Structural Conditions for Global Existence in Some Chemotaxis Systems with Different Logistics

This doctoral thesis develops a comprehensive mathematical analysis of a wide class of chemotaxis systems inspired by and extending the classical Keller–Segel model. The models investigated combine attraction–repulsion mechanisms, signal production and consumption, and logistic-type growth and decay effects, of both local and nonlocal type. The central aim is to identify structural conditions ensuring the global existence, boundedness, and qualitative stability of classical solutions, while also exploring the onset of blow-up phenomena. The study begins with two-signal systems incorporating both attractive and repulsive cues, where the interaction of opposing forces strongly influences system dynamics. By analyzing linear and nonlinear production laws, sensitivity functions, and threshold values of parameters, the thesis demonstrates how repulsive signaling can counteract destabilizing aggregation effects. Special attention is given to nonlinear diffusion and chemotactic sensitivities, as well as to logistic source terms with gradient-dependent damping. A novel contribution in this context is the introduction of a gradient-type nonlinearity in the logistic dampening term—a formulation that has only recently emerged in the literature and is shown here to play a decisive role in regularizing solutions and preventing singularity formation under appropriate conditions. Further investigations focus on signal-consumption models, motivated by biologically realistic settings where chemoattractants are degraded by cells. These systems are proven to be more stable than production-dominated counterparts, with global boundedness results established under explicit parametric thresholds involving chemotactic sensitivity, initial data, and damping coefficients. The thesis also explores indirect chemotaxis mechanisms, particularly in the context of immune–tumor interactions, where signaling is mediated by an intermediate variable. For these models, conditions for global boundedness and estimates on blow-up times are provided. Overall, the results highlight a recurring principle: the interplay between nonlinearities, cross-diffusion, and signal regulation dictates whether chemotactic populations evolve toward stable distributions or develop singularities. By constructing and analyzing a broad range of single- and multi-signal, local and nonlocal, parabolic and elliptic systems, this work establishes a framework that enriches both the mathematical theory of chemotaxis and its applications to biologically relevant phenomena such as tissue aggregation, immune responses, and tumor–immune interactions. While the primary focus lies on global behavior, selected numerical simulations illustrate and complement the analytical results, offering further insight into system dynamics. For solutions exhibiting blow-up behavior, estimates of the blow-up time are obtained through both theoretical analysis and numerical simulations. The thesis thus contributes both generalizations of classical results and novel perspectives on chemotaxis modeling, advancing the theoretical understanding of nonlinear partial differential equations in biological contexts.