Existence and Regularity of Solutions to Some Singular Parabolic Systems

Abstract

This thesis continues the work started with my previous supervisor, Dr. Shaohua Chen. In [15], the authors developed some tools that showed the boundedness or blowup of solutions to a semilinear parabolic system with homogeneous Neumann boundary conditions. This system, the so called ’Activator-Inhibitor Model’, is of interest as it is used to model biological processes and pattern formation. Similar tools were later adapted to deal with the same parabolic system in [3], in which the authors prove global boundedness of solutions under homogeneous Dirichlet conditions. This new problem is of mathematical interest as the solutions may grow singular near the boundary. Shortly after, a different system was considered in [4], where the authors proved global boundedness of solutions to a system featuring similar singular reaction terms. The goal of this thesis is twofold: first, the tools developed that allow us to tackle these sorts of problems will be demonstrated in detail to showcase its utility; the second is to then use these tools to generalize some of these previous results to a larger class of singular parabolic systems. In doing so, we expand the classical literature found in [14] and other notable works, where nonsingular equations are extensively treated. The motivation for the first should be clear. While there are numerous bodies of text treating nonsingular problems, there are no collections available dealing with these types of singularities exclusively. This is of practical use to other mathematicians studying partial differential equations. The motivation for the second is, perhaps, more practical. There are a growing number of models found in physics, chemistry and biology that may be generalized to a singular type system. Through allowing those individuals to treat these problems, we may gain valuable insights into the real world and how these processes behave.