Behaviour of solutions to some hyperbolic equations and systems

In this project, we consider the issue of existence, nonexistence, blow up, and decay rate estimates of solutions for certain initial boundary value viscoelastic and thermoelastic systems of a single or couple equations. The domain here is considering to be in regular bounded domains or in unbounded domains R^n as well. We aim is to establish our results, under some restrictions on the functions, without adding any extra dissipation. Our first objective here, is to discuss the blow up of solutions to initial boundary value viscoelastic systems of a single or couple of equations in bounded domains but for wide classes of relaxation functions. Moreover, Cauchy problem is to be considered with logarithmic nonlinearity. While in the second objective, we aim to establish decay results again with different types of relaxation functions and nonlinearities to Timoshenko-type systems of thermoelasticity of type III with a viscoelastic damping. At the end, we try to handle our system in the presence of time delay. Different types of time delay may be considered: constant, variable or distributive. Constructing appropriate Lyapunov functional together with the multiplier method, the energy methods, weighted spaces and the concavity method are powerful tools to reach our goals.