Aggregation Equation with Degenerate Diffusion

Recently, there has been a growing interest in the use of nonlocal partial differential equation (PDE) to model biological and physical phenomena. In this dissertation, we study the behavior of solutions to several nonlocal PDEs, which have both an aggregation term and a degenerate diffusion term.Chapter 1 and Chapter 2 of this dissertation are devoted to the study of the Patlak-Keller-Segel (PKS) equation and its variations. The PKS equation is a degenerate diffusion equation with a nonlocal aggregation term, which models the collective motion of cells attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow-up criteria are well known, the asymptotic behaviors of solutions are not completely clear. In Chapter 1, we investigate qualitative and asymptotic behavior of solutions for the PKS equation when the solution exists globally in time. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. Using maximum-principle type arguments as well as energy argument, we prove the finite propagation property of general solutions, and several results regarding asymptotic behaviors of solutions. In Chapter 2, we consider the PKS equation with general power-law interaction kernel, and focus on the cases where the solution blows up in finite time. We study radially symmetric finite time blow-up dynamics from both the numerical and asymptotic aspect, and show that the solution exhibits three kinds of blow-up behavior: self-similar with no mass concentrated at the core, imploding shock solution and near-self-similar blow-up with a fixed amount of mass concentrated at the core. Computation are performed for a variety of parameters using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement.Chapter 3 discusses the study on an aggregation-diffusion equation with smooth interaction kernel in the periodic domain. This equation represents the generalization to m>1 of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on m in (1,2] with particular emphasis on m=2. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For m=2, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter