The Dynamical Properties Of Memory Wave Equation And Nonhomogeneous Kirchhoff Equation

In the early 18 th century, people discussed the problem of string vibration equation and its solution, which marked the beginning of PDE. But, it did not cause the attention of many scholars and it was not until the 19 th century that developed rapidly. So far,there has been many scholars in the domestic and overseas which study the numerical method for the problem of partial differential equations and apply them into engineering practice. They have made outstanding contributions on the development of the theory of partial differential equation, enriching the content of the subject.In this paper, we mainly consider the dynamics properties of solutions for viscoelastic wave equation with long time memory and nonhomogeneous Kirchhoff equation with the Dirichlet boundary condition.According to the content, this article is divided into the three chaptersChapter 1 Preference, we introduce the research background and the main results of the problems.Chapter 2 This section consider the initial-boundary problem of a viscoelastic wave equation with long time memory term:In the condition of some assumptions, we can deduce the general results: there is a positive constantε0, k′1, k2> 0, allowing the weak solution of the problem to meethere H1=1t1H2(s)ds and H2(t) = t H′(ε0t).Chapter 3 This chapter study the following nonhomogeneous Kirchhoff equation with a small positive constant ε in condition of Dirichlet boundray?where ? is a bounded open set in RN,0 ≤ q ≤2n n-2, u0∈ D(A), u1∈ D(A12).On the basis of the hypothesis of this section, we deduce the exponential decay properties for the weak solution of the nonhomogeneous problems: there is the positive constants b and K which has no contact with t, satisfyingAt the same time we can deduce the blow-up properties of the weak solution and?nd the blow-wp ?nite time : ifThe weak solution will blow up in a limited time.