The effect of time delay to a nonlinear viscoelastic wave equation in the whole space

In this project, we discuss two major problems in viscoelasticity with time delay. In these problems, we deal with viscoelastic wave equations with linear or nonlinear dampings and nonlinear source terms in the whole space in the presence of the time delay. We aim to investigate or study the effect of the time delay on the existence and nonexistence issue and to set up conditions that give blow up or decay results. It is known that working in the whole-space domain is different from the bounded domain case. This difference is due to the fact that Poincars inequality and some embedding inequalities are no longer valid. This causes several difficulties where we have to overcome.