On the Long-time Behavior of some viscoelastic coupled-equation systems

In this project, we consider the issue of blow up and the decay rate estimates of solutions for certain initial boundary value viscoelastic systems of couple equations in regular bounded domains and in unbounded domains R^n as well. We aim is to establish our results, under some restrictions on the coupled functions, without adding any extra dissipation. In the first system or objective here is we discuss the blow up of solutions to initial boundary value viscoelastic systems of couple of equations in bounded domains. While in the second system we aim to establish a blow up or a decay result in the whole space in the presence of the density that control the weighted spaces. At the end, we have another system of the same type where the source term is general but with the space and the time variables. Constructing appropriate Lyapunov functional together with the multiplier method, the energy methods, weighted space and the concavity method are powerful tools to reach our goals.