The Attenuation Estimation Of Two Kinds Of Fixed-solution Problems For Nonhomogeneous Viscous Wave Equations

Viscoelastic mechanics is a cross-discipline subject of physics and mathematics. In the early years, viscoelastic materials did not cause wide attention in science and engineering, which led it to develop slowly. But in nearly forty years, viscoelastic mechanics and its corresponding mathematical theory have developed rapidly. In the forefront field of inter-national application mathematics, which has become a very active research subject. Most of equations studied in viscoelastic mechanics are partial differential equations. In partic-ular, energy decay result of viscoelastic wave equations caused the extensive attention of scholars. Therefore a series of books on partial differential equation as [27,28,29,30] have also entered the college or graduate students'class and become the students'professional compulsory courses.In this thesis, we mainly study the decay estimates of two kinds of definite problems for inhomogeneous viscoelastic wave equations. It is divided into two chapters according to contents:Chapter 1 In this paper, we consider the decay estimates for a Cauchy problem of inhomogeneous viscoelastic wave equation. where g is the relaxation function, and satisfies:g(t)> 0, In this section we will study the result of the energy decay when f(u)=-u. Unlike initial boundary problem, there is no boundary condition in Cauchy problems, so Poincare inequality can not be used in this paper, which add more difficulty for the estimate of inequalities. For the convenience of research question, we use the Fourier transform: to transform the PDE to ODE.Chapter 2 In this chapter, we consider uniform stability of the following initial- boundary problem of inhomogeneous viscoelastic wave equation where ? (?) Rn ia a bounded domain with a smooth boundary a?, and f(u)=|u|p-1u. In this section, we will study different decay behaviors of the solution, under different conditions of g. Firstly, we discuss the exponential decay result when g satisfies the general condition:g(t)> 0, g'(t)< 0, and Then we will discuss the general decay result when g satisfies additional condition g'(t)+H(g(t))?0, where H(·) ? C1(R+) is a strictly increasing convex function with H(0)= 0.