equation[J]. SIAM J. Math. Anal.1980,11(5): 793-812.
[75]Raposo C. A, Ferreira J, Santos M. L and Castro N. N. O. Exponential stability for the Timo-shenko system with two weak dampings[J]. Appl. Math. Lett.2005,18(5):535-541.
[76]Liu Zhuangyi and Zheng Songmu, Semigroups associated with dissipative systems[M], Chap-man and Hall/CRC,1999.
[77]Ammar-Khodja F, Benabdallah A, Munoz Rivera J. E and Racke R. Energy decay for Timo-shenko system of memory type[J]. J. Differential Equations,2003,194(1):82-115.
[78]Guesmia A and Messaoudi S. A. On the control of solutions of a viscoelastic equation[J]. Appl. Math. Comput.2008,206(2):589-597.
[79]Messaoudi S. A and Mustafa M.I. A stability result in a memory-type Timoshenko system[J]. Danam. Systems Appl. in press.
[80]Fernandez Sare H. D and Munoz Rivera J. E. Stabiliy of Timoshenko systems with past his-tory[J]. J. Math. Anal. Appl.2008,339(1):482-502.
[81]Alabau-Boussouira F. Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control [J]. Nonlinear Differential Equations Appl.2007,14(4-5):643-669.
[82]Liu Zhuangyi and Rao Bopeng, Energy decay rate of the thermoelastic Bresse system[J]. Z. Math. Phys.2009,60(1):54-69.
[83]Messaoudi S. A and Mustafa M.I. On the stabilization of the Timoshenko system by a weak nonlinear dissipation[J]. Math. Methods Appl. Sci.2009,32(4):454-469.
[84]Dafermos C.M. Asymptotic stability in viscoelasticity[J]. Arch. Ration. Mech. Anal.1970,37(4): 297-308.
[85]Zheng Songmu, Nonlinear Evolution Equations[M]. Pitman Monogr. Survey. Pure Appl. Math, Vol.133, CRC Press, USA,2004.
[86]Messaoudi S.A and Said-Houari B. Uniform decay in a Timoshenko-type system with past his-tory[J]. J. Math. Anal. Appl.2009,360(2):459-475.
[87]Pazy A. Semigroup of linear operators and appplications to partial differential equations[M]. Springer-Verlag, New York,1983.
[88]Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics[M]. Springer-Verlag, New York,1988.
[89]Pata V and Zucchi A. Attractors for a damped hyperbolic equation with linear memory[J]. Adv. Math. Sci. Appl.2001,11(2):505-529.
[90]Dafermos C. M. An abstract Volterra equation with applications to linear viscoelascity[J].J. Differential Equations,1990,7:554-569.
[91]Liu Weijiu, The exponential stabilization of higher-dimensional linear system of thermovis-coelasticity[J]. J. Math. Pures Appl.1998,77(4):355-386.
[92]Liu Weijiu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity[J]. ESAIM:Control Optim. Calc. Var.1998,3:23-48.
[93]Desch W and Miller R. K. Exponential stabilization of Volterra integral equations with singular kernels[J]. J. Integral Equations Applications,1998,1(3):397-433.
[94]Fabiano R. H and Ito K. An approximation framwork for equations in linear viscoelasticity with strongly singular kernels[J]. Quarterly of Applied Mathematics,1994,52(1):65-81.
[95]Liu Kangsheng and Liu Zuangyi. On the type of Co-semigroup associated with the abstract linear viscoelastic system[J]. ZAMP,1996,47(1):1-15.
[96]Leugering G. On boundary feedback stabilization of a viscoelastic membrance[J]. Dynamics and Stability of Systems,1989,4(1):71-79.
[97]Chen G. Energy decay estimates and exact boundary value controllability for the wave equa-tion in a bounded domain[J]. J. Math. Pures Appl.1979,58(3):249-273.
[98]Komornik V and Zuazua E. A direct method for the boundary stabilization of the wave equa-tion[J]. J. Math. Pures Appl.1990,69(1):33-54.
[99]Lagnese J. Boundary stabilization of linear elastodynamic systems, SIAM J. Control Optim. 1983,21(6):968-984.
[100]Leugering G. On boundary feedback stabilization of a viscoelastic beam[J]. Proc. Roy. Soc. Edinburgh Sect. A,1990,114:57-69.
[101]Navarro C. B. Asymptotic stability in linear thermoviscoelasticity[J]. J. Math. Anal. Appl. (1978),65(2):399-431.
[102]Messaoudi S.A and Tatar Nasser-eddine, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem[J]. Math. Methods Appl. Sci.2007,30(6):665-680.
[103]Messaoudi S.A and Tatar Nasser-eddine, Exponential and polynomial decay for a quasilinear viscoelastic problem[J]. Nonlinear Anal. TMA.2008,68(4):785-793.
[104]Messaoudi S. A and Mustafa M.I. On the control of solutions of viscoelastic equations with boundary feedback[J]. Nonlinear Analysis,2009,10(5):3132-3140.
[105]Desch W and Miller R. K. Exponential stabilization of Volterra integrodifferential equations in Hilbert space[J]. J. Differential Equations,1987,70(3):366-389.
[106]Hasse RW. Approaches to nuclear friction[J]. Reports on Progress in Physics,1978,41(7):1027-1101.
[107]Bonche P, Koonin S and Negele J.W. One-dimensional nuclear dynamics in the TDHF approxi-mation[J]. Physical Review C,1976,13(3):1226-1258.
[108]Sureau E. La matiere nucleaire:desetoiles aux noyaux[M]. Hermann:Paris,1998.
[109]Dafermos C.M. Global smooth solutions to the initial boundary value problem for the equa-tions of one-dimensional nonlinear thermoviscoelasticity[J] SIAM J. Math. Anal.1982,13(3): 397-408.
[110]Dafermos C.M and Hsiao L, Global smooth thermomechanical processes in one-dimensional nonlinear thermoelasticity[J]. Nonlinear Analysis Theory Methods and Applications,1982,6(5): 435-454.
[111]Ducomet B. Global existence for a simplified model of nuclear fluid in one dimension[J]. Jour-nal of Mathematical Fluid Mechanics,2000,2(1):1-15.
[112]Ducomet B. Asymptotic behaviour for a non-monotone fluid in one-dimension:the positive temperature case[J] Math. Appl. Sci.2001,24(8):543-559.
[113]Watson S.J. Unique global solvability for initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity[J]. Arch. Rat. Mech. Anal.2000,153(1):1-37.
[114]Racke R and Zheng Songmu, Global existence and asymptotic behaviour in nonlinear themo-viscoelasticity[J]. J. Differential Equations,1997,134(1):46-67.
[115]Qin Yuming, Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity[J]. IMA J. Appl. Math.2005,70(4):509-526.
[116]Qin Yuming, Ma T, Cavalcanti M. M and Andrade D. Exponential stability in H4 for the Navier-Stokes equations of viscous and heat-conductive fluid[J]. Comm. Pure Appl. Anal.2005, 4(3):635-664.
[117]Qin Yuming and Munoz Rivera J. E. Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas[J]. Proc. Roy. Soc. Edinburgh Sec. A,2002,132(3):685-709.
[118]Qin Yuming and Munoz Rivera J. E. Exponential stability and universal attractors for the Navier-Stokes equations of compressible fluids between two horizontal parallel plates in R3[J]. Appl. Numer. Math.2003,47(2):209-235.
[119]Qin Yuming and Fang Jian-an, Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density[J]. Nonlinear Anal. TMA.2006,65(4):892-917.
[120]Qin Yuming. Universal attractor in H4 for the nonlinear one-dimensional compressible Navier-stokes equations[J]. J. Differential Equations,2004,207(1):21-72.
[121]Qin Yuming and Lu Tzon-tzer. Global attractor for a nonlinear viscoelasticity[J]. J. Math. Anal. Appl.2008,341(2):975-997.
[122]Qin Yuming, Liu Hailiang and Song Changming. Global attractor for a nonlinear thermovis-coelastic system in shape memory alloys[J]. Proc. Roy. Soc. Edingburgh Sec. A,2008,138(5): 1103-1135.
[123]Ducomet B. Simplified models of quantum fluids in nuclear physics[J]. Proceedings of Partial Differential Equations and Applications. Math. Bohem.2001,126(2):323-336.
[124]Zheng Songmu and Shen Weixi, Global solutions to the Cauchy problem of quasilinear hyperbolic-parabolic coupled systems[J] Scientia Sinica,1987, A3(11):1133-1149.
[125]Wu Zhonglin and Xu Juanjuan, Global Existence in H2 for a one-dimensional non-monotone fluid[J]. Chinese Quant. J. Math.,2007,22(2):296-304.