Cauchy Problem For A Nonlinear Viscoelastic Wave Equation With Nonlinear Damping And Source Terms

In this paper,we mainly consider the following nonlinear viscoelastic wave equa-tion (?),where(?)>1,(?)>1,and(?):R~+?R~+is a positive nonincreasing function.In the first part of this article,using the ideas of the potential well theory,we prove that there are solutions of equation with positive initial energy that blow up in the region of the unbounded,and in the second part,using the continuation principle,we show a global existence result.Further,in the third part,we prove the energy decay estimate of the solutions.