Study On The Dynamic Properties Of Solutions Of Nonlinear Viscoelastic Equations

Viscoelastic mechanics theory studies what regularity the stress and inotropic strain satisfy when viscoelastic materials are under the function of load. 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 mathe-matical theory have developed rapidly. In the forefront field of international application mathematics, which has become a very active research subject. Most of equations stud-ied in viscoelastic mechanics are partial differential equations. In particular, energy decay result of viscoelastic wave equations caused the extensive attention of scholars.In this thesis, we mainly study general decay results of solutions for the viscoelastic system with a nonlinear source. It is divided into two chapters accord to contents:Chapter 1 In this paper, We consider with boundary control system of definite solution problems of nonlinear viscous fluctuations.(?)where u= (u1,…, un) are Lam e constants. Let divu=ux11+ ux22+… uxnn be a vectorfunction, u be the divergence of divn , ?=(?).We write(?)Here ? is a bounded domain of Rn(n > 1) with a smooth boundary a?. r > 0 and g is a positive non-increasing function defined on R+ .? :=a? with ? =?0??1,m(?0??1)= 0,?0, ?1 have positive measures and n is the unite outward normal to a?.Chapter 2 In this chapter, we consider homogeneous boundary nonlinear viscoelastic wave equation of definite solution problems.(?)Here ? is a bounded domain of Rn (n ? 1) with a smooth boundary a?, r > 0 and g is a positive non-increasing function defined on R+. Our goal is to study the decay rates of the energy associated with the equation , and to use itetation to get to the solutions of the equation.