Dynamics Properties Of The Solutions For Nonlinear Petrovsky-type And Kirchhoff-type Equations

After the appearance of Differential and Integral subject, people began to study the partial differential equations which existed in many mechanics and physics problems. From then on, the theory of the partial differential equations has been commonly used by mathematicians, mechanics and physicists for study.As the constant development of science and technology, the initial boundary value problem of the nonlinear partial differential equations can be more and more applied in a variety of applied science, such as applied mathematics, physics, control theory. It is one of the most active research subject that are studied in the field of nonlinear science. Furthermore, the initial boundary value problem of the nonlinear hyperbolic equation is one of the most important branches in partial differential equations. Consequently, it becomes of great importance to study the existence and dynamics properties of the solutions of the initial-boundary value problems.The present paper mainly employs the Faedo-Galerkin approximate methods, fixed point theory and so on, to investigate the existence and dynamics properties of the solutions for some initial-boundary vame problems of nonlinear Petrovsky-type and Kirchhoff-type equations. The obtained results are new, intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents and results of this paper.Chapter 2 We consider the initial boundary value problem for the following non-linear Petrovsky type equation where p> 2, m> 2,Ωis a bounded domain of Rn,n> 2 with smooth boundary (?)Ω, and v is the unit outward normal on (?)Ω,(?)vu denotes the normal derivation of u. Under suitable conditions of the initial data and the relaxation function, we prove that the solution with upper bounded initial energy blows up in finite time. Moreover, for the linear damping case, we show that the solution blows up in finite time by different method for nonpositive initial energy.Chapter 3 In this chapter, we are concerned with the energy decay of the solution to the initial boundary value problem for the Petrovsky type equation whereΩis a bounded domain of Rn, n> 1 with smooth boundary(?)Ω, v is the unit outward normal on (?)Ω, and (?)vu denotes the normal derivation of u. Our purpose lies in that the energy of the above equation exponentially and polynomially decays respectively under suitable conditions.Chapter 4 We consider the higher-order nonlinear Kirchhoff-type equation with initial conditions and homogeneous boundary conditions where p>q≥2,m≥1,Ωis a bounded domain of Rn, n≥1 with smooth bound-ary (?)Ω, v is the unit outward normal on (?)Ω, and ((?)iu)/((?)vi) denotes the i-order normal derivation of u, D denotes the gradient operator, that is Du= (ux1,ux2,…,uxn), and Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.