Qualitative analysis of a PDE model for chemotaxis with logarithmic sensitivity and logistic growth
This thesis examines the qualitative behavior of solutions to a PDE model for chemotaxis; that is the existence, uniqueness and asymptotic behavior of solutions. We study initial-boundary value problems for a chemotaxis model with logarithmic sensitivity and logistic growth for the cell population density, and nonlinear growth of the chemical concentration. Extensive work has been done for this particular model without logistic growth on both bounded and unbounded domains. However, the model with logistic growth on a bounded domain has not been studied before. This case is of particular interest given its relevance for modeling tumor angiogenesis. We first establish global well-posedness of strong solutions for large initial data with no-flux boundary conditions and, moreover, establish the qualitative result that both the population density and chemical concentration asymptotically converge to constant states. The population density in particular converges to its carrying capacity. We additionally prove that the vanishing chemical diffusivity limit holds in this regime. Finally, we provide numerical confirmation of the rigorous qualitative results, as well as numerical simulations that demonstrate a separation of scales phenomenon. We then establish global well-posedness of strong solutions for large initial data with dynamic boundary conditions. Moreover, the solutions will asymptotically approach the boundary data under mild and natural assumptions on the boundary functions. We additionally show the formation of a boundary profile in the singular chemical zero diffusive limit. Lastly, we provide numerical simulations that confirm the boundary layer formation, as well as convergence towards certain steady states of the solution when relaxing the assumptions on the boundary data. The main tool developed in these results is a particular Lyapunov functional that helps overcome the mathematical challenges of the non-conservation of the mass due to the logistic growth. These results give a complete study of this particular system on bounded domains with both zero-flux and dynamic moving boundary conditions.