【作者】 张再云；

【导师】 刘振海；

【作者基本信息】 中南大学 ， 应用数学， 2011， 博士

【摘要】 本文主要研究弹性波在不同介质中的传播及其稳定性分析.运用Galerkin逼近方法证明了边界值问题的解的存在性与唯一性,而且运用Nakao引理和乘子技巧证明了能量的稳定性,即能量的一般衰减性,包括能量的指数衰减和多项式衰减.第二章主要研究带有非线性阻尼边界条件的弹性波的边界值问题及其能量的衰减性更确切地说,我们研究上面方程的强解和弱解的存在性与唯一性及其能量的衰减性.在本章中,我们克服的主要困难有三点：第一,由于我们考虑的是非线性阻尼边界条件,通常的Galerkin逼近方法在这里失效,因此,需要通过变量代换转化为初值为零的等价问题；第二,由于边界条件上的非线性阻尼项g(u1)和非线性源项f(u)的出现,在通过极限的过程中,带来一些困难.为了克服这些困难,我们用紧性和单调性讨论：最后,因为局部耗散项b(x)u1,经典的能量方法在这里失效,因此运用扰动的能量方法和乘子技巧克服这个困难,从而得到能量的指数衰减.第三章研究了一类广泛的弹性波在弹性介质中的边界值问题及其稳定性分析.运用非线性半群方法得到强解或者弱解的存在性与唯一性,然后运用扰动的能量方法和乘子技巧得到能量的指数衰减,推广了第二章的主要结果.更确切地说,我们研究带有梯度项和非线性边界阻尼条件的Klein-Gordon型方程证明的主要困难除了跟上述第二章的情形一样的困难外,还有以下两点：第一,由于上述方程中出现了梯度项和非线性源项,逼近解的构造更加复杂,因此,运用非线性半群方法得到强解或者弱解的存在性与唯一性。更确切地说,我们把上述方程转化为抽象的柯西问题接下来,只要证明算子A在Hilbert空间Η=V×H中产生G0类压缩半群.也就是说,证明算子A在Hilbert空间Η=V×H中满足这里R(I+A)为I+A的值域,I为恒等算子.第二,由于非线性阻尼边界条件和耗散项的出现,能量估计更加困难更加具有技巧性,运用扰动的能量方法和乘子技巧得到能量的指数衰减.第四章研究了弹性波在粘弹性介质中的边界值问题及其稳定性分析.更确切地说,我们研究了带有非线性局部阻尼项和依赖于速度的粘弹性材料的密度的非线性粘弹性波方程由于方程中出现局部非线性阻尼项α(x)u1|u1|k,非线性源项bu|u|r和非线性项M(τ),松弛函数g(t),使得能量估计更加困难,分别运用Galerkin逼近方法和扰动的能量方法研究了解的存在性与唯一性及其能量的一般衰减性.而且,对于某些初值和对松弛函数的适当的假设,我们证明了能量的衰减率取决于松弛函数g(t)的衰减率.更确切地说,若松弛函数g(t)是指数性衰减为零,则能量也是指数性衰减为零,若松弛函数g(t)是多项式衰减为零,则能量也是多项式衰减为零,这个结果推广了早期文献的结果。详见第四章.第五章研究了弹性波在各向同性的不可压介质中的边界值问题及其稳定性分析.更确切地说,我们研究了下面的非线性弹性波方程证明的主要困难有两个：第一,由于各向同性和不可压的介质的性质,需要先验估计时,我们运用Sobolev嵌入理论和乘子技巧克服一些困难,从而得到解的存在性与唯一性：第二,由于局部的非线性耗散效应ρ(x,u1),进行能量估计时遇到一些困难.为了克服这些困难,运用Nakao引理和乘子技巧得到能量的一般衰减性.证明的关键在于如何得到：能量关于时间t是一般的衰减的结论(包括能量的指数衰减和能量的多项式衰减).这里我们采用了第二、三、四章的基本思想(乘子技巧),但是我们的能量估计方法是新颖的,而且由于各向同性的不可压介质中的弹性波以及非线性局部耗散效应,我们不能直接应用经典的能量估计方法,这里我们巧妙地构造了一系列的乘子和运用Nakao引理来克服这个困难.在最后一部分(附录),若忽略压力项,考虑到弹性波在各向异性介质中的传播,我们研究了弹性波在各向异性介质中的边界值问题及其稳定性分析.更确切地说,我们研究了如下的非线性弹性波方程运用非线性半群方法证明解的存在性与唯一性及其能量的指数衰减性.也就是说,我们分为两个步骤证明：第一步,我们证明解的存在性与唯一性.为了进一步分析,我们把上面的方程转化为下面的抽象的柯西问题基于非线性半群理论,我们将证明算子A产生Hilbert空间Η上的G0类压缩半群.第二步,我们证明能量的指数衰减.也就是说,我们证明下面的估计这里c,ω为正常数,s(t)=eA1为Hilbert空间Η上的C0类压缩半群.更多还原

【Abstract】 In this dissertation, we investigate the propagation and stability analysis of elastic waves in different medium. By using Galerkin’s approximation method, we prove the existence and uniqueness of solutions of boundary value problem (BVP). Moreover, by means of Nakao Lemma and the multiplier technique, we prove the energy stability, that is, general energy decay, including the exponential energy decay and polynomial energy decay.In chapter two, we investigate the BVP and stability analysis of the elastic waves with nonlinear boundary damping given by More precisely, we investigate existence, uniqueness and energy decay of strong solutions and weak solutions of the above equation. Main difficulties involved in studying our problem are as follows: First, due to the nonlinear boundary damping, the usual Galerkin’s approxi-mation method does not work here. Therefore, this approximation requites a change of variables to transformation our problem into an equivelent problem with the initial value equalizing zero.Second, we overcome some difficulties, such as the presence of nonlinear boundary damping g(ut) and nonlinear source term f(u) that bring up serious difficulties when passing the limit, which overcome combining arguments of compactly and monotonicity.Third, due to the locally dissipative term b(x)ut, the classic energy method does not work here. We apply the perturbed energy method and the multiplier technique to overcome difficulties and obtain the exponential energy decay.In chapter three, we investigate the BVP and stability analysis of the extended elastic waves in elasticity. We obtain the existence and uniqueness of strong solutions or weak solutions by means of nonlinear semigroup method. Then, we obtain the exponential energy decay by using of the perturbed energy method and multiplier technique and we extend our main results in chapter two. More precisely, we investigate the Klein-Gordon type with grade term and nonlinear boundary damping given by Except the same difficulties as the case in chapter two, main difficulties lie in two sides:First, in the presence of grade term and nonlinear source term, the construction of approximating solutions become complex. So, we obtain the existence and uniqueness of strong solutions or weak solutions by means of nonlinear semiroup method. More precisely, we formulate the above equation as an abstract Cauchy problem Next, we shall prove that operator A generate a Co semigroup of contractions on Hilbert spaceH=V×H. That is, it is sufficient to prove that Where R(I+A) is the rage of the operator I+A, I is the identity operator.Second, in the presence of noninear boundary damping and dissipative term, the energy estimate become difficult and skillful. We obtain the exponential energy decay by using of the perturbed energy method and the multiplier technique.In chapter four, we investigate the BVP and stabiltiy analysis of the elastic waves in viscoelasticity medium. More precisely, we investigate the nonlinear viscoelastic equation with nonlinear localized damping and velocity-depende-nt material density given byIn the presence of nonlinear localized damping a(x)ut|ut|k,nonlinear source term bu|u|r, nonlinear term M(τ) and the relaxation functiong(t), the energy estimate become difficult. We obtain the existence, uniqueness and general energy decay of the solutions by means of the Galerkin’s approximation method and the perturbed energy method respectively. Furthermore, for certain initial value and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. More precisely, the energy decays exponentially to zero provided the relaxation function g(t) decays exponentia-lly to zero. When the relaxation function g(t) decays polynomially, we show the energy decays polynomially to zero with the same rate of decay. This result improves the earlier ones in the literatures. More details are present in chapter four.In chapter five, we study the BVP and stability of the elastic waves in isotropic incompressible medium. More precisely, we investigate the nonlinear elastic wave equation as follows: There are two difficulties in our proof.First, dut to the properties of isotropic and incompressible, we apply the Sobolev embedding theory and the multiplier technique to overcome some difficulties when we need a prior estimate. Then, we obtain the existence and uniqueness of the solutions.Second, due to nonlinear localized dissipative effectsρ(x,ut), we overcome some difficulties when we show the energy estimates. Applying Nakao lemma and the multiplier technique, we obtain the general enegy decay. The key is how to get:the energy is general decay with respect to the time t, including the exponential energy decay and the polynomial energy decay. Here, we apply the basic ideas in chapter 2,3,4(i.e the multiplier technique), but our energy estimate method is new. Moreover, due to the elastic waves in isotropic incompressible medium and nonlinear localized dissipative effects, we cannot use the classic energy method estimate directly. Here, we skillfully construct a series of multipliers and use Nakao lemma to overcome some difficulties.At the last part(i.e Appendix), in absence of pressure term and in the presence ofρ(x, ut)= a(x)ut, we consider the propagation of the elastic waves in anisotropic medium. That is, we investigate the BVP and stability analysis of the elastic waves in anisotropic medium. More precisely, we investigate nonlinear elastic wave equation as follows: We prove the exitence, uniqueness and the exponential energy decay of the solutions by using nonlinear semigroup method. That is, we divide our proof into two steps.In step 1, we prove the exitence, uniqueness of the solutions. To facilitate our analysis, we formulate the above equation as an abstract Cauchy problem given by Based on nonlinear semigroup theory, we shall prove that the opertor A generates a C0 semigroup of contractions on Hilbert space H.In step 2, we prove the exponential energy decay of the solutions. That is, we prove the following estimate where C andωare positive constants, S(t)= eAt is a C0 semigroup of contractions on Hilbert spac H.更多还原