Global dynamics of the local and nonlocal Patlak-Keller-Segel chemotaxis systems

Chemotaxis is the directed movement of cells in response to chemical gradients. This thesis is devoted to studying two generic classes of two coupled parabolic equations modeling chemotaxis: a classical (local) Patlak-Keller-Segel chemotaxis model with/without growth (the proliferation and the reduction due to birth and death of cells) and a nonlocal gradient P-K-S chemotaxis model. Here, we first obtain new and deep characterizations of blowup mechanism for the chemotaxis models with/without growth. Then, under simple conditions on the growth source, we use these criteria to establish the global boundedness of the underlying models, which serves as a fundamental step to understand the dynamics of these models. Hence, it is possible to study the convergence of such solutions to bounded steady states with striking features such as spikes and transition layers. For the nonlocal model, part of our results follows this direction. More importantly, our results provide a full picture on how the sampling radius affects pattern formation and stability.;For the P-K-S chemotaxis systems with/without growth in n-D, it is recently known that blow-up is possible even in the presence of linear birth and superlinear degradation. Here, we derive new and interesting characterizations on the growth versus the boundedness. We show that the L r-boundedness of the cell density can guarantee its Linfinity-boundedness and hence its global boundedness, where r = n + epsilonor n/2 + epsilon, depending on whether the growth source is essentially linear (including no growth) or superlinear. Hence, a blowup solution also blows up in Lp-norm for any suitably large p. More detailed information on how the growth source affects the boundedness of the solution is derived. Later on, we use these criteria to derive some global boundedness and existence results: logistic growth in 2-D, cubic growth as initially proposed by Mimura and Tsujikawa in 3-D, (n--1) st growth in n-D with n ≥ 4, or cubic growth and convex domain in n-D with n ≥ 1. Therefore, in a chemotaxis-growth model, blow-up is impossible if the growth source is suitably strong. Finally, we stress that these results remove the commonly assumed convexity on the domain.;For the nonlocal chemotaxis model, we first correct the nonlocal gradient modified by Hillen-Painter-Schemiser from Othmer-Hillen, which fails to take care of the "boundary effects." Subsequently, we obtain the boundedness and the global existence of its solution in 1-D. As a byproduct, it offers a justification for the common belief that P-K-S models normally have no blow-ups in 1-D. Then we obtain the limiting equations when the sampling radius rho→ 0, as well as convergence to steady states when time t → infinity. Next, we show that the model has the ability to give rise to pattern formation if the chemotactic coefficient is larger than an expressible bifurcation value. Interestingly, the smaller the cell, the more likely pattern formation will occur. Then we establish a characterization of the limiting profiles for the nonconstant steady states as either spiky or of transition layer type. Finally, we obtain the full stability information for the nonconstant bifurcating solutions. Surprisingly, the stability is independent of the net creation rate of the chemical and is closely related to the cell radius. As a result, a critical degradation rate phenomenon is found: if the cell degradation rate lies below (above) a threshold/stabilizing value, the cell is stable (unstable). This threshold value is an increasing function of the cell radius. The large cells can compensate their degradation of the chemical, and become stable; however, for small cells to be stable, their degradation rate must be less than a threshold value.