Doğrusal olmayan sönümlü dalga denklemlerinin kararlılığı ve kararsızlığı üzerine

This thesis is devoted to the invastigation of the qualitative behaviours of solutions to the initial-boundary and initial value problems for various nonlinear damped wave equations. Depending on the type of nonlinearities involved these problems exhibit very different dynamics: The `unstable` ones contain trajectories, which develop singularities in finite time, while for the `stable` ones all solutions behave well for all times. In this manuscript we first derive sufficient conditions on the initial functions, which ensure the finite time blow-up of the corresponding solutions to the unstable problems. These results are first derived in an abstract setting, and then applied to equations like the semilinear and quasilinear structurally damped wave equations, damped plate equations, as well as semilinear Boussinesq and Kirchhoff type equations with damping. Also, we show the global nonexistence of solutions to damped quasilinear wave and Boussinesq type equations with a convective term. Later, for the stable structurally damped wave equations with super-critical nonlinear potentials we show the existence of finitely many determining Fourier modes. Finally, we present some simulations carried out for both the stable and the unstable semilinear structurally damped wave equations, and justify our scheme of numerical approximation.