The General Decay Of The Solution Of A Class Of Quasi-linear Viscoelastic Partial Differential Equations

In this paper,we establish a general decay rate for a quasilinear viscoelastic equation with inital boundary value problem(?)where ? is a bounded domain of Rn(n ? 1)with a smooth boundary(?)?,?>0.There are four chapters in this thesis.Chapter 1.IntroductionChapter 2.Preliminaries and main results under different assumptionsIn this chapter we prepare some notations and preliminaries and give the main results under different assumptions.Chapter 3.The proofs of the main results under assumption AThe relaxation function satisfies g'(t)?<?(t)gp(t),t?0,1?p<3/2.We show that the dissipation induced by the integral term is strong enough to stabilize the solution.In this chapter,we want to weaken the assumptions for C(t)in[12]and apply differential technique and integration technique,and show the general decay results in this literature.Chapter 4.The proofs of the main results under assumption BThe relaxation function satisfies g'(t)?-H(g(t)),where the convex function H ?C1(R+),which is a strictly increasing with H(0)= 0(this condition can not be covered by the condition in chapter 3).In this chapter,we adopt iteration technique to establish a general decay result of the above problem.And the exponential decay result and polynomial decay result in in[15]are special cases of this paper.