This thesis investigates (in)stability and long-term behavior in three distinct models: a living fluids model, a heterogeneous catalysis process, and the application of duality scales in complemented subspaces concerning partial differential equations. The living fluids model, represented by generalized Navier-Stokes equations, describes dense bacterial suspensions at low Reynolds numbers. A thorough analysis of linear and nonlinear stability and instability is conducted in the periodic L²-setting, identifying parameter sets linked to stability and instability. Additionally, it is demonstrated that this model possesses a global attractor of finite dimension with high regularity. The second model, originating from chemical engineering, addresses heterogeneous catalysis within a cylinder-shaped domain, where the catalyst is located on the lateral boundary. The study reveals stability and instability in the Lp-setting, influenced by the chosen chemical reaction at the boundary. Lastly, the thesis explores duality scales in Banach spaces, offering a refined understanding of duality. It is shown that, under specific conditions regarding a consistent projection P, the duality scale property is maintained when examining complemented subspaces, with applications to the Stokes operator.
