The focus of this dissertation is on global existence of solutions of some chemotaxis systems with logistic sources, subject to both homogeneous and nonlinear Neumann boundary conditions. It is known that blow-up phenomena can be prevented for some chemotaxis models with quadratic logistic sources under homogeneous Neumann boundary conditions. However, it is shown in this research that quadratic logistic sources are not optimal for preventing blow-up phenomena in some chemotaxis systems. Indeed, our first result demonstrates that Keller-Segel systems can avoid blow-up solutions under nonlinear Neumann boundary conditions with quadratic logistic sources. Moreover, the second result shows that blow-up solutions can be prevented in two spatial dimensions with sub-logistic sources. Additionally, the third result shows that sub-logistic sources are even sufficient to avoid blow-up solutions under nonlinear Neumann boundary conditions in two spacial dimensions. Furthermore, we investigate some nonlinear nonlocal sources Keller systems and the effect of sub-logistic sources under nonlinear Neumann boundary conditions and two-species with two chemicals models in two spacial dimensions. Finally, we show that the presence of logistic sources can prevent blow-up phenomenon in superlinear cross diffusion chemotaxis and in superlinear signal production chemotaxis models. Mathematical tools utilized in the research include variational methods, Moser-Alikakos iterations, regularity theory for elliptic and parabolic equations in Sobolev spaces and in Orlicz spaces, some elemental inequalities in Sobolev spaces and differential inequalities.
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