Mean-Field Models and Nonlocal Problems
Open Access
- Author:
- Tian, Chao
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- November 29, 2021
- Committee Members:
- Anna Mazzucato, Chair & Dissertation Advisor
Zhibiao Zhao, Outside Unit, Field & Minor Member
Wenrui Hao, Major Field Member
Qiang Du, Special Member
Xiantao Li, Major & Minor Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies - Keywords:
- Nonlocal-in-Time
Nonlinearity
Semigroup
Nonlocal noise
Mean-field - Abstract:
- % Place abstract below. In this thesis, we were motivated by the stochastic bi-stable mean-field model initiated by Dawson et al. and the nonlocal-in-time problem developed by Du et al. and observed that formally the stochastic nonlocal-in-time reaction-diffusion problem gives a continuous case of the stochastic bi-stable mean-field model. We generalized the bi-stable mean-field models by introducing an inter-group interaction mechanism, analyzed the well-posedness of this generalization, and derived the asymptotic expression of equilibrium from a coupled nonlinear system of compatible conditions. For $h = 0$, we computed the probability of systemic transitions of both the homogeneous case and the heterogeneous cases, from which we concluded that due to the small heterogeneities on model parameters in different groups, the system will become more likely to fail. Based on the results for linear nonlocal-in-time problem developed in by Du et al., we established the well-posedness theorems, the local limit results and the semi-group properties of the linear nonlocal-in-time problems with integrable kernels. With the Lipschitz continuous nonlinearity added, we proved the global existence and uniqueness of the solution of the nonlinear nonlocal-in-time problem. In addition, considering the nonlinearity of $u^p|u|^{\alpha}$ type and with integrable kernels, we established local existence and uniqueness results for solution in both $L^{\infty}(0,t_0)$ and $W^{1,\infty}(0,t_0)$ spaces. In particular, for small initial data, we also had the global existence and uniqueness of solutions in $L^{\infty}(0,T)$. And for the non-integrable kernel case, we formulated the local existence of the solutions of nonlinear nonlocal-in-time problem. We then obtained the well-posedness result of a class of the deterministic nonlocal-in-time reaction-diffusion problems with integrable nonlocal kernels. Due to the limitation of the solution spaces we have studied on the nonlinear problems, the related results for stochastic nonlocal reaction-diffusion problems are still open. However, based on the definition of the nonlocal-in-time operator, we introduced the nonlocal versions of the Brownian motions and the it$\hat{o}$ integrals, from which we established the well-posedness and the localization results of a stochastic nonlocal-in-time problem.