Existence And Asymptotic Property Of Solutions Of Pseudo-Hyperbolic Equation (s)

Pseudo-hyperbolic equation is a kind of higher-order partial differential equation which with mixed partial derivative in time and space. It’s often used in describing many physical phenomena. Such as in the study of nonlinear continuous dynamic system, the elv trans-mission in the animal nervous system, the vibration of a nonlinear elastic rod which satisfied the kelvin assumption and the air vibration with viscosity in the cylinder etc. Due to this kind of equation containing about second order partial derivative in time variables and mixed partial derivative in time and space variables, it is difficult to get the numerical solving. So it is necessary and meaningful to study the solution of pseudo-hyperbolic equation.In this thesis, we study the existence and uniqueness of solutions of a kind of pseudo-hyperbolic system by using the approach of operator semigroup, at the same time, we prove that all the eigenvectors of the system form an orthogonal basis in the state space H Moreover, we investigate the asymptotic property of this system in virtue of orthogonal basis theory.The thesis consists of three chapters.Chapter 1 is the preface.In Chapter 2, we consider the existence and asymptotic property of solutions of the following pseudo-hyperbolic system Here Ω is a bounded open area on Rn, Δ is the usually Laplace differential sign, "a" is a positive arbitrary constant.In order to study the system (1), we introduce the state space H= H1/0(Ω)×L2(Ω). For any (f1, g1), (f2,g2) € H, we define the product of it isWith the investigation of the existence and asymptotic property of solutions of this type of pseudo-hyperbolic system, three main lemma and theorems are obtained respectively.Lemma 2.1.1 Suppose that μk is the eigenvalue of the operator A, then the eigenvalue distribution of the system (2.2) is as follows:when a2-4(1+μk)μk≥ 0,when a2-4(1+μk)μk< 0, and all the eigenvectors of the system (2.2) form an orthonormal basis in the state space H.Theorem 2.1.2 In the state space H= H1/0(Ω) × L2(Ω), for any (u0,u1) € D(A) x H1/0(Ω), the system (2.2) has unique solution.Theorem 2.1.3 In the state space H= H1/0(Ω) × L2(Ω), for any (u0(x),u1(x))∈ D(A) × H1/0(Ω), the solution of system (2.1) tend to 0.In Chapter 3, we study the existence and asymptotic property of solutions of the fol-lowing system Here Ω is a bounded area on Rn, Δ is the usually Laplace differential sign, "a、b" are positive arbitrary constants.With the similar way we used in the second chapter, we study the existence and unique-ness of solutions of this kind of pseudo-hyperbolic equations and a corresponding certification shall be given. After analyzing the eigenvalue, we can get the conclusion that when the vol-ume of the bounded domain on Rn meets the conditions that , the solution of the system tend to 0.