Substantial分数阶偏泛函微分方程的动力学行为研究及其应用

Substantial fractional differential equations are more and more widely applied in engineering, life and other subjects since they can not only precisely characterize memory process and non-locality but also describe the couple between time and spatial. At the present, the study of Substantial fractional partial functional differential equations mainly focuses on numerical simulation, while the theoretical aspects of qualitative and stability are still in the preliminary stage. This project studies the dynamical behavior of a class of substantial fractional partial functional differential equation. Firstly, the well-posedness and long-time behavior of the substantial fractional partial functional differential equation is discussed. More precisely, the existence and uniqueness as well as the continuity of weak solutions, the stability of equilibrium solutions and the existence of pullback attractors are investigated. Secondly, the influence of random noise on the well-posedness and dynamic behavior of the system is studied. The existence and uniqueness of random attractors and its Hausdorff dimension are analyzed. Finally, these theoretical results are applied to the substantial fractional partial HIV virus model. We study the threshold dynamics of virus elimination, the existence as well as stability conditions of the equilibrium point of the model, reveal the changes of HIV virus in vivo and predict the disease trend at the individual level.

Substantial分数阶微分方程由于不但能准确刻画过程的记忆性和非局部性，还可以描述时间和空间的耦合，在工程、生命等学科中的应用越来越广泛。目前Substantial分数阶偏泛函微分方程的研究主要集中在数值模拟方面，而定性和稳定性的理论方面和仍处于初步阶段。本项目拟研究Substantial分数阶偏泛函微分方程解的动力学行为，讨论模型的适定性及其长时间动力学行为，即研究其弱解的存在唯一性和关于时间的连续性，分析平衡解的存在性及其指数稳定性和拉回吸引子的存在性。探讨随机因素对系统解的适定性、动力学行为的影响，研究随机吸引子的存在唯一性及其Hausdorff维数。然后将理论结果运用到Substantial分数阶偏HIV病毒动力学模型中，研究使得病毒消除的阈值动力学、平衡态的存在性及稳定性条件，揭示HIV病毒在体内的变化规律，预测个体水平的疾病进展趋势。

分数阶微积分是研究任意阶积分和微分的数学性质及其应用的领域。本项目研究分数阶偏泛函微分方程的动力学行为及其在传染病病毒动力学模型中的应用。分析了几类分数阶偏泛函微分方程解的动力学行为，研究了其弱解的存在唯一性，平衡解的稳定性及吸引集的存在性。建立了分数阶舱室模型，研究该模型解的存在唯一性、无病平衡点和地方病平衡点的存在性及其全局稳定稳定性，揭示传染病的变化规律，预测变化趋势。分数阶舱室模型的理论结果可以应用到其它传染病模型，为设计相应的数值格式进行数值模拟提供必要理论依据。这属于应用数学与理论生命科学的交叉，是面向实际问题的应用数学研究。讨论了分数阶偏泛函微分方程的适定性和渐近行为：证明了温和解的存在唯一性及其关于初值的连续依赖性。并将该理论结果应用于分数阶反应扩散方程，得到了指数吸引集的存在性。这些理论结果为分数阶微积分在粘弹性力学与牛顿流体力学、随机过程、混沌与湍流、控制系统、反常扩散以及高分材料的解链等领域的应用提供理论支撑。推广了实数域上的海涅归结原理到一般地拓扑空间。研究了具有无穷变化时滞Navier-Stokes方程解的全局稳定性(多项式稳定性，全局渐近稳定性和指数稳定性)和耗散性，研究了Pantograph方程，得到了吸引集的存在性，事实上该吸引集是单点集。