Blow Up and Global Existence for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Nonlinear Damping and Source terms

In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if p > m, and a global result for p ≤ m.


Introduction
A purely elastic material has a capacity to store mechanical energy with no dissipation (of the energy). A complete opposite to an elastic material is a purely viscous material. The important thing about viscous materials is that when the force is removed it does not return to its original shape. Materials which are outside the scope of these two theories will be those for which some, but not all, of the work done to deform them can be recovered. Such materials possess a capacity of storage and dissipation of mechanical energy. This is the case for viscoelastic material. The dynamic properties of viscoelastic materials are of great importance and interest as they appear in many applications to natural sciences. Many authors have given attention to this problem for quite a long time, especially in the last two decades, and have made a lot of progress.
In [1], Messaoudi considered the following initial-boundary value problem: where Ω was a bounded domain of ( )  [2], to certain solutions with positive initial energy. A similar result was also obtained by Wu [3] using a different method.
For the problem (1.1) in n R and with 2 m = , concerning Cauchy problems, Kafini and Messaoudi [4] established a blow up result for the problem and the initial data were compactly supported with negative energy such that In the absence of the viscoelastic term ( ) 0 g = , the problem has been extensively studied and results concerning existence and nonexistence have been established. In bounded domains, for the equation causes finite time blow up of solutions with negative initial energy (see [5]). In contrast, for 0 b = , the damping term 2 m t t a u u − assures global existence for arbitrary initial data (see [6]). The case of linear damping ( ) 2 m = and nonlinear source has been first considered by Levine [7] [8]. He showed that solutions with negative initial energy blew up in finite time. Furthermore, the interaction between the nonlinear damping ( ) 2 m > and the source terms was studied by Georgiev and Todorova [9], for a bounded domain with Dirichlet boundary conditions. For the same problem, Messaoudi [10] extended the blow up result to solutions with negative initial energy.
in a bounded domain and 2 p > . They established a local existence result and showed that the local solution was global and decays uniformly if the initial data were small enough.
In [12], Song and Xue considered with the following viscoelastic equation with strong damping: where Ω was a bounded domain of n R ( ) was a positive nonincreasing function. They showed, under suitable conditions on g , that there were solutions of (1.5) with arbitrarily high initial energy that blow up in a finite time. For the same problem (1.5), in [13], Song and Zhong showed that there were solutions of (1.5) with positive initial energy that blew up in finite time. For more related works, we refer the reader to [14]- [18].
In this work, we intend to study the following initial-boundary value problem: is a bounded domain with a smooth boundary ∂Ω , 2 m ≥ , p > 2, 1 2 , 0 Thus, the original memory term can be written as and we get a new system and boundary conditions The paper is organized as follows. In Section 2, we first prove the blow up result, and then in Section 3, we prove the global existence result.
For convenience, we denote the norm and scalar product in

Blow Up
In this section, we present some materials needed in the proof of our results, state a local existence result, which can be established, combining the argument of [21], and prove our main result. For this reason, we assume that By taking a derivative of ( ) By using Holder inequality and Young's inequality to estimate the last two terms on right hand side of (2.24), we obtain ( ) ( )