【作者】 马志勇；

【导师】 秦玉明；

【作者基本信息】 东华大学 ， 控制理论与控制工程， 2010， 博士

【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。更多还原

【Abstract】 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.更多还原