| Wave systems have been one of the important mathematical models in Partial Differ-ential Equation (PDE) and distributed parameter control theory. The wave systems with variable-coefficient principle part are closer, than constant-coefficient ones, to reflecting practical problems. So it is of important practical significance to investigate the stabi-lization of the wave systems with variable coefficients. The main contents of the second and third chapters in this thesis are to study the stabilization of some wave systems with variable coefficients in the domains of different types (bounded and unbounded), where the main method used is Riemannian geometry method. In the fourth chapter, we explore the blow-up problem of solutions for PDEs which is another aspect of the property of solutions for PDEs.This thesis consists of four chapters.Chapter 1 is the preface and is divided into two parts.In Section 1 of Chapter 1 the survey on the research background and the research advance of the related work is given. Riemannian geometry method, which plays its critical, domain role in dealing with the variable-coefficient problems of this thesis, is introduced. The main results obtained in this thesis are listed.Section 2 of Chapter 1 covers some necessary knowledge and notations of Riemannian geometry which will be needed to use Riemannian geometry method.Chapter 2 is devoted to the study on the energy decay for two wave equations with variable-coefficient principle part in the bounded domains. Riemannian geometry method is applied to deal with the variable-coefficient problems. The energy decay rates of different types for the corresponding wave systems are obtained. In Chapter 2, it assume that Ω is a bounded domain in Rn (n≥2) with smooth boundary Γ=Γ0∪Γ1. Γ0 is nonempty and relatively open in Γ,Γ0∩Γ1=0. div X denotes the divergence of the vector field X in the Euclidean metric.Section 1 of Chapter 2 is concerned with the following variable-coefficient wave equa- tion with semi-linear porous acoustic boundary conditions where A(x)= (aij(x)) are symmetric and positive definite matrices for all x∈Rn andaij(x) are smooth functions on Rn. denotes the outward unit normal vector along the boundary and vA≡Av. a:Ω'R,f, k, h:Γ1'R and φ,ρ,η:R'R are given functions. Applying Riemannian geometry method, the uniform decay rate of the energy for the above system is obtained provided that the given functions satisfy some assumptions and there exists an escape vector field for the Riemannian metric g on Ω. The main contribution of this section is to improve the corresponding result in the case of constant coefficients. This is not a trivial generalization. In Section 2 of Chapter 2, we investigate the following semi-linear wave equation of variable coefficients with a delay in the boundary feedback where A(x)= (aij(x)) is a matrix function with aij= aji of class C1, satisfying for some positive constants λ, A. denotes the outside unit normal vector of the boundary, vA=Av. h:Rn'R and f:R'R are continuous nonlinear functions. Here, τ>0 is a time delay,μ1,μ2 are positive real num-bers and the initial data (u0,u1,go) belongs to a suitable space. Under some conditions, the existence of the solution to the above system is proved by Calerkin approximation. Through combining the energy multipliers technique in the Riemannian metric and the method to deal with the delay problems, the exponential stability of the above system is obtained by introducing an equivalent energy functional of the above system. The main contribution of this section is still to improve the corresponding results in the case of constant coefficients. This is not a trivial generalization.Chapter 3 is devoted to the stabilization for the wave equations of variable coef-ficients with nonlinear dissipations in the unbounded domains. The variable-coefficient wave equations with two types of boundary conditions in the exterior domains are mainly investigated. In the end, we state the main ideas of proving the energy decay rate for the Cauchy problem of variable-coefficient wave equation with nonlinear dissipation. In Chapter 3, it assumed that A(x)= (aij(x)) is a symmetric, positive definite matrix for each x∈ R2, aij(x) are smooth functions on Rn.Section 1 of Chapter 3 deals with the following wave equation of variable coefficients with half-linear localized dissipation in an exterior domain where Ω is an exterior domain in Rn(n>2) such that V≡Rn\Ω is compact, the boundary (?)Ω is smooth, say C3 class. ρ(x,ut) is a nonlinear function likewhere R is a large positive number. Riemannian geometry method and the method to deal with the problem of unbounded domain in the case of constant coefficients are clever combined to achieve the polynomial decay rate of the above system.In Section 2 of Chapter 3, we consider the following wave equation of variable coeffi-cients with the nonlinear Neumann boundary dissipation in an exterior domainwhere Ω is an exterior domain in Rn(n>2) such that V=Rn\Ω is compact with smooth, say C2 class, boundary (?)Ω, which comprises two parts (?)Ω=Γ0∪Γ1,Γ0∩Γ1=0. is the outside unit normal vector field along the boundary (?)Q and vA= Av. p(x, ut) is still a nonlinear function like where R is a large positive number. g(ut) is a nonlinear function like |ut|rut. Riemannian geometry method and a trace estimate of the wave equation are applied to achieved the stabilization of the above system without any geometrical conditions on the shape of the dissipative portion of the boundary.The results obtained in Chapter 3 still improve the corresponding results in the case of constant coefficients. This is not a trivial generalization. As far as we know, there is almost no study on the energy decay of such variable-coefficient systems in an exterior domain.Chapter 4 deals with the following one-dimensional wave equation with nonlinear damping terms and source termswhere (0, L) is a bounded open interval in R, m≥1, p> 1. Under some restriction on the parameters and the initial data, it is proved that the solution of the above system blows up in finite time. |
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