Insights into Nonlinear Diffusion Problems and Implications for Biological Systems

Abstract

This thesis aims to study and analyze nonlinear partial differential equations used in modeling diffusion phenomena in physical and biological systems. The focus is placed on the porous medium equation, non-divergence form equations, and reaction-diffusion systems in the context of disease spread. The research employs mathematical techniques such as Banach's fixed-point theorem and self-similarity to analyze the existence and behavior of solutions. The results demonstrate the ability of these models to provide accurate descriptions of the complex dynamics of natural systems in heterogeneous environments, enhancing their application in fields such as environmental pollution monitoring and epidemic spread modeling