讲座题目：Nontrivial equilibrium solutions and general stability for stochastic evolution equations with pantograph delay and tempered fractional noise
主办单位：长江大学信息与数学学院
报告专家：王业娟（兰州大学教授、博士生导师）
报告时间：2023年4月12日19：00-20：00
腾讯会议ID：315-531-460
会议链接：https://meeting.tencent.com/dm/RZYOMwI20Cbm
专家简介：王业娟，兰州大学数学与统计学院教授，博士生导师。主要研究领域：动力系统在生物动力学、控制系统、大气科学和金融中的应用；非线性分析；偏微分（分数阶）方程、随机微分方程的理论、应用与数值模拟。2020 年获教育部自然科学二等奖。先后主持国家自然科学基金面上项目（数学和地学）、青年基金项目、留学回国人员基金、上海市优秀青年教师基金、中央高校基本科研业务费等。已刊出专著《Critical parabolic-type problems》，在《SIAM J.Math. Anal.》、《SIAM J. Numer. Anal.》、《J. Diff. Eqns.》、《J.Diff. Differ.Eqns.》、《Chaos》、《Quart. Appl. Math.》、《Disc.Contin. Dyna. Syst.》、《Eur. Phys. J. Plus》等杂志上发表学术论文50 多篇。目前担任美国数学会《数学评论》评论员。
摘要：In this paper, we investigate the asymptotic behavior of stochastic pantograph delay evolution equations driven by a tempered fractional Brownian motion (tfBm) with Hurst parameter H > 1/2. First of all, the global existence, uniqueness, and mean-square stability with general decay rate of mild solutions are established. In particular, we would like to point out that our analysis is not necessary to construct Lyapunov functions, but we deal directly with stability via the Banach fixed point theorem, the fractional power of operators, and the semigroup theory. It is worth emphasizing that a novel estimate of stochastic integrals with
respect to tfBm is presented, which greatly contributes to the stability analyses. Then after extending the factorization formula to the tfBm case, we construct the nontrivial equilibrium solution, defined for t \in R, by means of an approximation technique and a convergence analysis. Moreover, we analyze the Holder regularity in time and general stability (including both polynomial and logarithmic stability) of the nontrivial equilibrium solution in the sense of mean-square. As an example of application, the reaction diffusion neural network system with
pantograph delay is considered, and the nontrivial equilibrium solution and general stability of the system are proved under the Lipschitz assumption.
