Blow up and decay 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. In addition to the fact that the domain here is considered to be in regular bounded domains or in unbounded domains as well, we intend to try problems with variable exponents. Our 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 or general classes of relaxation functions. Moreover, Cauchy problem is to be considered with nonlinearity of variable exponents. 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 in porous media with a viscoelastic damping. At the end, we try to handle a system of plate equation with finite and infinite memories at the same time and to get general decay result. 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.