The Propagation For Elastic Waves In Different Medium And Stability Analysis

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(?) 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 effects?(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 of?(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 and?are positive constants, S(t)= eAt is a C0 semigroup of contractions on Hilbert spac H.